Singular Value Decomposition Part 2: Theorem, Proof, Algorithm

I’m just going to jump right into the definitions and rigor, so if you haven’t read the previous post motivating the singular value decomposition, go back and do that first. This post will be theorem, proof, algorithm, data. The data set we test on is a thousand-story CNN news data set. All of the data, code, and examples used in this post is in a github repository, as usual.

We start with the best-approximating k-dimensional linear subspace.

Definition: Let X = \{ x_1, \dots, x_m \} be a set of m points in \mathbb{R}^n. The best approximating k-dimensional linear subspace of X is the k-dimensional linear subspace V \subset \mathbb{R}^n which minimizes the sum of the squared distances from the points in X to V.

Let me clarify what I mean by minimizing the sum of squared distances. First we’ll start with the simple case: we have a vector x \in X, and a candidate line L (a 1-dimensional subspace) that is the span of a unit vector v. The squared distance from x to the line spanned by v is the squared length of x minus the squared length of the projection of x onto v. Here’s a picture.

vectormax

I’m saying that the pink vector z in the picture is the difference of the black and green vectors x-y, and that the “distance” from x to v is the length of the pink vector. The reason is just the Pythagorean theorem: the vector x is the hypotenuse of a right triangle whose other two sides are the projected vector y and the difference vector z.

Let’s throw down some notation. I’ll call \textup{proj}_v: \mathbb{R}^n \to \mathbb{R}^n the linear map that takes as input a vector x and produces as output the projection of x onto v. In fact we have a brief formula for this when v is a unit vector. If we call x \cdot v the usual dot product, then \textup{proj}_v(x) = (x \cdot v)v. That’s v scaled by the inner product of x and v. In the picture above, since the line L is the span of the vector v, that means that y = \textup{proj}_v(x) and z = x -\textup{proj}_v(x) = x-y.

The dot-product formula is useful for us because it allows us to compute the squared length of the projection by taking a dot product |x \cdot v|^2. So then a formula for the distance of x from the line spanned by the unit vector v is

\displaystyle (\textup{dist}_v(x))^2 = \left ( \sum_{i=1}^n x_i^2 \right ) - |x \cdot v|^2

This formula is just a restatement of the Pythagorean theorem for perpendicular vectors.

\displaystyle \sum_{i} x_i^2 = (\textup{proj}_v(x))^2 + (\textup{dist}_v(x))^2

In particular, the difference vector we originally called z has squared length \textup{dist}_v(x)^2. The vector y, which is perpendicular to z and is also the projection of x onto L, it’s squared length is (\textup{proj}_v(x))^2. And the Pythagorean theorem tells us that summing those two squared lengths gives you the squared length of the hypotenuse x.

If we were trying to find the best approximating 1-dimensional subspace for a set of data points X, then we’d want to minimize the sum of the squared distances for every point x \in X. Namely, we want the v that solves \min_{|v|=1} \sum_{x \in X} (\textup{dist}_v(x))^2.

With some slight algebra we can make our life easier. The short version: minimizing the sum of squared distances is the same thing as maximizing the sum of squared lengths of the projections. The longer version: let’s go back to a single point x and the line spanned by v. The Pythagorean theorem told us that

\displaystyle \sum_{i} x_i^2 = (\textup{proj}_v(x))^2 + (\textup{dist}_v(x))^2

The squared length of x is constant. It’s an input to the algorithm and it doesn’t change through a run of the algorithm. So we get the squared distance by subtracting (\textup{proj}_v(x))^2 from a constant number,

\displaystyle \sum_{i} x_i^2 - (\textup{proj}_v(x))^2 = (\textup{dist}_v(x))^2

which means if we want to minimize the squared distance, we can instead maximize the squared projection. Maximizing the subtracted thing minimizes the whole expression.

It works the same way if you’re summing over all the data points in X. In fact, we can say it much more compactly this way. If the rows of A are your data points, then Av contains as each entry the (signed) dot products x_i \cdot v. And the squared norm of this vector, |Av|^2, is exactly the sum of the squared lengths of the projections of the data onto the line spanned by v. The last thing is that maximizing a square is the same as maximizing its square root, so we can switch freely between saying our objective is to find the unit vector v that maximizes |Av| and that which maximizes |Av|^2.

At this point you should be thinking,

Great, we have written down an optimization problem: \max_{v : |v|=1} |Av|. If we could solve this, we’d have the best 1-dimensional linear approximation to the data contained in the rows of A. But (1) how do we solve that problem? And (2) you promised a k-dimensional approximating subspace. I feel betrayed! Swindled! Bamboozled!

Here’s the fantastic thing. We can solve the 1-dimensional optimization problem efficiently (we’ll do it later in this post), and (2) is answered by the following theorem.

The SVD Theorem: Computing the best k-dimensional subspace reduces to k applications of the one-dimensional problem.

We will prove this after we introduce the terms “singular value” and “singular vector.”

Singular values and vectors

As I just said, we can get the best k-dimensional approximating linear subspace by solving the one-dimensional maximization problem k times. The singular vectors of A are defined recursively as the solutions to these sub-problems. That is, I’ll call v_1 the first singular vector of A, and it is:

\displaystyle v_1 = \arg \max_{v, |v|=1} |Av|

And the corresponding first singular value, denoted \sigma_1(A), is the maximal value of the optimization objective, i.e. |Av_1|. (I will use this term frequently, that |Av| is the “objective” of the optimization problem.) Informally speaking, (\sigma_1(A))^2 represents how much of the data was captured by the first singular vector. Meaning, how close the vectors are to lying on the line spanned by v_1. Larger values imply the approximation is better. In fact, if all the data points lie on a line, then (\sigma_1(A))^2 is the sum of the squared norms of the rows of A.

Now here is where we see the reduction from the k-dimensional case to the 1-dimensional case. To find the best 2-dimensional subspace, you first find the best one-dimensional subspace (spanned by v_1), and then find the best 1-dimensional subspace, but only considering those subspaces that are the spans of unit vectors perpendicular to v_1. The notation for “vectors v perpendicular to v_1” is v \perp v_1. Restating, the second singular vector v _2 is defined as

\displaystyle v_2 = \arg \max_{v \perp v_1, |v| = 1} |Av|

And the SVD theorem implies the subspace spanned by \{ v_1, v_2 \} is the best 2-dimensional linear approximation to the data. Likewise \sigma_2(A) = |Av_2| is the second singular value. Its squared magnitude tells us how much of the data that was not “captured” by v_1 is captured by v_2. Again, if the data lies in a 2-dimensional subspace, then the span of \{ v_1, v_2 \} will be that subspace.

We can continue this process. Recursively define v_k, the k-th singular vector, to be the vector which maximizes |Av|, when v is considered only among the unit vectors which are perpendicular to \textup{span} \{ v_1, \dots, v_{k-1} \}. The corresponding singular value \sigma_k(A) is the value of the optimization problem.

As a side note, because of the way we defined the singular values as the objective values of “nested” optimization problems, the singular values are decreasing, \sigma_1(A) \geq \sigma_2(A) \geq \dots \geq \sigma_n(A) \geq 0. This is obvious: you only pick v_2 in the second optimization problem because you already picked v_1 which gave a bigger singular value, so v_2‘s objective can’t be bigger.

If you keep doing this, one of two things happen. Either you reach v_n and since the domain is n-dimensional there are no remaining vectors to choose from, the v_i are an orthonormal basis of \mathbb{R}^n. This means that the data in A contains a full-rank submatrix. The data does not lie in any smaller-dimensional subspace. This is what you’d expect from real data.

Alternatively, you could get to a stage v_k with k < n and when you try to solve the optimization problem you find that every perpendicular v has Av = 0. In this case, the data actually does lie in a k-dimensional subspace, and the first-through-k-th singular vectors you computed span this subspace.

Let’s do a quick sanity check: how do we know that the singular vectors v_i form a basis? Well formally they only span a basis of the column space of A, i.e. a basis of the subspace spanned by the data contained in the columns of A. But either way the point is that each v_{i+1} spans a new dimension from the previous v_1, \dots, v_i because we’re choosing v_{i+1} to be orthogonal to all the previous v_i. So the answer to our sanity check is “by construction.”

Back to the singular vectors, the discussion from the last post tells us intuitively that the data is probably never in a small subspace.  You never expect the process of finding singular vectors to stop before step n, and if it does you take a step back and ask if something deeper is going on. Instead, in real life you specify how much of the data you want to capture, and you keep computing singular vectors until you’ve passed the threshold. Alternatively, you specify the amount of computing resources you’d like to spend by fixing the number of singular vectors you’ll compute ahead of time, and settle for however good the k-dimensional approximation is.

Before we get into any code or solve the 1-dimensional optimization problem, let’s prove the SVD theorem.

Proof of SVD theorem.

Recall we’re trying to prove that the first k singular vectors provide a linear subspace W which maximizes the squared-sum of the projections of the data onto W. For k=1 this is trivial, because we defined v_1 to be the solution to that optimization problem. The case of k=2 contains all the important features of the general inductive step. Let W be any best-approximating 2-dimensional linear subspace for the rows of A. We’ll show that the subspace spanned by the two singular vectors v_1, v_2 is at least as good (and hence equally good).

Let w_1, w_2 be any orthonormal basis for W and let |Aw_1|^2 + |Aw_2|^2 be the quantity that we’re trying to maximize (and which W maximizes by assumption). Moreover, we can pick the basis vector w_2 to be perpendicular to v_1. To prove this we consider two cases: either v_1 is already perpendicular to W in which case it’s trivial, or else v_1 isn’t perpendicular to W and you can choose w_1 to be \textup{proj}_W(v_1) and choose w_2 to be any unit vector perpendicular to w_1.

Now since v_1 maximizes |Av|, we have |Av_1|^2 \geq |Aw_1|^2. Moreover, since w_2 is perpendicular to v_1, the way we chose v_2 also makes |Av_2|^2 \geq |Aw_2|^2. Hence the objective |Av_1|^2 + |Av_2|^2 \geq |Aw_1|^2 + |Aw_2|^2, as desired.

For the general case of k, the inductive hypothesis tells us that the first k terms of the objective for k+1 singular vectors is maximized, and we just have to pick any vector w_{k+1} that is perpendicular to all v_1, v_2, \dots, v_k, and the rest of the proof is just like the 2-dimensional case.

\square

Now remember that in the last post we started with the definition of the SVD as a decomposition of a matrix A = U\Sigma V^T? And then we said that this is a certain kind of change of basis? Well the singular vectors v_i together form the columns of the matrix V (the rows of V^T), and the corresponding singular values \sigma_i(A) are the diagonal entries of \Sigma. When A is understood we’ll abbreviate the singular value as \sigma_i.

To reiterate with the thoughts from last post, the process of applying A is exactly recovered by the process of first projecting onto the (full-rank space of) singular vectors v_1, \dots, v_k, scaling each coordinate of that projection according to the corresponding singular values, and then applying this U thing we haven’t talked about yet.

