The Inner Product as a Decision Rule

The standard inner product of two vectors has some nice geometric properties. Given two vectors $x, y \in \mathbb{R}^n$, where by $x_i$ I mean the $i$-th coordinate of $x$, the standard inner product (which I will interchangeably call the dot product) is defined by the formula

$\displaystyle \langle x, y \rangle = x_1 y_1 + \dots + x_n y_n$

This formula, simple as it is, produces a lot of interesting geometry. An important such property, one which is discussed in machine learning circles more than pure math, is that it is a very convenient decision rule.

In particular, say we’re in the Euclidean plane, and we have a line $L$ passing through the origin, with $w$ being a unit vector perpendicular to $L$ (“the normal” to the line).

If you take any vector $x$, then the dot product $\langle x, w \rangle$ is positive if $x$ is on the same side of $L$ as $w$, and negative otherwise. The dot product is zero if and only if $x$ is exactly on the line $L$, including when $x$ is the zero vector.

Left: the dot product of $w$ and $x$ is positive, meaning they are on the same side of $w$. Right: The dot product is negative, and they are on opposite sides.

Here is an interactive demonstration of this property. Click the image below to go to the demo, and you can drag the vector arrowheads and see the decision rule change.

Click above to go to the demo

The code for this demo is available in a github repository.

It’s always curious, at first, that multiplying and summing produces such geometry. Why should this seemingly trivial arithmetic do anything useful at all?

The core fact that makes it work, however, is that the dot product tells you how one vector projects onto another. When I say “projecting” a vector $x$ onto another vector $w$, I mean you take only the components of $x$ that point in the direction of $w$. The demo shows what the result looks like using the red (or green) vector.

In two dimensions this is easy to see, as you can draw the triangle which has $x$ as the hypotenuse, with $w$ spanning one of the two legs of the triangle as follows:

If we call $a$ the (vector) leg of the triangle parallel to $w$, while $b$ is the dotted line (as a vector, parallel to $L$), then as vectors $x = a + b$. The projection of $x$ onto $w$ is just $a$.

Another way to think of this is that the projection is $x$, modified by removing any part of $x$ that is perpendicular to $w$. Using some colorful language: you put your hands on either side of $x$ and $y$, and then you squish $x$ onto $y$ along the line perpendicular to $y$ (i.e., along $b$).

And if $y$ is a unit vector, then the length of $a$—that is, the length of the projection of $x$ onto $y$—is exactly the inner product product $\langle x, y \rangle$.

Moreover, if the angle between $x$ and $y$ is larger than 90 degrees, the projected vector will point in the opposite direction of $y$, so it’s really a “signed” length.

Left: the projection points in the same direction as $w$. Right: the projection points in the opposite direction.

And this is precisely why the decision rule works. This 90-degree boundary is the line perpendicular to $y$.

More technically said: Let $x, y \in \mathbb{R}^n$ be two vectors, and $\langle x,y \rangle$ their dot product. Define by $\| y \|$ the length of $y$, specifically $\sqrt{\langle y, y \rangle}$. Define by $\text{proj}_{y}(x)$ by first letting $y' = \frac{y}{\| y \|}$, and then let $\text{proj}_{y}(x) = \langle x,y' \rangle y'$. In words, you scale $y$ to a unit vector $y'$, use the result to compute the inner product, and then scale $y$ so that it’s length is $\langle x, y' \rangle$. Then

Theorem: Geometrically, $\text{proj}_y(x)$ is the projection of $x$ onto the line spanned by $y$.

This theorem is true for any $n$-dimensional vector space, since if you have two vectors you can simply apply the reasoning for 2-dimensions to the 2-dimensional plane containing $x$ and $y$. In that case, the decision boundary for a positive/negative output is the entire $n-1$ dimensional hyperplane perpendicular to $y$ (the projected vector).

In fact, the usual formula for the angle between two vectors, i.e. the formula $\langle x, y \rangle = \|x \| \cdot \| y \| \cos \theta$, is a restatement of the projection theorem in terms of trigonometry. The $\langle x, y' \rangle$ part of the projection formula (how much you scale the output) is equal to $\| x \| \cos \theta$. At the end of this post we have a proof of the cosine-angle formula above.

Part of why this decision rule property is so important is that this is a linear function, and linear functions can be optimized relatively easily. When I say that, I specifically mean that there are many known algorithms for optimizing linear functions, which don’t have obscene runtime or space requirements. This is a big reason why mathematicians and statisticians start the mathematical modeling process with linear functions. They’re inherently simpler.

