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## Where does the Sigmoid in Logistic Regression come from?

Note: The title of this post is circular. But I use/abuse it because of the post linked below.

I noticed on the Hacker News front page (and via multiple reshares on twitter), a discussion on why logistic regression uses a sigmoid. The article linked in the story talks about the log-odds ratio, and how it leads to the sigmoid (and gives a good intuitive plug on it).

However, I think that the more important question is – Why do you care about log-odds? Why do you use log-odds and not something else? The point of this quick post is to write out why using the log-odds is infact very well motivated in the first place, and once it is modeled by a linear function, what you get is the logistic function.

Beginning with log-odds would infact be begging the question, so let us try to understand.

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To motivate and in order to define the loss etc, suppose we had a linear classifier: $\hat{y} = h(\mathbf{x}) = sign(w_0 + \mathbf{w}^T\mathbf{x})$. This just means that for a vector input, we take the dot product with the parameters $\mathbf{w}$ and take the sign.

The learning problem would be to figure out a good direction $\mathbf{w}$ and a good location of the decision boundary $w_0$.

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We want to figure these out so as to minimize the expected 0-1 loss (or expected number of mistakes) for the classifier $h: \mathcal{X} \rightarrow \mathcal{Y}$. The 0-1 loss for a datapoint-label pair $\mathbf{x},y$ is simply:

$\displaystyle L(h(\mathbf{x}),y) = \begin{cases} 0, & \text{if } h(\mathbf{x}) = y \\ 1, & \text{if } h(\mathbf{x}) \neq y \end{cases}$

Now, the next question we would like to ask. What is the risk of this classifier that we want to minimize? The risk is the expected loss. That is, if we draw a random sample from the (unknown) distribution $p(\mathbf{x},y)$, what would be the expected error? More concretely:

$R(h) = \mathbb{E}_{\mathbf{x},y} [L(h(\mathbf{x}),y)]$

Writing out the expectation:

$\displaystyle R(h) =\int_x \sum_{c=1}^C L(h(\mathbf{x}),c) p(\mathbf{x}, y=c) d\mathbf{x}$

Using the chain rule this becomes:

$\displaystyle R(h) =\int_x \sum_{c=1}^C L(h(\mathbf{x}),c) p(y=c|\mathbf{x}) p(\mathbf{x})d\mathbf{x}$

It is important to understand this expression. This is not assuming anything about the data. However, it is this expression that we want to minimize if we want to get a good classifier. To minimize this expression, it suffices to simply minimize for the conditional risk for any point $\mathbf{x}$ (i.e. the middle part of the above expression):

$\displaystyle R(h|\mathbf{x}) =\sum_{c=1}^C L(h(\mathbf{x}),c) p(y=c|\mathbf{x})$

But this conditional risk can be written as:

$\displaystyle R(h|\mathbf{x}) =0 \times p(y=h(\mathbf{x})|\mathbf{x}) + 1 \times \sum_{c \neq h(\mathbf{x})} p(y=c|\mathbf{x})$

Note that, $\displaystyle \sum_{c \neq h(\mathbf{x})} p(y=c|\mathbf{x}) = 1 - p(y=h(\mathbf{x})|\mathbf{x})$

Therefore, the conditional risk is simply:

$\displaystyle R(h|\mathbf{x}) = 1 - p(y=h(\mathbf{x})|\mathbf{x})$

Now, it is this conditional risk that we want to minimize given a point $\mathbf{x}$. And in order to do so, looking at the expression above, the classifier must make the following decision:

$\displaystyle h(\mathbf{x}) = \arg\max_c p(y=c| \mathbf{x})$

It is again important to note that so far we have made absolutely no assumptions about the data. So the above classifier is the best classifier that we can have in terms of generalization, in the sense of what might be the expected loss on a new sample point. Such a classifier is called the Bayes Classifier or sometimes called the Plug-in classifier.

But the optimal decision rule mentioned above i.e. $\displaystyle h(\mathbf{x}) = \arg\max_c p(y=c| \mathbf{x})$ is equivalent to saying that:

$\displaystyle h(\mathbf{x}) = c^\ast \iff \frac{p(y = c^\ast|\mathbf{x})}{p(y=c|\mathbf{x})} \geq 1 \text{ }\forall c$

by taking log, this would be:

$\displaystyle h(\mathbf{x}) = c^\ast \iff \log \frac{p(y = c^\ast|\mathbf{x})}{p(y=c|\mathbf{x})} \geq 0 \text{ }\forall c$

If, we were only dealing with binary classification, this would imply:

$\displaystyle h(\mathbf{x}) = 1 \iff \log \frac{p(y = 1|\mathbf{x})}{p(y=0|\mathbf{x})} \geq 0$

Notice that by making no assumptions about the data, simply by writing out the conditional risk, the log-odds ratio has fallen out directly. This is not an accident, because the optimal bayes classifier has this form for binary classification. But the question still remains, how do we model this log-odds ratio? The simplest option is to consider a linear model (there is no reason to stick to a linear model, but due to some reasons, one being convexity, we stick to a linear model):

$\displaystyle log \frac{p(y = 1|\mathbf{x})}{p(y=0|\mathbf{x})} = w_0 + \mathbf{w}^T \mathbf{x} = 0$

Now, we know that $p(y=1|\mathbf{x}) = 1 - p(y=0|\mathbf{x})$, plugging this in the above, and exponentiating, we have:

$\displaystyle log \frac{p(y = 1|\mathbf{x})}{1 - p(y=0|\mathbf{x})} = \exp(w_0 + \mathbf{w}^T \mathbf{x}) = 1$

Rearranging, yields the familiar logistic model (and the sigmoid):

$\displaystyle p(y = 1|\mathbf{x}) = \frac{1}{1+ \exp(- w_0 - \mathbf{w}^T\mathbf{x})} = \frac{1}{2}$.