So let’s determine what U has to be. The way we picked v_i to make A diagonal gives us an immediate suggestion: use the Av_i as the columns of U. Indeed, define u_i = Av_i, the images of the singular vectors under A. We can swiftly show the u_i form a basis of the image of A. The reason is because if v = \sum_i c_i v_i (using all n of the singular vectors v_i), then by linearity Av = \sum_{i} c_i Av_i = \sum_i c_i u_i. It is also easy to see why the u_i are orthogonal (prove it as an exercise). Let’s further make sure the u_i are unit vectors and redefine them as u_i = \frac{1}{\sigma_i}Av_i

If you put these thoughts together, you can say exactly what A does to any given vector x. Since the v_i form an orthonormal basis, x = \sum_i (x \cdot v_i) v_i, and then applying A gives

\displaystyle \begin{aligned}Ax &= A \left ( \sum_i (x \cdot v_i) v_i \right ) \\  &= \sum_i (x \cdot v_i) A_i v_i \\ &= \sum_i (x \cdot v_i) \sigma_i u_i \end{aligned}

If you’ve been closely reading this blog in the last few months, you’ll recognize a very nice way to write the last line of the above equation. It’s an outer product. So depending on your favorite symbols, you’d write this as either A = \sum_{i} \sigma_i u_i \otimes v_i or A = \sum_i \sigma_i u_i v_i^T. Or, if you like expressing things as matrix factorizations, as A = U\Sigma V^T. All three are describing the same object.

Let’s move on to some code.

A black box example

Before we implement SVD from scratch (an urge that commands me from the depths of my soul!), let’s see a black-box example that uses existing tools. For this we’ll use the numpy library.

Recall our movie-rating matrix from the last post:

movieratings

The code to compute the svd of this matrix is as simple as it gets:

from numpy.linalg import svd

movieRatings = [
    [2, 5, 3],
    [1, 2, 1],
    [4, 1, 1],
    [3, 5, 2],
    [5, 3, 1],
    [4, 5, 5],
    [2, 4, 2],
    [2, 2, 5],
]

U, singularValues, V = svd(movieRatings)

Printing these values out gives

[[-0.39458526  0.23923575 -0.35445911 -0.38062172 -0.29836818 -0.49464816 -0.30703202 -0.29763321]
 [-0.15830232  0.03054913 -0.15299759 -0.45334816  0.31122898  0.23892035 -0.37313346  0.67223457]
 [-0.22155201 -0.52086121  0.39334917 -0.14974792 -0.65963979  0.00488292 -0.00783684  0.25934607]
 [-0.39692635 -0.08649009 -0.41052882  0.74387448 -0.10629499  0.01372565 -0.17959298  0.26333462]
 [-0.34630257 -0.64128825  0.07382859 -0.04494155  0.58000668 -0.25806239  0.00211823 -0.24154726]
 [-0.53347449  0.19168874  0.19949342 -0.03942604  0.00424495  0.68715732 -0.06957561 -0.40033035]
 [-0.31660464  0.06109826 -0.30599517 -0.19611823 -0.01334272  0.01446975  0.85185852  0.19463493]
 [-0.32840223  0.45970413  0.62354764  0.1783041   0.17631186 -0.39879476  0.06065902  0.25771578]]
[ 15.09626916   4.30056855   3.40701739]
[[-0.54184808 -0.67070995 -0.50650649]
 [-0.75152295  0.11680911  0.64928336]
 [ 0.37631623 -0.73246419  0.56734672]]

Now this is a bit weird, because the matrices U, V are the wrong shape! Remember, there are only supposed to be three vectors since the input matrix has rank three. So what gives? This is a distinction that goes by the name “full” versus “reduced” SVD. The idea goes back to our original statement that U \Sigma V^T is a decomposition with U, V^T both orthogonal and square matrices. But in the derivation we did in the last section, the U and V were not square. The singular vectors v_i could potentially stop before even becoming full rank.

In order to get to square matrices, what people sometimes do is take the two bases v_1, \dots, v_k and u_1, \dots, u_k and arbitrarily choose ways to complete them to a full orthonormal basis of their respective vector spaces. In other words, they just make the matrix square by filling it with data for no reason other than that it’s sometimes nice to have a complete basis. We don’t care about this. To be honest, I think the only place this comes in useful is in the desire to be particularly tidy in a mathematical formulation of something.

We can still work with it programmatically. By fudging around a bit with numpy’s shapes to get a diagonal matrix, we can reconstruct the input rating matrix from the factors.

Sigma = np.vstack([
    np.diag(singularValues),
    np.zeros((5, 3)),
])

print(np.round(movieRatings - np.dot(U, np.dot(Sigma, V)), decimals=10))

And the output is, as one expects, a matrix of all zeros. Meaning that we decomposed the movie rating matrix, and built it back up from the factors.

We can actually get the SVD as we defined it (with rectangular matrices) by passing a special flag to numpy’s svd.

U, singularValues, V = svd(movieRatings, full_matrices=False)
print(U)
print(singularValues)
print(V)

Sigma = np.diag(singularValues)
print(np.round(movieRatings - np.dot(U, np.dot(Sigma, V)), decimals=10))

And the result

[[-0.39458526  0.23923575 -0.35445911]
 [-0.15830232  0.03054913 -0.15299759]
 [-0.22155201 -0.52086121  0.39334917]
 [-0.39692635 -0.08649009 -0.41052882]
 [-0.34630257 -0.64128825  0.07382859]
 [-0.53347449  0.19168874  0.19949342]
 [-0.31660464  0.06109826 -0.30599517]
 [-0.32840223  0.45970413  0.62354764]]
[ 15.09626916   4.30056855   3.40701739]
[[-0.54184808 -0.67070995 -0.50650649]
 [-0.75152295  0.11680911  0.64928336]
 [ 0.37631623 -0.73246419  0.56734672]]
[[-0. -0. -0.]
 [-0. -0.  0.]
 [ 0. -0.  0.]
 [-0. -0. -0.]
 [-0. -0. -0.]
 [-0. -0. -0.]
 [-0. -0. -0.]
 [ 0. -0. -0.]]

This makes the reconstruction less messy, since we can just multiply everything without having to add extra rows of zeros to \Sigma.

What do the singular vectors and values tell us about the movie rating matrix? (Besides nothing, since it’s a contrived example) You’ll notice that the first singular vector \sigma_1 > 15 while the other two singular values are around 4. This tells us that the first singular vector covers a large part of the structure of the matrix. I.e., a rank-1 matrix would be a pretty good approximation to the whole thing. As an exercise to the reader, write a program that evaluates this claim (how good is “good”?).

The greedy optimization routine

Now we’re going to write SVD from scratch. We’ll first implement the greedy algorithm for the 1-d optimization problem, and then we’ll perform the inductive step to get a full algorithm. Then we’ll run it on the CNN data set.

The method we’ll use to solve the 1-dimensional problem isn’t necessarily industry strength (see this document for a hint of what industry strength looks like), but it is simple conceptually. It’s called the power method. Now that we have our decomposition of theorem, understanding how the power method works is quite easy.

Let’s work in the language of a matrix decomposition A = U \Sigma V^T, more for practice with that language than anything else (using outer products would give us the same result with slightly different computations). Then let’s observe A^T A, wherein we’ll use the fact that U is orthonormal and so U^TU is the identity matrix:

\displaystyle A^TA = (U \Sigma V^T)^T(U \Sigma V^T) = V \Sigma U^TU \Sigma V^T = V \Sigma^2 V^T

So we can completely eliminate U from the discussion, and look at just V \Sigma^2 V^T. And what’s nice about this matrix is that we can compute its eigenvectors, and eigenvectors turn out to be exactly the singular vectors. The corresponding eigenvalues are the squared singular values. This should be clear from the above derivation. If you apply (V \Sigma^2 V^T) to any v_i, the only parts of the product that aren’t zero are the ones involving v_i with itself, and the scalar \sigma_i^2 factors in smoothly. It’s dead simple to check.

Theorem: Let x be a random unit vector and let B = A^TA = V \Sigma^2 V^T. Then with high probability, \lim_{s \to \infty} B^s x is in the span of the first singular vector v_1. If we normalize B^s x to a unit vector at each s, then furthermore the limit is v_1.

Proof. Start with a random unit vector x, and write it in terms of the singular vectors x = \sum_i c_i v_i. That means Bx = \sum_i c_i \sigma_i^2 v_i. If you recursively apply this logic, you get B^s x = \sum_i c_i \sigma_i^{2s} v_i. In particular, the dot product of (B^s x) with any v_j is c_i \sigma_j^{2s}.

What this means is that so long as the first singular value \sigma_1 is sufficiently larger than the second one \sigma_2, and in turn all the other singular values, the part of B^s x  corresponding to v_1 will be much larger than the rest. Recall that if you expand a vector in terms of an orthonormal basis, in this case B^s x expanded in the v_i, the coefficient of B^s x on v_j is exactly the dot product. So to say that B^sx converges to being in the span of v_1 is the same as saying that the ratio of these coefficients, |(B^s x \cdot v_1)| / |(B^s x \cdot v_j)| \to \infty for any j. In other words, the coefficient corresponding to the first singular vector dominates all of the others. And so if we normalize, the coefficient of B^s x corresponding to v_1 tends to 1, while the rest tend to zero.

Indeed, this ratio is just (\sigma_1 / \sigma_j)^{2s} and the base of this exponential is bigger than 1.

\square

If you want to be a little more precise and find bounds on the number of iterations required to converge, you can. The worry is that your random starting vector is “too close” to one of the smaller singular vectors v_j, so that if the ratio of \sigma_1 / \sigma_j is small, then the “pull” of v_1 won’t outweigh the pull of v_j fast enough. Choosing a random unit vector allows you to ensure with high probability that this doesn’t happen. And conditioned on it not happening (or measuring “how far the event is from happening” precisely), you can compute a precise number of iterations required to converge. The last two pages of these lecture notes have all the details.

We won’t compute a precise number of iterations. Instead we’ll just compute until the angle between B^{s+1}x and B^s x is very small. Here’s the algorithm

import numpy as np
from numpy.linalg import norm

from random import normalvariate
from math import sqrt


def randomUnitVector(n):
    unnormalized = [normalvariate(0, 1) for _ in range(n)]
    theNorm = sqrt(sum(x * x for x in unnormalized))
    return [x / theNorm for x in unnormalized]


def svd_1d(A, epsilon=1e-10):
    ''' The one-dimensional SVD '''

    n, m = A.shape
    x = randomUnitVector(m)
    lastV = None
    currentV = x
    B = np.dot(A.T, A)

    iterations = 0
    while True:
        iterations += 1
        lastV = currentV
        currentV = np.dot(B, lastV)
        currentV = currentV / norm(currentV)

        if abs(np.dot(currentV, lastV)) > 1 - epsilon:
            print("converged in {} iterations!".format(iterations))
            return currentV

We start with a random unit vector x, and then loop computing x_{t+1} = Bx_t, renormalizing at each step. The condition for stopping is that the magnitude of the dot product between x_t and x_{t+1} (since they’re unit vectors, this is the cosine of the angle between them) is very close to 1.

And using it on our movie ratings example:

if __name__ == "__main__":
    movieRatings = np.array([
        [2, 5, 3],
        [1, 2, 1],
        [4, 1, 1],
        [3, 5, 2],
        [5, 3, 1],
        [4, 5, 5],
        [2, 4, 2],
        [2, 2, 5],
    ], dtype='float64')

    print(svd_1d(movieRatings))

With the result

converged in 6 iterations!
[-0.54184805 -0.67070993 -0.50650655]

Note that the sign of the vector may be different from numpy’s output because we start with a random vector to begin with.