In fact, there are many techniques in machine learning—a prominent one is the so-called Kernel Trick—that exist solely to take data that is not inherently linear in nature (cannot be fruitfully analyzed by linear methods) and transform it into a dataset that is. Using the Kernel Trick as an example to foreshadow some future posts on Support Vector Machines, the idea is to take data which cannot be separated by a line, and transform it (usually by adding new coordinates) so that it can. Then the decision rule, computed in the larger space, is just a dot product. Irene Papakonstantinou neatly demonstrates this with paper folding and scissors. The tradeoff is that the size of the ambient space increases, and it might increase so much that it makes computation intractable. Luckily, the Kernel Trick avoids this by remembering where the data came from, so that one can take advantage of the smaller space to compute what would be the inner product in the larger space.

Next time we’ll see how this decision rule shows up in an optimization problem: finding the “best” hyperplane that separates an input set of red and blue points into monochromatic regions (provided that is possible). Finding this separator is core subroutine of the Support Vector Machine technique, and therein lie interesting algorithms. After we see the core SVM algorithm, we’ll see how the Kernel Trick fits into the method to allow nonlinear decision boundaries.

Proof of the cosine angle formula

Theorem: The inner product $\langle v, w \rangle$ is equal to $\| v \| \| w \| \cos(\theta)$, where $\theta$ is the angle between the two vectors.

Note that this angle is computed in the 2-dimensional subspace spanned by $v, w$, viewed as a typical flat plane, and this is a 2-dimensional plane regardless of the dimension of $v, w$.

Proof. If either $v$ or $w$ is zero, then both sides of the equation are zero and the theorem is trivial, so we may assume both are nonzero. Label a triangle with sides $v,w$ and the third side $v-w$. Now the length of each side is $\| v \|, \| w\|,$ and $\| v-w \|$, respectively. Assume for the moment that $\theta$ is not 0 or 180 degrees, so that this triangle is not degenerate.

The law of cosines allows us to write

$\displaystyle \| v - w \|^2 = \| v \|^2 + \| w \|^2 - 2 \| v \| \| w \| \cos(\theta)$

Moreover, The left hand side is the inner product of $v-w$ with itself, i.e. $\| v - w \|^2 = \langle v-w , v-w \rangle$. We’ll expand $\langle v-w, v-w \rangle$ using two facts. The first is trivial from the formula, that inner product is symmetric: $\langle v,w \rangle = \langle w, v \rangle$. Second is that the inner product is linear in each input. In particular for the first input: $\langle x + y, z \rangle = \langle x, z \rangle + \langle y, z \rangle$ and $\langle cx, z \rangle = c \langle x, z \rangle$. The same holds for the second input by symmetry of the two inputs. Hence we can split up $\langle v-w, v-w \rangle$ as follows.

\displaystyle \begin{aligned} \langle v-w, v-w \rangle &= \langle v, v-w \rangle - \langle w, v-w \rangle \\ &= \langle v, v \rangle - \langle v, w \rangle - \langle w, v \rangle + \langle w, w \rangle \\ &= \| v \|^2 - 2 \langle v, w \rangle + \| w \|^2 \\ \end{aligned}

Combining our two offset equations, we can subtract $\| v \|^2 + \| w \|^2$ from each side and get

$\displaystyle -2 \|v \| \|w \| \cos(\theta) = -2 \langle v, w \rangle,$

Which, after dividing by $-2$, proves the theorem if $\theta \not \in \{0, 180 \}$.

Now if $\theta = 0$ or 180 degrees, the vectors are parallel, so we can write one as a scalar multiple of the other. Say $w = cv$ for $c \in \mathbb{R}$. In that case, $\langle v, cv \rangle = c \| v \| \| v \|$. Now $\| w \| = | c | \| v \|$, since a norm is a length and is hence non-negative (but $c$ can be negative). Indeed, if $v, w$ are parallel but pointing in opposite directions, then $c < 0$, so $\cos(\theta) = -1$, and $c \| v \| = - \| w \|$. Otherwise $c > 0$ and $\cos(\theta) = 1$. This allows us to write $c \| v \| \| v \| = \| w \| \| v \| \cos(\theta)$, and this completes the final case of the theorem.

$\square$

The Reasonable Effectiveness of the Multiplicative Weights Update Algorithm

Christos Papadimitriou, who studies multiplicative weights in the context of biology.

Hard to believe

Sanjeev Arora and his coauthors consider it “a basic tool [that should be] taught to all algorithms students together with divide-and-conquer, dynamic programming, and random sampling.” Christos Papadimitriou calls it “so hard to believe that it has been discovered five times and forgotten.” It has formed the basis of algorithms in machine learning, optimization, game theory, economics, biology, and more.

What mystical algorithm has such broad applications? Now that computer scientists have studied it in generality, it’s known as the Multiplicative Weights Update Algorithm (MWUA). Procedurally, the algorithm is simple. I can even describe the core idea in six lines of pseudocode. You start with a collection of $n$ objects, and each object has a weight.