The recursive step, getting from v_1 to the entire SVD, is equally straightforward. Say you start with the matrix A and you compute v_1. You can use v_1 to compute u_1 and \sigma_1(A). Then you want to ensure you’re ignoring all vectors in the span of v_1 for your next greedy optimization, and to do this you can simply subtract the rank 1 component of A corresponding to v_1. I.e., set A' = A - \sigma_1(A) u_1 v_1^T. Then it’s easy to see that \sigma_1(A') = \sigma_2(A) and basically all the singular vectors shift indices by 1 when going from A to A'. Then you repeat.

If that’s not clear enough, here’s the code.

def svd(A, epsilon=1e-10):
    n, m = A.shape
    svdSoFar = []

    for i in range(m):
        matrixFor1D = A.copy()

        for singularValue, u, v in svdSoFar[:i]:
            matrixFor1D -= singularValue * np.outer(u, v)

        v = svd_1d(matrixFor1D, epsilon=epsilon)  # next singular vector
        u_unnormalized = np.dot(A, v)
        sigma = norm(u_unnormalized)  # next singular value
        u = u_unnormalized / sigma

        svdSoFar.append((sigma, u, v))

    # transform it into matrices of the right shape
    singularValues, us, vs = [np.array(x) for x in zip(*svdSoFar)]

    return singularValues, us.T, vs

And we can run this on our movie rating matrix to get the following

>>> theSVD = svd(movieRatings)
>>> theSVD[0]
array([ 15.09626916,   4.30056855,   3.40701739])
>>> theSVD[1]
array([[ 0.39458528, -0.23923093,  0.35446407],
       [ 0.15830233, -0.03054705,  0.15299815],
       [ 0.221552  ,  0.52085578, -0.39336072],
       [ 0.39692636,  0.08649568,  0.41052666],
       [ 0.34630257,  0.64128719, -0.07384286],
       [ 0.53347448, -0.19169154, -0.19948959],
       [ 0.31660465, -0.0610941 ,  0.30599629],
       [ 0.32840221, -0.45971273, -0.62353781]])
>>> theSVD[2]
array([[ 0.54184805,  0.67071006,  0.50650638],
       [ 0.75151641, -0.11679644, -0.64929321],
       [-0.37632934,  0.73246611, -0.56733554]])

Checking this against our numpy output shows it’s within a reasonable level of precision (considering the power method took on the order of ten iterations!)

>>> np.round(np.abs(npSVD[0]) - np.abs(theSVD[1]), decimals=5)
array([[ -0.00000000e+00,  -0.00000000e+00,   0.00000000e+00],
       [  0.00000000e+00,  -0.00000000e+00,   0.00000000e+00],
       [  0.00000000e+00,  -1.00000000e-05,   1.00000000e-05],
       [  0.00000000e+00,   0.00000000e+00,  -0.00000000e+00],
       [  0.00000000e+00,  -0.00000000e+00,   1.00000000e-05],
       [ -0.00000000e+00,   0.00000000e+00,  -0.00000000e+00],
       [  0.00000000e+00,  -0.00000000e+00,   0.00000000e+00],
       [ -0.00000000e+00,   1.00000000e-05,  -1.00000000e-05]])
>>> np.round(np.abs(npSVD[2]) - np.abs(theSVD[2]), decimals=5)
array([[  0.00000000e+00,   0.00000000e+00,  -0.00000000e+00],
       [ -1.00000000e-05,  -1.00000000e-05,   1.00000000e-05],
       [  1.00000000e-05,   0.00000000e+00,  -1.00000000e-05]])
>>> np.round(np.abs(npSVD[1]) - np.abs(theSVD[0]), decimals=5)
array([ 0.,  0., -0.])

So there we have it. We added an extra little bit to the svd function, an argument k which stops computing the svd after it reaches rank k.

CNN stories

One interesting use of the SVD is in topic modeling. Topic modeling is the process of taking a bunch of documents (news stories, or emails, or movie scripts, whatever) and grouping them by topic, where the algorithm gets to choose what counts as a “topic.” Topic modeling is just the name that natural language processing folks use instead of clustering.

The SVD can help one model topics as follows. First you construct a matrix A called a document-term matrix whose rows correspond to words in some fixed dictionary and whose columns correspond to documents. The (i,j) entry of A contains the number of times word i shows up in document j. Or, more precisely, some quantity derived from that count, like a normalized count. See this table on wikipedia for a list of options related to that. We’ll just pick one arbitrarily for use in this post.

The point isn’t how we normalize the data, but what the SVD of A = U \Sigma V^T means in this context. Recall that the domain of A, as a linear map, is a vector space whose dimension is the number of stories. We think of the vectors in this space as documents, or rather as an “embedding” of the abstract concept of a document using the counts of how often each word shows up in a document as a proxy for the semantic meaning of the document. Likewise, the codomain is the space of all words, and each word is embedded by which documents it occurs in. If we compare this to the movie rating example, it’s the same thing: a movie is the vector of ratings it receives from people, and a person is the vector of ratings of various movies.

Say you take a rank 3 approximation to A. Then you get three singular vectors v_1, v_2, v_3 which form a basis for a subspace of words, i.e., the “idealized” words. These idealized words are your topics, and you can compute where a “new word” falls by looking at which documents it appears in (writing it as a vector in the domain) and saying its “topic” is the closest of the v_1, v_2, v_3. The same process applies to new documents. You can use this to cluster existing documents as well.

The dataset we’ll use for this post is a relatively small corpus of a thousand CNN stories picked from 2012. Here’s an excerpt from one of them

$ cat data/cnn-stories/story479.txt 
3 things to watch on Super Tuesday
Here are three things to watch for: Romney's big day. He's been the off-and-on frontrunner throughout the race, but a big Super Tuesday could begin an end game toward a sometimes hesitant base coalescing behind former Massachusetts Gov. Mitt Romney. Romney should win his home state of Massachusetts, neighboring Vermont and Virginia, ...

So let’s first build this document-term matrix, with the normalized values, and then we’ll compute it’s SVD and see what the topics look like.

Step 1 is cleaning the data. We used a bunch of routines from the nltk library that boils down to this loop:

    for filename, documentText in documentDict.items():
        tokens = tokenize(documentText)
        tagged_tokens = pos_tag(tokens)
        wnl = WordNetLemmatizer()
        stemmedTokens = [wnl.lemmatize(word, wordnetPos(tag)).lower()
                         for word, tag in tagged_tokens]

This turns the Super Tuesday story into a list of words (with repetition):

["thing", "watch", "three", "thing", "watch", "big", ... ]

If you’ll notice the name Romney doesn’t show up in the list of words. I’m only keeping the words that show up in the top 100,000 most common English words, and then lemmatizing all of the words to their roots. It’s not a perfect data cleaning job, but it’s simple and good enough for our purposes.

Now we can create the document term matrix.

def makeDocumentTermMatrix(data):
    words = allWords(data)  # get the set of all unique words

    wordToIndex = dict((word, i) for i, word in enumerate(words))
    indexToWord = dict(enumerate(words))
    indexToDocument = dict(enumerate(data))

    matrix = np.zeros((len(words), len(data)))
    for docID, document in enumerate(data):
        docWords = Counter(document['words'])
        for word, count in docWords.items():
            matrix[wordToIndex[word], docID] = count

    return matrix, (indexToWord, indexToDocument)

This creates a matrix with the raw integer counts. But what we need is a normalized count. The idea is that a common word like “thing” shows up disproportionately more often than “election,” and we don’t want raw magnitude of a word count to outweigh its semantic contribution to the classification. This is the applied math part of the algorithm design. So what we’ll do (and this technique together with SVD is called latent semantic indexing) is normalize each entry so that it measures both the frequency of a term in a document and the relative frequency of a term compared to the global frequency of that term. There are many ways to do this, and we’ll just pick one. See the github repository if you’re interested.

So now lets compute a rank 10 decomposition and see how to cluster the results.

    data = load()
    matrix, (indexToWord, indexToDocument) = makeDocumentTermMatrix(data)
    matrix = normalize(matrix)
    sigma, U, V = svd(matrix, k=10)

This uses our svd, not numpy’s. Though numpy’s routine is much faster, it’s fun to see things work with code written from scratch. The result is too large to display here, but I can report the singular values.

>>> sigma
array([ 42.85249098,  21.85641975,  19.15989197,  16.2403354 ,
        15.40456779,  14.3172779 ,  13.47860033,  13.23795002,
        12.98866537,  12.51307445])

Now we take our original inputs and project them onto the subspace spanned by the singular vectors. This is the part that represents each word (resp., document) in terms of the idealized words (resp., documents), the singular vectors. Then we can apply a simple k-means clustering algorithm to the result, and observe the resulting clusters as documents.

    projectedDocuments = np.dot(matrix.T, U)
    projectedWords = np.dot(matrix, V.T)

    documentCenters, documentClustering = cluster(projectedDocuments)
    wordCenters, wordClustering = cluster(projectedWords)

    wordClusters = [
        [indexToWord[i] for (i, x) in enumerate(wordClustering) if x == j]
        for j in range(len(set(wordClustering)))
    ]

    documentClusters = [
        [indexToDocument[i]['text']
         for (i, x) in enumerate(documentClustering) if x == j]
        for j in range(len(set(documentClustering)))
    ]   

And now we can inspect individual clusters. Right off the bat we can tell the clusters aren’t quite right simply by looking at the sizes of each cluster.

>>> Counter(wordClustering)
Counter({1: 9689, 2: 1051, 8: 680, 5: 557, 3: 321, 7: 225, 4: 174, 6: 124, 9: 123})
>>> Counter(documentClustering)
Counter({7: 407, 6: 109, 0: 102, 5: 87, 9: 85, 2: 65, 8: 55, 4: 47, 3: 23, 1: 15})

What looks wrong to me is the size of the largest word cluster. If we could group words by topic, then this is saying there’s a topic with over nine thousand words associated with it! Inspecting it even closer, it includes words like “vegan,” “skunk,” and “pope.” On the other hand, some word clusters are spot on. Examine, for example, the fifth cluster which includes words very clearly associated with crime stories.

>>> wordClusters[4]
['account', 'accuse', 'act', 'affiliate', 'allegation', 'allege', 'altercation', 'anything', 'apartment', 'arrest', 'arrive', 'assault', 'attorney', 'authority', 'bag', 'black', 'blood', 'boy', 'brother', 'bullet', 'candy', 'car', 'carry', 'case', 'charge', 'chief', 'child', 'claim', 'client', 'commit', 'community', 'contact', 'convenience', 'court', 'crime', 'criminal', 'cry', 'dead', 'deadly', 'death', 'defense', 'department', 'describe', 'detail', 'determine', 'dispatcher', 'district', 'document', 'enforcement', 'evidence', 'extremely', 'family', 'father', 'fear', 'fiancee', 'file', 'five', 'foot', 'friend', 'front', 'gate', 'girl', 'girlfriend', 'grand', 'ground', 'guilty', 'gun', 'gunman', 'gunshot', 'hand', 'happen', 'harm', 'head', 'hear', 'heard', 'hoodie', 'hour', 'house', 'identify', 'immediately', 'incident', 'information', 'injury', 'investigate', 'investigation', 'investigator', 'involve', 'judge', 'jury', 'justice', 'kid', 'killing', 'lawyer', 'legal', 'letter', 'life', 'local', 'man', 'men', 'mile', 'morning', 'mother', 'murder', 'near', 'nearby', 'neighbor', 'newspaper', 'night', 'nothing', 'office', 'officer', 'online', 'outside', 'parent', 'person', 'phone', 'police', 'post', 'prison', 'profile', 'prosecute', 'prosecution', 'prosecutor', 'pull', 'racial', 'racist', 'release', 'responsible', 'return', 'review', 'role', 'saw', 'scene', 'school', 'scream', 'search', 'sentence', 'serve', 'several', 'shoot', 'shooter', 'shooting', 'shot', 'slur', 'someone', 'son', 'sound', 'spark', 'speak', 'staff', 'stand', 'store', 'story', 'student', 'surveillance', 'suspect', 'suspicious', 'tape', 'teacher', 'teen', 'teenager', 'told', 'tragedy', 'trial', 'vehicle', 'victim', 'video', 'walk', 'watch', 'wear', 'whether', 'white', 'witness', 'young']

As sad as it makes me to see that ‘black’ and ‘slur’ and ‘racial’ appear in this category, it’s a reminder that naively using the output of a machine learning algorithm can perpetuate racism.

Here’s another interesting cluster corresponding to economic words:

>>> wordClusters[6]
['agreement', 'aide', 'analyst', 'approval', 'approve', 'austerity', 'average', 'bailout', 'beneficiary', 'benefit', 'bill', 'billion', 'break', 'broadband', 'budget', 'class', 'combine', 'committee', 'compromise', 'conference', 'congressional', 'contribution', 'core', 'cost', 'currently', 'cut', 'deal', 'debt', 'defender', 'deficit', 'doc', 'drop', 'economic', 'economy', 'employee', 'employer', 'erode', 'eurozone', 'expire', 'extend', 'extension', 'fee', 'finance', 'fiscal', 'fix', 'fully', 'fund', 'funding', 'game', 'generally', 'gleefully', 'growth', 'hamper', 'highlight', 'hike', 'hire', 'holiday', 'increase', 'indifferent', 'insistence', 'insurance', 'job', 'juncture', 'latter', 'legislation', 'loser', 'low', 'lower', 'majority', 'maximum', 'measure', 'middle', 'negotiation', 'offset', 'oppose', 'package', 'pass', 'patient', 'pay', 'payment', 'payroll', 'pension', 'plight', 'portray', 'priority', 'proposal', 'provision', 'rate', 'recession', 'recovery', 'reduce', 'reduction', 'reluctance', 'repercussion', 'rest', 'revenue', 'rich', 'roughly', 'sale', 'saving', 'scientist', 'separate', 'sharp', 'showdown', 'sign', 'specialist', 'spectrum', 'spending', 'strength', 'tax', 'tea', 'tentative', 'term', 'test', 'top', 'trillion', 'turnaround', 'unemployed', 'unemployment', 'union', 'wage', 'welfare', 'worker', 'worth']

One can also inspect the stories, though the clusters are harder to print out here. Interestingly the first cluster of documents are stories exclusively about Trayvon Martin. The second cluster is mostly international military conflicts. The third cluster also appears to be about international conflict, but what distinguishes it from the first cluster is that every story in the second cluster discusses Syria.

>>> len([x for x in documentClusters[1] if 'Syria' in x]) / len(documentClusters[1])
0.05555555555555555
>>> len([x for x in documentClusters[2] if 'Syria' in x]) / len(documentClusters[2])
1.0

Anyway, you can explore the data more at your leisure (and tinker with the parameters to improve it!).

Issues with the power method

Though I mentioned that the power method isn’t an industry strength algorithm I didn’t say why. Let’s revisit that before we finish. The problem is that the convergence rate of even the 1-dimensional problem depends on the ratio of the first and second singular values, \sigma_1 / \sigma_2. If that ratio is very close to 1, then the convergence will take a long time and need many many matrix-vector multiplications.

One way to alleviate that is to do the trick where, to compute a large power of a matrix, you iteratively square B. But that requires computing a matrix square (instead of a bunch of matrix-vector products), and that requires a lot of time and memory if the matrix isn’t sparse. When the matrix is sparse, you can actually do the power method quite quickly, from what I’ve heard and read.

But nevertheless, the industry standard methods involve computing a particular matrix decomposition that is not only faster than the power method, but also numerically stable. That means that the algorithm’s runtime and accuracy doesn’t depend on slight changes in the entries of the input matrix. Indeed, you can have two matrices where \sigma_1 / \sigma_2 is very close to 1, but changing a single entry will make that ratio much larger. The power method depends on this, so it’s not numerically stable. But the industry standard technique is not. This technique involves something called Householder reflections. So while the power method was great for a proof of concept, there’s much more work to do if you want true SVD power.

Until next time!

Singular Value Decomposition Part 1: Perspectives on Linear Algebra

The singular value decomposition (SVD) of a matrix is a fundamental tool in computer science, data analysis, and statistics. It’s used for all kinds of applications from regression to prediction, to finding approximate solutions to optimization problems. In this series of two posts we’ll motivate, define, compute, and use the singular value decomposition to analyze some data. (Jump to the second post)

I want to spend the first post entirely on motivation and background. As part of this, I think we need a little reminder about how linear algebra equivocates linear subspaces and matrices. I say “I think” because what I’m going to say seems rarely spelled out in detail. Indeed, I was confused myself when I first started to read about linear algebra applied to algorithms, machine learning, and data science, despite having a solid understanding of linear algebra from a mathematical perspective. The concern is the connection between matrices as transformations and matrices as a “convenient” way to organize data.

Data vs. maps

Linear algebra aficionados like to express deep facts via statements about matrix factorization. That is, they’ll say something opaque like (and this is the complete statement for SVD we’ll get to in the post):

The SVD of an m \times n matrix A with real values is a factorization of A as U\Sigma V^T, where U is an m \times m orthogonal matrix, V is an n \times n orthogonal matrix, and \Sigma is a diagonal matrix with nonnegative real entries on the diagonal.

Okay, I can understand the words individually, but what does it mean in terms of the big picture? There are two seemingly conflicting interpretations of matrices that muddle our vision.

The first is that A is a linear map from some n-dimensional vector space to an m-dimensional one. Let’s work with real numbers and call the domain vector space \mathbb{R}^n and the codomain \mathbb{R}^m. In this interpretation the factorization expresses a change of basis in the domain and codomain. Specifically, V expresses a change of basis from the usual basis of \mathbb{R}^n to some other basis, and U does the same for the co-domain \mathbb{R}^m.

That’s fine and dandy, so long as the data that makes up A is the description of some linear map we’d like to learn more about. Years ago on this blog we did exactly this analysis of a linear map modeling a random walk through the internet, and we ended up with Google’s PageRank algorithm. However, in most linear algebra applications A actually contains data in its rows or columns. That’s the second interpretation. That is, each row of A is a data point in \mathbb{R}^n, and there are m total data points, and they represent observations of some process happening in the world. The data points are like people rating movies or weather sensors measuring wind and temperature. If you’re not indoctrinated with the linear algebra world view, you might not naturally think of this as a mapping of vectors from one vector space to another.

Most of the time when people talk about linear algebra (even mathematicians), they’ll stick entirely to the linear map perspective or the data perspective, which is kind of frustrating when you’re learning it for the first time. It seems like the data perspective is just a tidy convenience, that it just “makes sense” to put some data in a table. In my experience the singular value decomposition is the first time that the two perspectives collide, and (at least in my case) it comes with cognitive dissonance.

The way these two ideas combine is that the data is thought of as the image of the basis vectors of \mathbb{R}^n under the linear map specified by A. Here is an example to make this concrete. Let’s say I want to express people rating movies. Each row will correspond to the ratings of a movie, and each column will correspond to a person, and the i,j entry of the matrix A is the rating person j gives to movie i.

movieratings

In reality they’re rated on a scale from 1 to 5 stars, but to keep things simple we’ll just say that the ratings can be any real numbers (they just happened to pick integers). So this matrix represents a linear map. The domain is \mathbb{R}^3, and the basis vectors are called people, and the codomain is \mathbb{R}^8, whose basis vectors are movies.

moviemapping.png

Now the data set is represented by A(\vec e_{\textup{Aisha}}),A(\vec e_{\textup{Bob}}),A(\vec e_{\textup{Chandrika}}), and by the definition of how a matrix represents a linear map, the entires of these vectors are exactly the columns of A. If the codomain is really big, then the image of A is a small-dimensional linear subspace of the codomain. This is an important step, that we’ve increased our view from just the individual data points to all of their linear combinations as a subspace.

Why is this helpful at all? This is where we start to see the modeling assumptions of linear algebra show through. If we’re trying to use this matrix to say something about how people rate movies (maybe we want to predict how a new person will rate these movies), we would need to be able to represent that person as a linear combination of Aisha, Bob, and Chandrika. Likewise, if we had a new movie and we wanted to use this matrix to say anything about it, we’d have to represent the movie as a linear combination of the existing movies.

Of course, I don’t literally mean that a movie (as in, the bits comprising a file containing a movie) can be represented as a linear combination of other movies. I mean that we can represent a movie formally as a linear combination in some abstract vector space for the task at hand. In other words, we’re representing those features of the movie that influence its rating abstractly as a vector. We don’t have a legitimate mathematical way to understand that, so the vector is a proxy.

It’s totally unclear what this means in terms of real life, except that you can hope (or hypothesize, or verify), that if the rating process of movies is “linear” in nature then this formal representation will accurately reflect the real world. It’s like how physicists all secretly know that mathematics doesn’t literally dictate the laws of nature, because humans made up math in their heads and if you poke nature too hard the math breaks down, but it’s so damn convenient to describe hypotheses (and so damn accurate), that we can’t avoid using it to design airplanes. And we haven’t found anything better than math for this purpose.

Likewise, movie ratings aren’t literally a linear map, but if we pretend they are we can make algorithms that predict how people rate movies with pretty good accuracy. So if you know that Skyfall gets ratings 1,2, and 1 from Aisha, Bob, and Chandrika, respectively, then a new person would rate Skyfall based on a linear combination of how well they align with these three people. In other words, up to a linear combination, in this example Aisha, Bob, and Chandrika epitomize the process of rating movies.

And now we get to the key: factoring the matrix via SVD provides an alternative and more useful way to represent the process of people rating movies. By changing the basis of one or both vector spaces involved, we isolate the different (orthogonal) characteristics of the process. In the context of our movie example, “factorization” means the following:

  1. Come up with a special list of vectors v_1, v_2, \dots, v_8 so that every movie can be written as a linear combination of the v_i.
  2. Do the analogous thing for people to get p_1, p_2, p_3.
  3. Do (1) and (2) in such a way that the map A is diagonal with respect to both new bases simultaneously.

One might think of the v_i as “idealized movies” and the p_j as “idealized critics.” If you want to use this data to say things about the world, you’d be making the assumption that any person can be written as a linear combination of the p_j and any movie can be written as a linear combination of the v_i. These are the rows/columns of U, V from the factorization. To reiterate, these linear combinations are only with respect to the task of rating movies. And they’re “special” because they make the matrix diagonal.