Set all the object weights to be 1.
For some large number of rounds:
Pick an object at random proportionally to the weights
Some event happens
Increase the weight of the chosen object if it does well in the event
Otherwise decrease the weight

The name “multiplicative weights” comes from how we implement the last step: if the weight of the chosen object at step $t$ is $w_t$ before the event, and $G$ represents how well the object did in the event, then we’ll update the weight according to the rule:

$\displaystyle w_{t+1} = w_t (1 + G)$

Think of this as increasing the weight by a small multiple of the object’s performance on a given round.

Here is a simple example of how it might be used. You have some money you want to invest, and you have a bunch of financial experts who are telling you what to invest in every day. So each day you pick an expert, and you follow their advice, and you either make a thousand dollars, or you lose a thousand dollars, or something in between. Then you repeat, and your goal is to figure out which expert is the most reliable.

This is how we use multiplicative weights: if we number the experts $1, \dots, N$, we give each expert a weight $w_i$ which starts at 1. Then, each day we pick an expert at random (where experts with larger weights are more likely to be picked) and at the end of the day we have some gain or loss $G$. Then we update the weight of the chosen expert by multiplying it by $(1 + G / 1000)$. Sometimes you have enough information to update the weights of experts you didn’t choose, too. The theoretical guarantees of the algorithm say we’ll find the best expert quickly (“quickly” will be concrete later).

In fact, let’s play a game where you, dear reader, get to decide the rewards for each expert and each day. I programmed the multiplicative weights algorithm to react according to your choices. Click the image below to go to the demo.

This core mechanism of updating weights can be interpreted in many ways, and that’s part of the reason it has sprouted up all over mathematics and computer science. Just a few examples of where this has led:

1. In game theory, weights are the “belief” of a player about the strategy of an opponent. The most famous algorithm to use this is called Fictitious Play, and others include EXP3 for minimizing regret in the so-called “adversarial bandit learning” problem.
2. In machine learning, weights are the difficulty of a specific training example, so that higher weights mean the learning algorithm has to “try harder” to accommodate that example. The first result I’m aware of for this is the Perceptron (and similar Winnow) algorithm for learning hyperplane separators. The most famous is the AdaBoost algorithm.
3. Analogously, in optimization, the weights are the difficulty of a specific constraint, and this technique can be used to approximately solve linear and semidefinite programs. The approximation is because MWUA only provides a solution with some error.
4. In mathematical biology, the weights represent the fitness of individual alleles, and filtering reproductive success based on this and updating weights for successful organisms produces a mechanism very much like evolution. With modifications, it also provides a mechanism through which to understand sex in the context of evolutionary biology.
5. The TCP protocol, which basically defined the internet, uses additive and multiplicative weight updates (which are very similar in the analysis) to manage congestion.
6. You can get easy $\log(n)$-approximation algorithms for many NP-hard problems, such as set cover.

Additional, more technical examples can be found in this survey of Arora et al.

In the rest of this post, we’ll implement a generic Multiplicative Weights Update Algorithm, we’ll prove it’s main theoretical guarantees, and we’ll implement a linear program solver as an example of its applicability. As usual, all of the code used in the making of this post is available in a Github repository.

The generic MWUA algorithm

Let’s start by writing down pseudocode and an implementation for the MWUA algorithm in full generality.

In general we have some set $X$ of objects and some set $Y$ of “event outcomes” which can be completely independent. If these sets are finite, we can write down a table $M$ whose rows are objects, whose columns are outcomes, and whose $i,j$ entry $M(i,j)$ is the reward produced by object $x_i$ when the outcome is $y_j$. We will also write this as $M(x, y)$ for object $x$ and outcome $y$. The only assumption we’ll make on the rewards is that the values $M(x, y)$ are bounded by some small constant $B$ (by small I mean $B$ should not require exponentially many bits to write down as compared to the size of $X$). In symbols, $M(x,y) \in [0,B]$. There are minor modifications you can make to the algorithm if you want negative rewards, but for simplicity we will leave that out. Note the table $M$ just exists for analysis, and the algorithm does not know its values. Moreover, while the values in $M$ are static, the choice of outcome $y$ for a given round may be nondeterministic.

The MWUA algorithm randomly chooses an object $x \in X$ in every round, observing the outcome $y \in Y$, and collecting the reward $M(x,y)$ (or losing it as a penalty). The guarantee of the MWUA theorem is that the expected sum of rewards/penalties of MWUA is not much worse than if one had picked the best object (in hindsight) every single round.

Let’s describe the algorithm in notation first and build up pseudocode as we go. The input to the algorithm is the set of objects, a subroutine that observes an outcome, a black-box reward function, a learning rate parameter, and a number of rounds.

def MWUA(objects, observeOutcome, reward, learningRate, numRounds):
...