If the world was logical (and I’m not saying it is) then maybe v_1 would correspond to some idealized notion of “action movie,” and p_1 would correspond to some idealized notion of “action movie lover.” Then it makes sense why the mapping would be diagonal in this basis: an action movie lover only loves action movies, so p_1 gives a rating of zero to everything except v_1. A movie is represented by how it decomposes (linearly) into “idealized” movies. To make up some arbitrary numbers, maybe Skyfall is 2/3 action movie, 1/5 dystopian sci-fi, and -6/7 comedic romance. Likewise a person would be represented by how they decompose (via linear combination) into a action movie lover, rom-com lover, etc.

To be completely clear, the singular value decomposition does not find the ideal sci-fi movie. The “ideal”ness of the singular value decomposition is with respect to the inherent geometric structure of the data coupled with the assumptions of linearity. Whether this has anything at all to do with how humans classify movies is a separate question, and the answer is almost certainly no.

With this perspective we’re almost ready to talk about the singular value decomposition. I just want to take a moment to write down a list of the assumptions that we’d need if we want to ensure that, given a data set of movie ratings, we can use linear algebra to make exact claims about world.

  1. All people rate movies via the same linear map.
  2. Every person can be expressed (for the sole purpose of movie ratings) as linear combinations of “ideal” people. Likewise for movies.
  3. The “idealized” movies and people can be expressed as linear combinations of the movies/people in our particular data set.
  4. There are no errors in the ratings.

One could have a deep and interesting discussion about the philosophical (or ethical, or cultural) aspects of these assumptions. But since the internet prefers to watch respectful discourse burn, we’ll turn to algorithms instead.

Approximating subspaces

In our present context, the singular value decomposition (SVD) isn’t meant to be a complete description of a mapping in a new basis, as we said above. Rather, we want to use it to approximate the mapping A by low-dimensional linear things. When I say “low-dimensional linear things” I mean that given A, we’d want to find another matrix B which is measurably similar to A in some way, and has low rank compared to A.

How do we know that A isn’t already low rank? The reasons is that data with even the tiniest bit of noise is full rank with overwhelming probability. A concrete way to say this is that the space of low-rank matrices has small dimension (in the sense of a manifold) inside the space of all matrices. So perturbing even a single entry by an infinitesimally small amount would increase the rank.

We don’t need to understand manifolds to understand the SVD, though. For our example of people rating movies the full-rank property should be obvious. The noise and randomness and arbitrariness in human preferences certainly destroys any “perfect” linear structure we could hope to find, and in particular that means the data set itself, i.e. the image of A, is a large-dimensional subspace of the codomain.

Finding a low-rank approximation can be thought of as “smoothing” the noise out of the data. And this works particularly well when the underlying process is close to a linear map. That is, when the data is close to being contained entirely in a single subspace of relatively low-dimension. One way to think of why this might be the case is that if the process you’re observing is truly linear, but the data you get is corrupted by small amounts of noise. Then A will be close to low rank in a measurable sense (to be defined mathematically in the sequel post) and the low-rank approximation B will be a more efficient, accurate, and generalizable surrogate for A.

In terms of our earlier list of assumptions about when you can linear algebra to solve problems, for the SVD we can add “approximately” to the first three assumptions, and “not too many errors” to the fourth. If those assumptions hold, SVD will give us a matrix B which accurately represents the process being measured. Conversely, if SVD does well, then you have some evidence that the process is linear-esque.

To be more specific with notation, if A is a matrix representing some dataset via the image of A and you provide a small integer k, then the singular value decomposition computes the rank k matrix B_k which best approximates A. And since now we’re comfortable identifying a data matrix with the subspace defined by its image, this is the same thing as finding the k-dimensional subspace of the image of A which is the best approximation of the data (i.e., the image of B_k). We’ll quantify what we mean by “best approximation” in the next post.

That’s it, as far as intuitively understanding what the SVD is. I should add that the SVD doesn’t only allow one to compute a rank k approximation, it actually allows you to set k=n and get an exact representation of A. We just won’t use it for that purpose in this series.

The second bit of intuition is the following. It’s only slightly closer to rigor, but somehow this little insight really made SVD click for me personally:

The SVD is what you get when you iteratively solve the greedy optimization problem of fitting data to a line.

By that I mean, you can compute the SVD by doing the following:

  1. What’s the best line fitting my data?
  2. Okay, ignoring that first line, what’s the next best line?
  3. Okay, ignoring all the lines in the span of those first two lines, what’s the next best line?
  4. Ignoring all the lines in the span of the first three lines, what’s the next best line?
  5. (repeat)

It should be shocking that this works. For most problems, in math and in life, the greedy algorithm is far from optimal. When it happens, once every blue moon, that the greedy algorithm is the best solution to a natural problem (and not obviously so, or just approximately so), it’s our intellectual duty to stop what we’re doing, sit up straight, and really understand and appreciate it. These wonders transcend political squabbles and sports scores. And we’ll start the next post immediately by diving into this greedy optimization problem.

The geometric perspective

There are two other perspectives I want to discuss here, though it may be more appropriate for a reader who is not familiar with the SVD to wait to read this after the sequel to this post. I’m just going to relate my understanding (in terms of the greedy algorithm and data approximations) to the geometric and statistical perspectives on the SVD.

Michael Nielsen wrote a long and detailed article presenting some ideas about a “new medium” in which to think about mathematics. He demonstrates is framework by looking at the singular value decomposition for 2×2 matrices. His explanation for the intuition behind the SVD is that you can take any matrix (linear map) and break it up into three pieces: a rotation about the origin, a rescaling of each coordinate, followed by another rotation about the origin. While I have the utmost respect for Nielsen (his book on quantum mechanics is the best text in the field), this explanation never quite made SVD click for me personally. It seems like a restatement of the opaque SVD definition (as a matrix factorization) into geometric terms. Indeed, an orthogonal matrix is a rotation, and a diagonal matrix is a rescaling of each coordinate.

To me, the key that’s missing from this explanation is the emphasis on the approximation. What makes the SVD so magical isn’t that the factorization exists in the first place, but rather that the SVD has these layers of increasingly good approximation. Though the terminology will come in the next post, these layers are the (ordered) singular vectors and singular values. And moreover, that the algorithmic process of constructing these layers necessarily goes in order from strongest approximation to weakest.

Another geometric perspective that highlights this is that the rank-k approximation provided by the SVD is a geometric projection of a matrix onto the space of rank at-most-k matrices with respect to the “spectral norm” on matrices (the spectral norm of A is the largest eigenvalue of A^TA). The change of basis described above makes this projection very easy: given a singular value decomposition you just take the top k singular vectors. Indeed, the Eckart-Young theorem formalizes this with the statement that the rank-k SVD B_k minimizes the distance (w.r.t. the spectral norm) between the original matrix A and any rank k matrix. So you can prove that SVD gives you the best rank k approximation of A by some reasonable measure.

Next time: algorithms

Next time we’ll connect all this to the formal definitions and rigor. We’ll study the greedy algorithm approach, and then we’ll implement the SVD and test it on some data.

Until then!

Tensorphobia and the Outer Product

Variations on a theme

Back in 2014 I wrote a post called How to Conquer Tensorphobia that should end up on Math \cap Programming’s “greatest hits” album. One aspect of tensors I neglected to discuss was the connection between the modern views of tensors and the practical views of linear algebra. I feel I need to write this because every year or two I forget why it makes sense.

The basic question is:

What the hell is going on with the outer product of vectors?

The simple answer, the one that has never satisfied me, is that the outer product of v,w is the matrix vw^T whose i,j entry is the product v_i w_j. This doesn’t satisfy me because it’s an explanation by fiat. It lacks motivation and you’re supposed to trust (or verify) things magically work out. To me this definition is like the definition of matrix multiplication: having it dictated to you before you understand why it makes sense is a cop out. Math isn’t magic, it needs to make perfect sense.

The answer I like, and the one I have to re-derive every few years because I never wrote it down, is a little bit longer.

To borrow a programming term, the basic confusion is a type error. We start with two vectors, v,w in some vector space V (let’s say everything is finite dimensional), and we magically turn them into a matrix. Let me reiterate for dramatic effect: we start with vectors, which I have always thought of as objects, things you can stretch, rotate, or give as a gift to your cousin Mauricio. While a matrix is a mapping, a thing that takes vectors as input and spits out new vectors. Sure, you can play with the mappings, feed them kibble and wait for them to poop or whatever. And sure, sometimes vectors are themselves maps, like in the vector space of real-valued functions (where the word “matrix” is a stretch, since it’s infinite dimensional).

But to take two vectors and *poof* get a mapping of all vectors, it’s a big jump. And part of the discomfort is that it feels random. To be happy we have to understand that the construction is natural, or even better canonical, meaning this should be the only way to turn two vectors into a linear map. Then the definition would make sense.

So let’s see how we can do that. Just to be clear, everything we do in this post will be for finite-dimensional vector spaces over \mathbb{R}, but we’ll highlight the caveats when they come up.

Dual vector spaces

The first step is understanding how to associate a vector with a linear map in a “natural” or “canonical” way. There is one obvious candidate: if you give me a vector v, I can make a linear map by taking the dot product of v with the input. So I can associate

\displaystyle v \mapsto \langle v,- \rangle

The dash is a placeholder for the input. Another way to say it is to define \varphi_v(w) = \langle v,w \rangle and say the association takes v \mapsto \varphi_v. So this “association,” taking v to the inner product, is itself  a mapping from V to “maps from V to \mathbb{R}.” Note that \varphi_v(w) is linear in v and w because that’s part of the definition of an inner product.

To avoid saying “maps from V to \mathbb{R}” all the time, we’ll introduce some notation.

Definition: Let V be a vector space over a field k. The set of k-linear maps from V \to k is called \textup{Hom}(V,k). More generally, the set of k-linear maps from V to another k-vector space W is called \textup{Hom}(V,W).

“Hom” stands for “homomorphism,” and in general it just means “maps with the structure I care about.” For this post k = \mathbb{R}, but most of what we say here will be true for any field. If you go deeply into this topic, it matters whether k is algebraically closed, or has finite characteristic, but for simplicity we’ll ignore all of that. We’ll also ignore the fact that these maps are called linear functionals and this is where the name “functional analysis” comes from. All we really want to do is understand the definition of the outer product.

Another bit of notation for brevity:

Definition: Let V be a \mathbb{R}-vector space. The dual vector space for V, denoted V^*, is \textup{Hom}(V, \mathbb{R}).

So the “vector-to-inner-product” association we described above is a map V \to V^*. It takes in v \in V and spits out \varphi_v \in V^*.

Now here’s where things start to get canonical (interesting). First, V^* is itself a vector space. This is an easy exercise, and the details are not too important for us, but I’ll say the key: if you want to add two functions, you just add their (real number) outputs. In fact we can say more:

Theorem: V and V^* are isomorphic as vector spaces, and the map v \mapsto \varphi_v is the canonical isomorphism.