We define for object $x$ a nonnegative number $w_x$ we call a “weight.” The weights will change over time so we’ll also sub-script a weight with a round number $t$, i.e. $w_{x,t}$ is the weight of object $x$ in round $t$. Initially, all the weights are $1$. Then MWUA continues in rounds. We start each round by drawing an example randomly with probability proportional to the weights. Then we observe the outcome for that round and the reward for that round.

# draw: [float] -> int
# pick an index from the given list of floats proportionally
# to the size of the entry (i.e. normalize to a probability
# distribution and draw according to the probabilities).
def draw(weights):
choice = random.uniform(0, sum(weights))
choiceIndex = 0

for weight in weights:
choice -= weight
if choice <= 0:
return choiceIndex

choiceIndex += 1

# MWUA: the multiplicative weights update algorithm
def MWUA(objects, observeOutcome, reward, learningRate numRounds):
weights = [1] * len(objects)
for t in numRounds:
chosenObjectIndex = draw(weights)
chosenObject = objects[chosenObjectIndex]

outcome = observeOutcome(t, weights, chosenObject)
thisRoundReward = reward(chosenObject, outcome)

...


Sampling objects in this way is the same as associating a distribution $D_t$ to each round, where if $S_t = \sum_{x \in X} w_{x,t}$ then the probability of drawing $x$, which we denote $D_t(x)$, is $w_{x,t} / S_t$. We don’t need to keep track of this distribution in the actual run of the algorithm, but it will help us with the mathematical analysis.

Next comes the weight update step. Let’s call our learning rate variable parameter $\varepsilon$. In round $t$ say we have object $x_t$ and outcome $y_t$, then the reward is $M(x_t, y_t)$. We update the weight of the chosen object $x_t$ according to the formula:

$\displaystyle w_{x_t, t} = w_{x_t} (1 + \varepsilon M(x_t, y_t) / B)$

In the more general event that you have rewards for all objects (if not, the reward-producing function can output zero), you would perform this weight update on all objects $x \in X$. This turns into the following Python snippet, where we hide the division by $B$ into the choice of learning rate:

# MWUA: the multiplicative weights update algorithm
def MWUA(objects, observeOutcome, reward, learningRate, numRounds):
weights = [1] * len(objects)
for t in numRounds:
chosenObjectIndex = draw(weights)
chosenObject = objects[chosenObjectIndex]

outcome = observeOutcome(t, weights, chosenObject)
thisRoundReward = reward(chosenObject, outcome)

for i in range(len(weights)):
weights[i] *= (1 + learningRate * reward(objects[i], outcome))


One of the amazing things about this algorithm is that the outcomes and rewards could be chosen adaptively by an adversary who knows everything about the MWUA algorithm (except which random numbers the algorithm generates to make its choices). This means that the rewards in round $t$ can depend on the weights in that same round! We will exploit this when we solve linear programs later in this post.

But even in such an oppressive, exploitative environment, MWUA persists and achieves its guarantee. And now we can state that guarantee.

Theorem (from Arora et al): The cumulative reward of the MWUA algorithm is, up to constant multiplicative factors, at least the cumulative reward of the best object minus $\log(n)$, where $n$ is the number of objects. (Exact formula at the end of the proof)

The core of the proof, which we’ll state as a lemma, uses one of the most elegant proof techniques in all of mathematics. It’s the idea of constructing a potential function, and tracking the change in that potential function over time. Such a proof usually has the mysterious script:

1. Define potential function, in our case $S_t$.
2. State what seems like trivial facts about the potential function to write $S_{t+1}$ in terms of $S_t$, and hence get general information about $S_T$ for some large $T$.
3. Theorem is proved.
4. Wait, what?

Clearly, coming up with a useful potential function is a difficult and prized skill.

In this proof our potential function is the sum of the weights of the objects in a given round, $S_t = \sum_{x \in X} w_{x, t}$. Now the lemma.

Lemma: Let $B$ be the bound on the size of the rewards, and $0 < \varepsilon < 1/2$ a learning parameter. Recall that $D_t(x)$ is the probability that MWUA draws object $x$ in round $t$. Write the expected reward for MWUA for round $t$ as the following (using only the definition of expected value):

$\displaystyle R_t = \sum_{x \in X} D_t(x) M(x, y_t)$

Then the claim of the lemma is:

$\displaystyle S_{t+1} \leq S_t e^{\varepsilon R_t / B}$

Proof. Expand $S_{t+1} = \sum_{x \in X} w_{x, t+1}$ using the definition of the MWUA update:

$\displaystyle \sum_{x \in X} w_{x, t+1} = \sum_{x \in X} w_{x, t}(1 + \varepsilon M(x, y_t) / B)$

Now distribute $w_{x, t}$ and split into two sums:

$\displaystyle \dots = \sum_{x \in X} w_{x, t} + \frac{\varepsilon}{B} \sum_{x \in X} w_{x,t} M(x, y_t)$

Using the fact that $D_t(x) = \frac{w_{x,t}}{S_t}$, we can replace $w_{x,t}$ with $D_t(x) S_t$, which allows us to get $R_t$

\displaystyle \begin{aligned} \dots &= S_t + \frac{\varepsilon S_t}{B} \sum_{x \in X} D_t(x) M(x, y_t) \\ &= S_t \left ( 1 + \frac{\varepsilon R_t}{B} \right ) \end{aligned}

And then using the fact that $(1 + x) \leq e^x$ (Taylor series), we can bound the last expression by $S_te^{\varepsilon R_t / B}$, as desired.

$\square$

Now using the lemma, we can get a hold on $S_T$ for a large $T$, namely that

$\displaystyle S_T \leq S_1 e^{\varepsilon \sum_{t=1}^T R_t / B}$

If $|X| = n$ then $S_1=n$, simplifying the above. Moreover, the sum of the weights in round $T$ is certainly greater than any single weight, so that for every fixed object $x \in X$,

$\displaystyle S_T \geq w_{x,T} \leq (1 + \varepsilon)^{\sum_t M(x, y_t) / B}$

Squeezing $S_t$ between these two inequalities and taking logarithms (to simplify the exponents) gives

$\displaystyle \left ( \sum_t M(x, y_t) / B \right ) \log(1+\varepsilon) \leq \log n + \frac{\varepsilon}{B} \sum_t R_t$

Multiply through by $B$, divide by $\varepsilon$, rearrange, and use the fact that when $0 < \varepsilon < 1/2$ we have $\log(1 + \varepsilon) \geq \varepsilon - \varepsilon^2$ (Taylor series) to get

$\displaystyle \sum_t R_t \geq \left [ \sum_t M(x, y_t) \right ] (1-\varepsilon) - \frac{B \log n}{\varepsilon}$

The bracketed term is the payoff of object $x$, and MWUA’s payoff is at least a fraction of that minus the logarithmic term. The bound applies to any object $x \in X$, and hence to the best one. This proves the theorem.

$\square$

Briefly discussing the bound itself, we see that the smaller the learning rate is, the closer you eventually get to the best object, but by contrast the more the subtracted quantity $B \log(n) / \varepsilon$ hurts you. If your target is an absolute error bound against the best performing object on average, you can do more algebra to determine how many rounds you need in terms of a fixed $\delta$. The answer is roughly: let $\varepsilon = O(\delta / B)$ and pick $T = O(B^2 \log(n) / \delta^2)$. See this survey for more.

MWUA for linear programs

Now we’ll approximately solve a linear program using MWUA. Recall that a linear program is an optimization problem whose goal is to minimize (or maximize) a linear function of many variables. The objective to minimize is usually given as a dot product $c \cdot x$, where $c$ is a fixed vector and $x = (x_1, x_2, \dots, x_n)$ is a vector of non-negative variables the algorithm gets to choose. The choices for $x$ are also constrained by a set of $m$ linear inequalities, $A_i \cdot x \geq b_i$, where $A_i$ is a fixed vector and $b_i$ is a scalar for $i = 1, \dots, m$. This is usually summarized by putting all the $A_i$ in a matrix, $b_i$ in a vector, as

$x_{\textup{OPT}} = \textup{argmin}_x \{ c \cdot x \mid Ax \geq b, x \geq 0 \}$

We can further simplify the constraints by assuming we know the optimal value $Z = c \cdot x_{\textup{OPT}}$ in advance, by doing a binary search (more on this later). So, if we ignore the hard constraint $Ax \geq b$, the “easy feasible region” of possible $x$‘s includes $\{ x \mid x \geq 0, c \cdot x = Z \}$.

In order to fit linear programming into the MWUA framework we have to define two things.

1. The objects: the set of linear inequalities $A_i \cdot x \geq b_i$.
2. The rewards: the error of a constraint for a special input vector $x_t$.

Number 2 is curious (why would we give a reward for error?) but it’s crucial and we’ll discuss it momentarily.

The special input $x_t$ depends on the weights in round $t$ (which is allowed, recall). Specifically, if the weights are $w = (w_1, \dots, w_m)$, we ask for a vector $x_t$ in our “easy feasible region” which satisfies

$\displaystyle (A^T w) \cdot x_t \geq w \cdot b$

For this post we call the implementation of procuring such a vector the “oracle,” since it can be seen as the black-box problem of, given a vector $\alpha$ and a scalar $\beta$ and a convex region $R$, finding a vector $x \in R$ satisfying $\alpha \cdot x \geq \beta$. This allows one to solve more complex optimization problems with the same technique, swapping in a new oracle as needed. Our choice of inputs, $\alpha = A^T w, \beta = w \cdot b$, are particular to the linear programming formulation.