Confessions of a mathematician: we’re sweeping some complexity under the rug. When we upgraded our vector space to an inner product space, we fixed a specific (but arbitrary) inner product on V. For finite dimensional vector spaces it makes no difference, because every finite-dimensional \mathbb{R}-inner product space is isomorphic to \mathbb{R}^n with the usual inner product. But the theorem is devastatingly false for infinite-dimensional vector spaces. There are two reasons: (1) there are many (non-canonical) choices of inner products and (2) the mapping for any given inner product need not span V^*. Luckily we’re in finite dimensions so we can ignore all that. [Edit: see Emilio’s comments for a more detailed discussion of what’s being swept under the rug, and how we’re ignoring the categorical perspective when we say “natural” and “canonical.”]

Before we make sense of the isomorphism let’s talk more about V^*. First off, it’s not even entirely obvious that V^* is finite-dimensional. On one hand, if v_1, \dots, v_n is a basis of V then we can quickly prove that \varphi_{v_1}, \dots ,\varphi_{v_n} are linearly independent in V^*. Indeed, if they weren’t then there’d be some linear combination a_1 \varphi_{v_1} + \dots + a_n \varphi_{v_n} that is the zero function, meaning that for every vector w, the following is zero

\displaystyle a_1 \langle v_1, w \rangle + \dots + a_n \langle v_n, w \rangle = 0.

But since the inner product is linear in both arguments we get that \langle a_1 v_1 + \dots + a_n v_n , w \rangle = 0 for every w. And this can only happen when a_1v_1 + \dots + a_nv_n is the zero vector (prove this).

One consequence is that the linear map v \mapsto \varphi_v is injective. So we can think of V as “sitting inside” V^*. Now here’s a very slick way to show that the \varphi_{v_i} span all of V^*. First we can assume our basis v_1, \dots, v_n is actually an orthonormal basis with respect to our inner product (this is without loss of generality). Then we write any linear map f \in V^* as

f = f(v_1)\varphi_{v_1} + \dots + f(v_n) \varphi_{v_n}

To show these two are actually equal, it’s enough to show they agree on a basis for V. That is, if you plug in v_1 to the function on the left- and right-hand side of the above, you’ll get the same thing. The orthonormality of the basis makes it work, since all the irrelevant inner products are zero.

In case you missed it, that completes the proof that V and V^* are isomorphic. Now when I say that the isomorphism is “canonical,” I mean that if you’re willing to change the basis of V and V^*, then v \mapsto \varphi_v is the square identity matrix, i.e. the only isomorphism between any two finite vector spaces (up to a change of basis).

Tying in tensors

At this point we have a connection between single vectors v and linear maps V^* whose codomain has dimension 1. If we want to understand the outer product, we need a connection between pairs of vectors and matrices, i.e. \textup{Hom}(V,V). In other words, we’d like to find a canonical isomorphism between V \times V and \textup{Hom}(V,V). But already it’s not possible because the spaces have different dimensions. If \textup{dim}(V) = n then the former has dimension 2n and the latter has dimension n^2. So any “natural” relation between these spaces has to be a way to embed V \times V \subset \textup{Hom}(V,V) as a subspace via some injective map.

There are two gaping problems with this approach. First, the outer product is not linear as a map from V \times V \to \textup{Hom}(V,V). To see this, take any v,w \in V, pick any scalar \lambda \in \mathbb{R}. Scaling the pair (v,w) means scaling both components to (\lambda v, \lambda w), and so the outer product is the matrix (\lambda v)(\lambda w^T) = \lambda^2 vw^T.

The second problem is that the only way to make V \times V a subspace of \textup{Hom}(V,V) (up to a change of basis) is to map v,w to the first two rows of a matrix with zeros elsewhere. This is canonical but it doesn’t have the properties that the outer product promises us. Indeed, the outer product let’s us uniquely decompose a matrix as a “sum of rank 1 matrices,” but we don’t get a unique decomposition of a matrix as a sum of these two-row things. We also don’t even get a well-defined rank by decomposing into a sum of two-row matrices (you can get cancellation by staggering the sum). This injection is decisively useless.

It would seem like we’re stuck, until we think back to our association between V and V^*. If we take one of our two vectors, say v, and pair it with w, we can ask how (\varphi_v, w) could be turned into a linear map in \textup{Hom}(V,V). A few moments of guessing and one easily discovers the map

\displaystyle x \mapsto \varphi_v(x) w = \langle v,x \rangle w

In words, we’re scaling w by the inner product of x and v. In geometric terms, we project onto v and scale w by the signed length of that projection. Let’s call this map \beta_{v,w}(x), so that the association maps (v,w) \mapsto \beta_{v,w}. The thought process of “easily discovering” this is to think, “What can you do with a function \varphi_v and an input x? Plug it in. Then what can you do with the resulting number and a vector w? Scale w.”

If you look closely you’ll see we’ve just defined the outer product. This is because the outer product works by saying uv^T is a matrix, which acts on a vector x by doing (uv^T)x = u(v^T x). But the important thing is that, because V and V^* are canonically isomorphic, this is a mapping

\displaystyle V \times V = V^* \times V \to \textup{Hom}(V,V)

Now again, this mapping is not linear. In fact, it’s bilinear, and if there’s one thing we know about bilinear maps, it’s that tensors are their gatekeepers. If you recall our previous post on tensorphobia, this means that this bilinear map “factors through” the tensor product in a canonical way. So the true heart of this association (v,w) \mapsto \beta_{v,w} is a map B: V \otimes V \to \textup{Hom}(V,V) defined by

\displaystyle B(v \otimes w) = \beta_{v,w}

And now the punchline,

Theorem: B is an isomorphism of vector spaces.

Proof. If v_1, \dots, v_n is a basis for V then it’s enough to show that \beta_{v_i, v_j} forms a basis for \textup{Hom}(V,V). Since we already know \dim(\textup{Hom}(V,V)) = n^2 and there are n^2 of the \beta_{v_i, v_j}, all we need to do is show that the \beta‘s are linearly independent. For brevity let me remove the v‘s and call \beta_{i,j} =\beta_{v_i, v_j}.

Suppose they are not linearly independent. Then there is some choice of scalars a_{i,j} so that the linear combination below is the identically zero function

\displaystyle \sum_{i,j=1}^n a_{i,j}\beta_{i,j} = 0

In other words, if I plug in any v_i from my (orthonormal) basis, the result is zero. So let’s plug in v_1.

\displaystyle \begin{aligned} 0 &= \sum_{i,j=1}^n a_{i,j} \beta_{i,j}(v_1) \\ &= \sum_{i,j=1}^n a_{i,j} \langle v_i, v_1 \rangle v_j \\ &= \sum_{j=1}^n a_{1,j} \langle v_1, v_1 \rangle v_j \\ &= \sum_{j=1}^n a_{1,j} v_j \end{aligned}

The orthonormality makes all of the \langle v_i ,v_1 \rangle = 0 when i \neq 1, so we get a linear combination of the v_j being zero. Since the v_i form a basis, it must be that all the a_{1,j} = 0. The same thing happens when you plug in v_2 or any other v_k, and so all the a_{i,j} are zero, proving linear independence.

\square

This theorem immediately implies some deep facts, such as that every matrix can be uniquely decomposed as a sum of the \beta_{i,j}‘s. Moreover, facts like the \beta_{i,j}‘s being rank 1 are immediate: by definition the maps scale a single vector by some number. So of course the image will be one-dimensional. Finding a useful basis \{ v_i \} with which to decompose a matrix is where things get truly fascinating, and we’ll see that next time when we study the singular value decomposition.

In the mean time, this understanding generalizes nicely (via induction/recursion) to higher dimensional tensors. And you don’t need to talk about slices or sub-tensors or lose your sanity over n-tuples of indices.

Lastly, all of this duality stuff provides a “coordinate-free” way to think about the transpose of a linear map. We can think of the “transpose” operation as a linear map -^T : \textup{Hom}(V,W) \to \textup{Hom}(W^*,V^*) which (even in infinite dimensions) has the following definition. If f \in \textup{Hom}(V,W) then f^T is a linear map taking h \in W^* to f^T(h) \in V^*. The latter is a function in V^*, so we need to say what it does on inputs v \in V. The only definition that doesn’t introduce any type errors is (f^T(h))(v) = h(f(v)). A more compact way to say this is that f^T(h) = h \circ f.

Until next time!

Big Dimensions, and What You Can Do About It

Data is abundant, data is big, and big is a problem. Let me start with an example. Let’s say you have a list of movie titles and you want to learn their genre: romance, action, drama, etc. And maybe in this scenario IMDB doesn’t exist so you can’t scrape the answer. Well, the title alone is almost never enough information. One nice way to get more data is to do the following:

  1. Pick a large dictionary of words, say the most common 100,000 non stop-words in the English language.
  2. Crawl the web looking for documents that include the title of a film.
  3. For each film, record the counts of all other words appearing in those documents.
  4. Maybe remove instances of “movie” or “film,” etc.

After this process you have a length-100,000 vector of integers associated with each movie title. IMDB’s database has around 1.5 million listed movies, and if we have a 32-bit integer per vector entry, that’s 600 GB of data to get every movie.

One way to try to find genres is to cluster this (unlabeled) dataset of vectors, and then manually inspect the clusters and assign genres. With a really fast computer we could simply run an existing clustering algorithm on this dataset and be done. Of course, clustering 600 GB of data takes a long time, but there’s another problem. The geometric intuition that we use to design clustering algorithms degrades as the length of the vectors in the dataset grows. As a result, our algorithms perform poorly. This phenomenon is called the “curse of dimensionality” (“curse” isn’t a technical term), and we’ll return to the mathematical curiosities shortly.

A possible workaround is to try to come up with faster algorithms or be more patient. But a more interesting mathematical question is the following:

Is it possible to condense high-dimensional data into smaller dimensions and retain the important geometric properties of the data?

This goal is called dimension reduction. Indeed, all of the chatter on the internet is bound to encode redundant information, so for our movie title vectors it seems the answer should be “yes.” But the questions remain, how does one find a low-dimensional condensification? (Condensification isn’t a word, the right word is embedding, but embedding is overloaded so we’ll wait until we define it) And what mathematical guarantees can you prove about the resulting condensed data? After all, it stands to reason that different techniques preserve different aspects of the data. Only math will tell.

In this post we’ll explore this so-called “curse” of dimensionality, explain the formality of why it’s seen as a curse, and implement a wonderfully simple technique called “the random projection method” which preserves pairwise distances between points after the reduction. As usual, and all the code, data, and tests used in the making of this post are on Github.

Some curious issues, and the “curse”

We start by exploring the curse of dimensionality with experiments on synthetic data.