Two remarks on this choice of inputs. First, the vector $A^T w$ is a weighted average of the constraints in $A$, and $w \cdot b$ is a weighted average of the thresholds. So this this inequality is a “weighted average” inequality (specifically, a convex combination, since the weights are nonnegative). In particular, if no such $x$ exists, then the original linear program has no solution. Indeed, given a solution $x^*$ to the original linear program, each constraint, say $A_1 x^*_1 \geq b_1$, is unaffected by left-multiplication by $w_1$.

Second, and more important to the conceptual understanding of this algorithm, the choice of rewards and the multiplicative updates ensure that easier constraints show up less prominently in the inequality by having smaller weights. That is, if we end up overly satisfying a constraint, we penalize that object for future rounds so we don’t waste our effort on it. The byproduct of MWUA—the weights—identify the hardest constraints to satisfy, and so in each round we can put a proportionate amount of effort into solving (one of) the hard constraints. This is why it makes sense to reward error; the error is a signal for where to improve, and by over-representing the hard constraints, we force MWUA’s attention on them.

At the end, our final output is an average of the $x_t$ produced in each round, i.e. $x^* = \frac{1}{T}\sum_t x_t$. This vector satisfies all the constraints to a roughly equal degree. We will skip the proof that this vector does what we want, but see these notes for a simple proof. We’ll spend the rest of this post implementing the scheme outlined above.

Implementing the oracle

Fix the convex region $R = \{ c \cdot x = Z, x \geq 0 \}$ for a known optimal value $Z$. Define $\textup{oracle}(\alpha, \beta)$ as the problem of finding an $x \in R$ such that $\alpha \cdot x \geq \beta$.

For the case of this linear region $R$, we can simply find the index $i$ which maximizes $\alpha_i Z / c_i$. If this value exceeds $\beta$, we can return the vector with that value in the $i$-th position and zeros elsewhere. Otherwise, the problem has no solution.

To prove the “no solution” part, say $n=2$ and you have $x = (x_1, x_2)$ a solution to $\alpha \cdot x \geq \beta$. Then for whichever index makes $\alpha_i Z / c_i$ bigger, say $i=1$, you can increase $\alpha \cdot x$ without changing $c \cdot x = Z$ by replacing $x_1$ with $x_1 + (c_2/c_1)x_2$ and $x_2$ with zero. I.e., we’re moving the solution $x$ along the line $c \cdot x = Z$ until it reaches a vertex of the region bounded by $c \cdot x = Z$ and $x \geq 0$. This must happen when all entries but one are zero. This is the same reason why optimal solutions of (generic) linear programs occur at vertices of their feasible regions.

The code for this becomes quite simple. Note we use the numpy library in the entire codebase to make linear algebra operations fast and simple to read.

def makeOracle(c, optimalValue):
n = len(c)

def oracle(weightedVector, weightedThreshold):
def quantity(i):
return weightedVector[i] * optimalValue / c[i] if c[i] > 0 else -1

biggest = max(range(n), key=quantity)
if quantity(biggest) < weightedThreshold:
raise InfeasibleException

return numpy.array([optimalValue / c[i] if i == biggest else 0 for i in range(n)])

return oracle


Implementing the core solver

The core solver implements the discussion from previously, given the optimal value of the linear program as input. To avoid too many single-letter variable names, we use linearObjective instead of $c$.

def solveGivenOptimalValue(A, b, linearObjective, optimalValue, learningRate=0.1):
m, n = A.shape  # m equations, n variables
oracle = makeOracle(linearObjective, optimalValue)

def reward(i, specialVector):
...

def observeOutcome(_, weights, __):
...

numRounds = 1000
weights, cumulativeReward, outcomes = MWUA(
range(m), observeOutcome, reward, learningRate, numRounds
)
averageVector = sum(outcomes) / numRounds

return averageVector


First we make the oracle, then the reward and outcome-producing functions, then we invoke the MWUA subroutine. Here are those two functions; they are closures because they need access to $A$ and $b$. Note that neither $c$ nor the optimal value show up here.

    def reward(i, specialVector):
constraint = A[i]
threshold = b[i]
return threshold - numpy.dot(constraint, specialVector)

def observeOutcome(_, weights, __):
weights = numpy.array(weights)
weightedVector = A.transpose().dot(weights)
weightedThreshold = weights.dot(b)
return oracle(weightedVector, weightedThreshold)


Implementing the binary search, and an example

Finally, the top-level routine. Note that the binary search for the optimal value is sophisticated (though it could be more sophisticated). It takes a max range for the search, and invokes the optimization subroutine, moving the upper bound down if the linear program is feasible and moving the lower bound up otherwise.

def solve(A, b, linearObjective, maxRange=1000):
optRange = [0, maxRange]

while optRange[1] - optRange[0] > 1e-8:
proposedOpt = sum(optRange) / 2
print("Attempting to solve with proposedOpt=%G" % proposedOpt)