In two dimensions, take a circle centered at the origin with radius 1 and its bounding square.

circle.png

The circle fills up most of the area in the square, in fact it takes up exactly \pi out of 4 which is about 78%. In three dimensions we have a sphere and a cube, and the ratio of sphere volume to cube volume is a bit smaller, 4 \pi /3 out of a total of 8, which is just over 52%. What about in a thousand dimensions? Let’s try by simulation.

import random

def randUnitCube(n):
   return [(random.random() - 0.5)*2 for _ in range(n)]

def sphereCubeRatio(n, numSamples):
   randomSample = [randUnitCube(n) for _ in range(numSamples)]
   return sum(1 for x in randomSample if sum(a**2 for a in x) &lt;= 1) / numSamples 

The result is as we computed for small dimension,

 &gt;&gt;&gt; sphereCubeRatio(2,10000)
0.7857
&gt;&gt;&gt; sphereCubeRatio(3,10000)
0.5196

And much smaller for larger dimension

&gt;&gt;&gt; sphereCubeRatio(20,100000) # 100k samples
0.0
&gt;&gt;&gt; sphereCubeRatio(20,1000000) # 1M samples
0.0
&gt;&gt;&gt; sphereCubeRatio(20,2000000)
5e-07

Forget a thousand dimensions, for even twenty dimensions, a million samples wasn’t enough to register a single random point inside the unit sphere. This illustrates one concern, that when we’re sampling random points in the d-dimensional unit cube, we need at least 2^d samples to ensure we’re getting a even distribution from the whole space. In high dimensions, this face basically rules out a naive Monte Carlo approximation, where you sample random points to estimate the probability of an event too complicated to sample from directly. A machine learning viewpoint of the same problem is that in dimension d, if your machine learning algorithm requires a representative sample of the input space in order to make a useful inference, then you require 2^d samples to learn.

Luckily, we can answer our original question because there is a known formula for the volume of a sphere in any dimension. Rather than give the closed form formula, which involves the gamma function and is incredibly hard to parse, we’ll state the recursive form. Call V_i the volume of the unit sphere in dimension i. Then V_0 = 1 by convention, V_1 = 2 (it’s an interval), and V_n = \frac{2 \pi V_{n-2}}{n}. If you unpack this recursion you can see that the numerator looks like (2\pi)^{n/2} and the denominator looks like a factorial, except it skips every other number. So an even dimension would look like 2 \cdot 4 \cdot \dots \cdot n, and this grows larger than a fixed exponential. So in fact the total volume of the sphere vanishes as the dimension grows! (In addition to the ratio vanishing!)

def sphereVolume(n):
   values = [0] * (n+1)
   for i in range(n+1):
      if i == 0:
         values[i] = 1
      elif i == 1:
         values[i] = 2
      else:
         values[i] = 2*math.pi / i * values[i-2]

   return values[-1]

This should be counterintuitive. I think most people would guess, when asked about how the volume of the unit sphere changes as the dimension grows, that it stays the same or gets bigger.  But at a hundred dimensions, the volume is already getting too small to fit in a float.

&gt;&gt;&gt; sphereVolume(20)
0.025806891390014047
&gt;&gt;&gt; sphereVolume(100)
2.3682021018828297e-40
&gt;&gt;&gt; sphereVolume(1000)
0.0

The scary thing is not just that this value drops, but that it drops exponentially quickly. A consequence is that, if you’re trying to cluster data points by looking at points within a fixed distance r of one point, you have to carefully measure how big r needs to be to cover the same proportional volume as it would in low dimension.

Here’s a related issue. Say I take a bunch of points generated uniformly at random in the unit cube.

from itertools import combinations

def distancesRandomPoints(n, numSamples):
   randomSample = [randUnitCube(n) for _ in range(numSamples)]
   pairwiseDistances = [dist(x,y) for (x,y) in combinations(randomSample, 2)]
   return pairwiseDistances

In two dimensions, the histogram of distances between points looks like this

2d-distances.png

However, as the dimension grows the distribution of distances changes. It evolves like the following animation, in which each frame is an increase in dimension from 2 to 100.

distances-animation.gif

The shape of the distribution doesn’t appear to be changing all that much after the first few frames, but the center of the distribution tends to infinity (in fact, it grows like \sqrt{n}). The variance also appears to stay constant. This chart also becomes more variable as the dimension grows, again because we should be sampling exponentially many more points as the dimension grows (but we don’t). In other words, as the dimension grows the average distance grows and the tightness of the distribution stays the same. So at a thousand dimensions the average distance is about 26, tightly concentrated between 24 and 28. When the average is a thousand, the distribution is tight between 998 and 1002. If one were to normalize this data, it would appear that random points are all becoming equidistant from each other.

So in addition to the issues of runtime and sampling, the geometry of high-dimensional space looks different from what we expect. To get a better understanding of “big data,” we have to update our intuition from low-dimensional geometry with analysis and mathematical theorems that are much harder to visualize.

The Johnson-Lindenstrauss Lemma

Now we turn to proving dimension reduction is possible. There are a few methods one might first think of, such as look for suitable subsets of coordinates, or sums of subsets, but these would all appear to take a long time or they simply don’t work.

Instead, the key technique is to take a random linear subspace of a certain dimension, and project every data point onto that subspace. No searching required. The fact that this works is called the Johnson-Lindenstrauss Lemma. To set up some notation, we’ll call d(v,w) the usual distance between two points.

Lemma [Johnson-Lindenstrauss (1984)]: Given a set X of n points in \mathbb{R}^d, project the points in X to a randomly chosen subspace of dimension c. Call the projection \rho. For any \varepsilon > 0, if c is at least \Omega(\log(n) / \varepsilon^2), then with probability at least 1/2 the distances between points in X are preserved up to a factor of (1+\varepsilon). That is, with good probability every pair v,w \in X will satisfy

\displaystyle \| v-w \|^2 (1-\varepsilon) \leq \| \rho(v) - \rho(w) \|^2 \leq \| v-w \|^2 (1+\varepsilon)

Before we do the proof, which is quite short, it’s important to point out that the target dimension c does not depend on the original dimension! It only depends on the number of points in the dataset, and logarithmically so. That makes this lemma seem like pure magic, that you can take data in an arbitrarily high dimension and put it in a much smaller dimension.

On the other hand, if you include all of the hidden constants in the bound on the dimension, it’s not that impressive. If your data have a million dimensions and you want to preserve the distances up to 1% (\varepsilon = 0.01), the bound is bigger than a million! If you decrease the preservation \varepsilon to 10% (0.1), then you get down to about 12,000 dimensions, which is more reasonable. At 45% the bound drops to around 1,000 dimensions. Here’s a plot showing the theoretical bound on c in terms of \varepsilon for n fixed to a million.

boundplot

 

But keep in mind, this is just a theoretical bound for potentially misbehaving data. Later in this post we’ll see if the practical dimension can be reduced more than the theory allows. As we’ll see, an algorithm run on the projected data is still effective even if the projection goes well beyond the theoretical bound. Because the theorem is known to be tight in the worst case (see the notes at the end) this speaks more to the robustness of the typical algorithm than to the robustness of the projection method.

A second important note is that this technique does not necessarily avoid all the problems with the curse of dimensionality. We mentioned above that one potential problem is that “random points” are roughly equidistant in high dimensions. Johnson-Lindenstrauss actually preserves this problem because it preserves distances! As a consequence, you won’t see strictly better algorithm performance if you project (which we suggested is possible in the beginning of this post). But you will alleviate slow runtimes if the runtime depends exponentially on the dimension. Indeed, if you replace the dimension d with the logarithm of the number of points \log n, then 2^d becomes linear in n, and 2^{O(d)} becomes polynomial.

Proof of the J-L lemma

Let’s prove the lemma.

Proof. To start we make note that one can sample from the uniform distribution on dimension-c linear subspaces of \mathbb{R}^d by choosing the entries of a c \times d matrix A independently from a normal distribution with mean 0 and variance 1. Then, to project a vector x by this matrix (call the projection \rho), we can compute

\displaystyle \rho(x) = \frac{1}{\sqrt{c}}A x

Now fix \varepsilon > 0 and fix two points in the dataset x,y. We want an upper bound on the probability that the following is false

\displaystyle \| x-y \|^2 (1-\varepsilon) \leq \| \rho(x) - \rho(y) \|^2 \leq \| x-y \|^2 (1+\varepsilon)

Since that expression is a pain to work with, let’s rearrange it by calling u = x-y, and rearranging (using the linearity of the projection) to get the equivalent statement.

\left | \| \rho(u) \|^2 - \|u \|^2 \right | \leq \varepsilon \| u \|^2

And so we want a bound on the probability that this event does not occur, meaning the inequality switches directions.

Once we get such a bound (it will depend on c and \varepsilon) we need to ensure that this bound is true for every pair of points. The union bound allows us to do this, but it also requires that the probability of the bad thing happening tends to zero faster than 1/\binom{n}{2}. That’s where the \log(n) will come into the bound as stated in the theorem.

Continuing with our use of u for notation, define X to be the random variable \frac{c}{\| u \|^2} \| \rho(u) \|^2. By expanding the notation and using the linearity of expectation, you can show that the expected value of X is c, meaning that in expectation, distances are preserved. We are on the right track, and just need to show that the distribution of X, and thus the possible deviations in distances, is tightly concentrated around c. In full rigor, we will show

\displaystyle \Pr [X \geq (1+\varepsilon) c] < e^{-(\varepsilon^2 - \varepsilon^3) \frac{c}{4}}

Let A_i denote the i-th column of A. Define by X_i the quantity \langle A_i, u \rangle / \| u \|. This is a weighted average of the entries of A_i by the entries of u. But since we chose the entries of A from the normal distribution, and since a weighted average of normally distributed random variables is also normally distributed (has the same distribution), X_i is a N(0,1) random variable. Moreover, each column is independent. This allows us to decompose X as

X = \frac{k}{\| u \|^2} \| \rho(u) \|^2 = \frac{\| Au \|^2}{\| u \|^2}

Expanding further,

X = \sum_{i=1}^c \frac{\| A_i u \|^2}{\|u\|^2} = \sum_{i=1}^c X_i^2

Now the event X \leq (1+\varepsilon) c can be expressed in terms of the nonegative variable e^{\lambda X}, where 0 < \lambda < 1/2 is parameter, to get

\displaystyle \Pr[X \geq (1+\varepsilon) c] = \Pr[e^{\lambda X} \geq e^{(1+\varepsilon)c \lambda}]

This will become useful because the sum X = \sum_i X_i^2 will split into a product momentarily. First we apply Markov’s inequality, which says that for any nonnegative random variable Y, \Pr[Y \geq t] \leq \mathbb{E}[Y] / t. This lets us write

\displaystyle \Pr[e^{\lambda X} \geq e^{(1+\varepsilon) c \lambda}] \leq \frac{\mathbb{E}[e^{\lambda X}]}{e^{(1+\varepsilon) c \lambda}}

Now we can split up the exponent \lambda X into \sum_{i=1}^c \lambda X_i^2, and using the i.i.d.-ness of the X_i^2 we can rewrite the RHS of the inequality as

\left ( \frac{\mathbb{E}[e^{\lambda X_1^2}]}{e^{(1+\varepsilon)\lambda}} \right )^c

A similar statement using -\lambda is true for the (1-\varepsilon) part, namely that

\displaystyle \Pr[X \leq (1-\varepsilon)c] \leq \left ( \frac{\mathbb{E}[e^{-\lambda X_1^2}]}{e^{-(1-\varepsilon)\lambda}} \right )^c

The last thing that’s needed is to bound \mathbb{E}[e^{\lambda X_i^2}], but since X_i^2 \sim N(0,1), we can use the known density function for a normal distribution, and integrate to get the exact value \mathbb{E}[e^{\lambda X_1^2}] = \frac{1}{\sqrt{1-2\lambda}}. Including this in the bound gives us a closed-form bound in terms of \lambda, c, \varepsilon. Using standard calculus the optimal \lambda \in (0,1/2) is \lambda = \varepsilon / 2(1+\varepsilon). This gives

\displaystyle \Pr[X \geq (1+\varepsilon) c] \leq ((1+\varepsilon)e^{-\varepsilon})^{c/2}

Using the Taylor series expansion for e^x, one can show the bound 1+\varepsilon < e^{\varepsilon - (\varepsilon^2 - \varepsilon^3)/2}, which simplifies the final upper bound to e^{-(\varepsilon^2 - \varepsilon^3) c/4}.

Doing the same thing for the (1-\varepsilon) version gives an equivalent bound, and so the total bound is doubled, i.e. 2e^{-(\varepsilon^2 - \varepsilon^3) c/4}.

As we said at the beginning, applying the union bound means we need

\displaystyle 2e^{-(\varepsilon^2 - \varepsilon^3) c/4} < \frac{1}{\binom{n}{2}}

Solving this for c gives c \geq \frac{8 \log m}{\varepsilon^2 - \varepsilon^3}, as desired.

\square

Projecting in Practice

Let’s write a python program to actually perform the Johnson-Lindenstrauss dimension reduction scheme. This is sometimes called the Johnson-Lindenstrauss transform, or JLT.

First we define a random subspace by sampling an appropriately-sized matrix with normally distributed entries, and a function that performs the projection onto a given subspace (for testing).

import random
import math
import numpy

def randomSubspace(subspaceDimension, ambientDimension):
   return numpy.random.normal(0, 1, size=(subspaceDimension, ambientDimension))

def project(v, subspace):
   subspaceDimension = len(subspace)
   return (1 / math.sqrt(subspaceDimension)) * subspace.dot(v)

We have a function that computes the theoretical bound on the optimal dimension to reduce to.

def theoreticalBound(n, epsilon):
   return math.ceil(8*math.log(n) / (epsilon**2 - epsilon**3))

And then performing the JLT is simply matrix multiplication

def jlt(data, subspaceDimension):
   ambientDimension = len(data[0])
   A = randomSubspace(subspaceDimension, ambientDimension)
   return (1 / math.sqrt(subspaceDimension)) * A.dot(data.T).T

The high-dimensional dataset we’ll use comes from a data mining competition called KDD Cup 2001. The dataset we used deals with drug design, and the goal is to determine whether an organic compound binds to something called thrombin. Thrombin has something to do with blood clotting, and I won’t pretend I’m an expert. The dataset, however, has over a hundred thousand features for about 2,000 compounds. Here are a few approximate target dimensions we can hope for as epsilon varies.

&gt;&gt;&gt; [((1/x),theoreticalBound(n=2000, epsilon=1/x))
       for x in [2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20]]
[('0.50', 487), ('0.33', 821), ('0.25', 1298), ('0.20', 1901),
 ('0.17', 2627), ('0.14', 3477), ('0.12', 4448), ('0.11', 5542),
 ('0.10', 6757), ('0.07', 14659), ('0.05', 25604)]

Going down from a hundred thousand dimensions to a few thousand is by any measure decreases the size of the dataset by about 95%. We can also observe how the distribution of overall distances varies as the size of the subspace we project to varies.

The animation proceeds from 5000 dimensions down to 2 (when the plot is at its bulkiest closer to zero).

The animation proceeds from 5000 dimensions down to 2 (when the plot is at its bulkiest closer to zero).

The last three frames are for 10, 5, and 2 dimensions respectively. As you can see the histogram starts to beef up around zero. To be honest I was expecting something a bit more dramatic like a uniform-ish distribution. Of course, the distribution of distances is not all that matters. Another concern is the worst case change in distances between any two points before and after the projection. We can see that indeed when we project to the dimension specified in the theorem, that the distances are within the prescribed bounds.

def checkTheorem(oldData, newData, epsilon):
   numBadPoints = 0

   for (x,y), (x2,y2) in zip(combinations(oldData, 2), combinations(newData, 2)):
      oldNorm = numpy.linalg.norm(x2-y2)**2
      newNorm = numpy.linalg.norm(x-y)**2

      if newNorm == 0 or oldNorm == 0:
         continue

      if abs(oldNorm / newNorm - 1) &gt; epsilon:
         numBadPoints += 1

   return numBadPoints

if __name__ == &quot;__main__&quot;
   from data import thrombin
   train, labels = thrombin.load() 

   numPoints = len(train)
   epsilon = 0.2
   subspaceDim = theoreticalBound(numPoints, epsilon)
   ambientDim = len(train[0])
   newData = jlt(train, subspaceDim)

   print(checkTheorem(train, newData, epsilon))

This program prints zero every time I try running it, which is the poor man’s way of saying it works “with high probability.” We can also plot statistics about the number of pairs of data points that are distorted by more than \varepsilon as the subspace dimension shrinks. We ran this on the following set of subspace dimensions with \varepsilon = 0.1 and took average/standard deviation over twenty trials:

   dims = [1000, 750, 500, 250, 100, 75, 50, 25, 10, 5, 2]

The result is the following chart, whose x-axis is the dimension projected to (so the left hand is the most extreme projection to 2, 5, 10 dimensions), the y-axis is the number of distorted pairs, and the error bars represent a single standard deviation away from the mean.

thrombin-worst-case

This chart provides good news about this dataset because the standard deviations are low. It tells us something that mathematicians often ignore: the predictability of the tradeoff that occurs once you go past the theoretically perfect bound. In this case, the standard deviations tell us that it’s highly predictable. Moreover, since this tradeoff curve measures pairs of points, we might conjecture that the distortion is localized around a single set of points that got significantly “rattled” by the projection. This would be an interesting exercise to explore.

Now all of these charts are really playing with the JLT and confirming the correctness of our code (and hopefully our intuition). The real question is: how well does a machine learning algorithm perform on the original data when compared to the projected data? If the algorithm only “depends” on the pairwise distances between the points, then we should expect nearly identical accuracy in the unprojected and projected versions of the data. To show this we’ll use an easy learning algorithm, the k-nearest-neighbors clustering method. The problem, however, is that there are very few positive examples in this particular dataset. So looking for the majority label of the nearest k neighbors for any k > 2 unilaterally results in the “all negative” classifier, which has 97% accuracy. This happens before and after projecting.

To compensate for this, we modify k-nearest-neighbors slightly by having the label of a predicted point be 1 if any label among its nearest neighbors is 1. So it’s not a majority vote, but rather a logical OR of the labels of nearby neighbors. Our point in this post is not to solve the problem well, but rather to show how an algorithm (even a not-so-good one) can degrade as one projects the data into smaller and smaller dimensions. Here is the code.

def nearestNeighborsAccuracy(data, labels, k=10):
   from sklearn.neighbors import NearestNeighbors
   trainData, trainLabels, testData, testLabels = randomSplit(data, labels) # cross validation
   model = NearestNeighbors(n_neighbors=k).fit(trainData)
   distances, indices = model.kneighbors(testData)
   predictedLabels = []

   for x in indices:
      xLabels = [trainLabels[i] for i in x[1:]]
      predictedLabel = max(xLabels)
      predictedLabels.append(predictedLabel)

   totalAccuracy = sum(x == y for (x,y) in zip(testLabels, predictedLabels)) / len(testLabels)
   falsePositive = (sum(x == 0 and y == 1 for (x,y) in zip(testLabels, predictedLabels)) /
      sum(x == 0 for x in testLabels))
   falseNegative = (sum(x == 1 and y == 0 for (x,y) in zip(testLabels, predictedLabels)) /
      sum(x == 1 for x in testLabels))

   return totalAccuracy, falsePositive, falseNegative

And here is the accuracy of this modified k-nearest-neighbors algorithm run on the thrombin dataset. The horizontal line represents the accuracy of the produced classifier on the unmodified data set. The x-axis represents the dimension projected to (left-hand side is the lowest), and the y-axis represents the accuracy. The mean accuracy over fifty trials was plotted, with error bars representing one standard deviation. The complete code to reproduce the plot is in the Github repository [link link link].

thrombin-knn-accuracy

Likewise, we plot the proportion of false positive and false negatives for the output classifier. Note that a “positive” label made up only about 2% of the total data set. First the false positives

thrombin-knn-fp

Then the false negatives

thrombin-knn-fn

As we can see from these three charts, things don’t really change that much (for this dataset) even when we project down to around 200-300 dimensions. Note that for these parameters the “correct” theoretical choice for dimension was on the order of 5,000 dimensions, so this is a 95% savings from the naive approach, and 99.75% space savings from the original data. Not too shabby.

Notes

The \Omega(\log(n)) worst-case dimension bound is asymptotically tight, though there is some small gap in the literature that depends on \varepsilon. This result is due to Noga Alon, the very last result (Section 9) of this paper.

We did dimension reduction with respect to preserving the Euclidean distance between points. One might naturally wonder if you can achieve the same dimension reduction with a different metric, say the taxicab metric or a p-norm. In fact, you cannot achieve anything close to logarithmic dimension reduction for the taxicab (l_1) metric. This result is due to Brinkman-Charikar in 2004.

The code we used to compute the JLT is not particularly efficient. There are much more efficient methods. One of them, borrowing its namesake from the Fast Fourier Transform, is called the Fast Johnson-Lindenstrauss Transform. The technique is due to Ailon-Chazelle from 2009, and it involves something called “preconditioning a sparse projection matrix with a randomized Fourier transform.” I don’t know precisely what that means, but it would be neat to dive into that in a future post.

The central focus in this post was whether the JLT preserves distances between points, but one might be curious as to whether the points themselves are well approximated. The answer is an enthusiastic no. If the data were images, the projected points would look nothing like the original images. However, it appears the degradation tradeoff is measurable (by some accounts perhaps linear), and there appears to be some work (also this by the same author) when restricting to sparse vectors (like word-association vectors).

Note that the JLT is not the only method for dimensionality reduction. We previously saw principal component analysis (applied to face recognition), and in the future we will cover a related technique called the Singular Value Decomposition. It is worth noting that another common technique specific to nearest-neighbor is called “locality-sensitive hashing.” Here the goal is to project the points in such a way that “similar” points land very close to each other. Say, if you were to discretize the plane into bins, these bins would form the hash values and you’d want to maximize the probability that two points with the same label land in the same bin. Then you can do things like nearest-neighbors by comparing bins.

Another interesting note, if your data is linearly separable (like the examples we saw in our age-old post on Perceptrons), then you can use the JLT to make finding a linear separator easier. First project the data onto the dimension given in the theorem. With high probability the points will still be linearly separable. And then you can use a perceptron-type algorithm in the smaller dimension. If you want to find out which side a new point is on, you project and compare with the separator in the smaller dimension.

Beyond its interest for practical dimensionality reduction, the JLT has had many other interesting theoretical consequences. More generally, the idea of “randomly projecting” your data onto some small dimensional space has allowed mathematicians to get some of the best-known results on many optimization and learning problems, perhaps the most famous of which is called MAX-CUT; the result is by Goemans-Williamson and it led to a mathematical constant being named after them, \alpha_{GW} =.878567 \dots. If you’re interested in more about the theory, Santosh Vempala wrote a wonderful (and short!) treatise dedicated to this topic.

randomprojectionbook