# Because the binary search starts so high, it results in extreme
# reward values that must be tempered by a slow learning rate. Exercise
# to the reader: determine absolute bounds for the rewards, and set
# this learning rate in a more principled fashion.
learningRate = 1 / max(2 * proposedOpt * c for c in linearObjective)
learningRate = min(learningRate, 0.1)

try:
result = solveGivenOptimalValue(A, b, linearObjective, proposedOpt, learningRate)
optRange[1] = proposedOpt
except InfeasibleException:
optRange[0] = proposedOpt

return result


Finally, a simple example:

A = numpy.array([[1, 2, 3], [0, 4, 2]])
b = numpy.array([5, 6])
c = numpy.array([1, 2, 1])

x = solve(A, b, c)
print(x)
print(c.dot(x))
print(A.dot(x) - b)


The output:

Attempting to solve with proposedOpt=500
Attempting to solve with proposedOpt=250
Attempting to solve with proposedOpt=125
Attempting to solve with proposedOpt=62.5
Attempting to solve with proposedOpt=31.25
Attempting to solve with proposedOpt=15.625
Attempting to solve with proposedOpt=7.8125
Attempting to solve with proposedOpt=3.90625
Attempting to solve with proposedOpt=1.95312
Attempting to solve with proposedOpt=2.92969
Attempting to solve with proposedOpt=3.41797
Attempting to solve with proposedOpt=3.17383
Attempting to solve with proposedOpt=3.05176
Attempting to solve with proposedOpt=2.99072
Attempting to solve with proposedOpt=3.02124
Attempting to solve with proposedOpt=3.00598
Attempting to solve with proposedOpt=2.99835
Attempting to solve with proposedOpt=3.00217
Attempting to solve with proposedOpt=3.00026
Attempting to solve with proposedOpt=2.99931
Attempting to solve with proposedOpt=2.99978
Attempting to solve with proposedOpt=3.00002
Attempting to solve with proposedOpt=2.9999
Attempting to solve with proposedOpt=2.99996
Attempting to solve with proposedOpt=2.99999
Attempting to solve with proposedOpt=3.00001
Attempting to solve with proposedOpt=3
Attempting to solve with proposedOpt=3  # note %G rounds the printed values
Attempting to solve with proposedOpt=3
Attempting to solve with proposedOpt=3
Attempting to solve with proposedOpt=3
Attempting to solve with proposedOpt=3
Attempting to solve with proposedOpt=3
Attempting to solve with proposedOpt=3
Attempting to solve with proposedOpt=3
Attempting to solve with proposedOpt=3
Attempting to solve with proposedOpt=3
[ 0.     0.987  1.026]
3.00000000425
[  5.20000072e-02   8.49831849e-09]


So there we have it. A fiendishly clever use of multiplicative weights for solving linear programs.

Discussion

One of the nice aspects of MWUA is it’s completely transparent. If you want to know why a decision was made, you can simply look at the weights and look at the history of rewards of the objects. There’s also a clear interpretation of what is being optimized, as the potential function used in the proof is a measure of both quality and adaptability to change. The latter is why MWUA succeeds even in adversarial settings, and why it makes sense to think about MWUA in the context of evolutionary biology.

This even makes one imagine new problems that traditional algorithms cannot solve, but which MWUA handles with grace. For example, imagine trying to solve an “online” linear program in which over time a constraint can change. MWUA can adapt to maintain its approximate solution.

The linear programming technique is known in the literature as the Plotkin-Shmoys-Tardos framework for covering and packing problems. The same ideas extend to other convex optimization problems, including semidefinite programming.

If you’ve been reading this entire post screaming “This is just gradient descent!” Then you’re right and wrong. It bears a striking resemblance to gradient descent (see this document for details about how special cases of MWUA are gradient descent by another name), but the adaptivity for the rewards makes MWUA different.

Even though so many people have been advocating for MWUA over the past decade, it’s surprising that it doesn’t show up in the general math/CS discourse on the internet or even in many algorithms courses. The Arora survey I referenced is from 2005 and the linear programming technique I demoed is originally from 1991! I took algorithms classes wherever I could, starting undergraduate in 2007, and I didn’t even hear a whisper of this technique until midway through my PhD in theoretical CS (I did, however, study fictitious play in a game theory class). I don’t have an explanation for why this is the case, except maybe that it takes more than 20 years for techniques to make it to the classroom. At the very least, this is one good reason to go to graduate school. You learn the things (and where to look for the things) which haven’t made it to classrooms yet.

Until next time!

A Spectral Analysis of Moore Graphs

For fixed integers $r > 0$, and odd $g$, a Moore graph is an $r$-regular graph of girth $g$ which has the minimum number of vertices $n$ among all such graphs with the same regularity and girth.

(Recall, A the girth of a graph is the length of its shortest cycle, and it’s regular if all its vertices have the same degree)

Problem (Hoffman-Singleton): Find a useful constraint on the relationship between $n$ and $r$ for Moore graphs of girth $5$ and degree $r$.

Note: Excluding trivial Moore graphs with girth $g=3$ and degree $r=2$, there are only two known Moore graphs: (a) the Petersen graph and (b) this crazy graph:

The solution to the problem shows that there are only a few cases left to check.

Solution: It is easy to show that the minimum number of vertices of a Moore graph of girth $5$ and degree $r$ is $1 + r + r(r-1) = r^2 + 1$. Just consider the tree:

This is the tree example for $r = 3$, but the argument should be clear for any $r$ from the branching pattern of the tree: $1 + r + r(r-1)$

Provided $n = r^2 + 1$, we will prove that $r$ must be either $3, 7,$ or $57$. The technique will be to analyze the eigenvalues of a special matrix derived from the Moore graph.

Let $A$ be the adjacency matrix of the supposed Moore graph with these properties. Let $B = A^2 = (b_{i,j})$. Using the girth and regularity we know:

• $b_{i,i} = r$ since each vertex has degree $r$.
• $b_{i,j} = 0$ if $(i,j)$ is an edge of $G$, since any walk of length 2 from $i$ to $j$ would be able to use such an edge and create a cycle of length 3 which is less than the girth.
• $b_{i,j} = 1$ if $(i,j)$ is not an edge, because (using the tree idea above), every two vertices non-adjacent vertices have a unique neighbor in common.

Let $J_n$ be the $n \times n$ matrix of all 1’s and $I_n$ the identity matrix. Then

$\displaystyle B = rI_n + J_n - I_n - A.$

We use this matrix equation to generate two equations whose solutions will restrict $r$. Since $A$ is a real symmetric matrix is has an orthonormal basis of eigenvectors $v_1, \dots, v_n$ with eigenvalues $\lambda_1 , \dots, \lambda_n$. Moreover, by regularity we know one of these vectors is the all 1’s vector, with eigenvalue $r$. Call this $v_1 = (1, \dots, 1), \lambda_1 = r$. By orthogonality of $v_1$ with the other $v_i$, we know that $J_nv_i = 0$. We also know that, since $A$ is an adjacency matrix with zeros on the diagonal, the trace of $A$ is $\sum_i \lambda_i = 0$.

Multiply the matrices in the equation above by any $v_i$, $i > 1$ to get

\displaystyle \begin{aligned}A^2v_i &= rv_i - v_i - Av_i \\ \lambda_i^2v_i &= rv_i - v_i - \lambda_i v_i \end{aligned}

Rearranging and factoring out $v_i$ gives $\lambda_i^2 - \lambda_i - (r+1) = 0$. Let $z = 4r - 3$, then the non-$r$ eigenvalues must be one of the two roots: $\mu_1 = (-1 + \sqrt{z}) / 2$ or $\mu_2 = (-1 - \sqrt{z})/2$.

Say that $\mu_1$ occurs $a$ times and $\mu_2$ occurs $b$ times, then $n = a + b + 1$. So we have the following equations.

\displaystyle \begin{aligned} a + b + 1 &= n \\ r + a \mu_1 + b\mu_2 &= 0 \end{aligned}

From this equation you can easily derive that $\sqrt{z}$ is an integer, and as a consequence $r = (m^2 + 3) / 4$ for some integer $m$. With a tiny bit of extra algebra, this gives

$\displaystyle m(m^3 - 2m - 16(a-b)) = 15$

Implying that $m$ divides $15$, meaning $m \in \{ 1, 3, 5, 15\}$, and as a consequence $r \in \{ 1, 3, 7, 57\}$.

$\square$

Discussion: This is a strikingly clever use of spectral graph theory to answer a question about combinatorics. Spectral graph theory is precisely that, the study of what linear algebra can tell us about graphs. For an deeper dive into spectral graph theory, see the guest post I wrote on With High Probability.

If you allow for even girth, there are a few extra (infinite families of) Moore graphs, see Wikipedia for a list.

With additional techniques, one can also disprove the existence of any Moore graphs that are not among the known ones, with the exception of a possible Moore graph of girth $5$ and degree $57$ on $n = 3250$ vertices. It is unknown whether such a graph exists, but if it does, it is known that

You should go out and find it or prove it doesn’t exist.

Hungry for more applications of linear algebra to combinatorics and computer science? The book Thirty-Three Miniatures is a fantastically entertaining book of linear algebra gems (it’s where I found the proof in this post). The exposition is lucid, and the chapters are short enough to read on my daily train commute.

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.

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 row space of $A$, i.e. a basis of the subspace spanned by the data contained in the rows 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:

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!