As noted in the post linked in the beginning, the logistic model, $\sigma(x) = \frac{1}{1+ e^{-x}}$, which for any $x$ is, $0 \leq \sigma(x) \leq 1$, and is monotonic $\sigma(-\inf) = 0, \sigma(+\inf) = 1$.

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This derivation shows that the log-odds is not an arbitrary choice, infact a very natural choice. The sigmoid is simply a consequence of modeling the log-odds with a linear function (infact logistic regression is arguably the simplest example of a log-linear model, if we had structured outputs, a natural extension of such a model would be the Conditional Random Field. The choice of using a linear function is simply to make the optimization convex, amongst some other favourable properties).

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Note: This post was inspired by some very succinct notes by Gregory Shakhnarovich from his Machine Learning class, that I both took and served as a TA for.

## A Note on the Graph Laplacian

A short note on how the Graph Laplacian is a natural analogue of the Laplacian in vector analysis.

For a twice differentiable function $f$ on the euclidean space, the Laplacian of $f$, $\Delta f$ is the div(grad($f$)) or the divergence of the gradient of $f$. Let’s consider the laplacian in two dimensions:

$\displaystyle \Delta f(x,y) = \nabla \cdot \nabla f(x,y) = \frac{d^2 f(x,y)}{dx^2} + \frac{d^2 f(x,y)}{dy^2}$

To get an intuition about what it does, it is useful to consider a finite difference approximation to the above:

$\displaystyle \Delta f(x,y) \approx \frac{f(x+h,y) + f(x-h,y) + f(x,y+h) + f(x,y-h) - 4f(x,y)}{h^2}$

The above makes it clear that the Laplacian behaves like a local averaging operator. Infact the above finite difference approximation is used in image processing (say, when h is set to one pixel) to detect sharp changes such as edges, as the above is close to zero in smooth image regions.

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The finite difference approximation intuitively suggests that a discrete laplace operator should be natural. At the same time, the Graph Theory literature defines the Graph Laplacian as the matrix $L = D - A$ (where $D$ is the degree matrix and $A$ the adjacency matrix of the graph $G$, both are defined below). It is not obvious that this matrix $L$ follows from the div(grad(f)) definition mentioned above. The purpose of this note is to explicate in short on how this is natural, once the notion of gradient and divergence operators have been defined over a graph.

First of all the two most natural matrices that can be associated with a graph are defined below:

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There are two natural matrices that are used to specify a simple graph $G = (V,E)$ with vertex set $V =\{1, \dots, n \}$ and edge set $E = \{1, \dots, m \}$: The Adjacency matrix $A$ and the Incidence matrix $\nabla$.

The Adjacency matrix is defined as a $n \times n$ matrix such that:

$\displaystyle A_{u,v} = \begin{cases} 1 & \mbox{if } uv \in E \\ 0 & \mbox{otherwise} \end{cases}$

To define the $m \times n$ incidence matrix $\nabla$, we need to fix an orientation of the edges (for our purpose an arbitrary orientation suffices), then:

$\displaystyle \nabla_{e,v} = \begin{cases} -1 & \text{if } v \text{ is initial vertex of edge } e \\ +1 & \text{if } v \text{ is terminal vertex of edge } e \\ 0 & \text{if } e \text{ is not incident on vertex } v \end{cases}$

Now suppose we have a real valued function $f$ over $V$, $f: V \to \mathbb{R}$. For $G$, we view this as a column vector indexed by the vertices.

Similarly let $g$ be a real valued function over the edges $E$, $g: E \to \mathbb{R}$ (again, we view this as a row vector indexed by the edges).

Now, consider the following map: $f \mapsto \nabla f$ (which is now a vector indexed by $E$). The value of this map at any edge $(\nabla f) (e)$ is simply the difference of the values of $f$ at the two end points of the edge $e$ i.e. $f_u - f_v$. This makes it somewhat intuitive that the matrix $\nabla$ is some sort of a difference operator on $G$. Given this “difference operator” or “discrete differential” view, $(\nabla f)$ can then be viewed as the gradient that measures the change of the function $f$ along the edges of $G$. Thus the map $f \mapsto \nabla f$ is just the gradient operator.

Like the above, now we consider the following map: $g \mapsto \nabla^{T} g^T$ (which is now a vector indexed by $V$). Recall that $g$ was defined on the edges. The value of this map at any vertex $(g \nabla)(v)$ is $\displaystyle \sum_{e \text{ exits } v} g_e - \sum_{e \text{ enters } v} g_e$. If we were to view $g$ as describing some kind of a flow on the edges, then $(g \nabla)(v)$ is the net outbound flow on the vertex $v$, which is just the divergence. Thus the map $f \mapsto g \nabla$ is just the divergence operator.

Now consider $f \mapsto \nabla^{T} \nabla f$ (note that this makes sense since $\nabla f$ is a column vector), going by the above, this should be the divergence of the gradient. Thus, the analogy with the “real” Laplacian makes sense and the matrix $L_G = \nabla^{T} \nabla$ is appropriately called the Graph Laplacian.

Using the definition of $\nabla$, a simple computation yields the following for $L_G$:

$\displaystyle L_{G{u,v}} = \begin{cases} -1 & \mbox{if } uv \in E \\ deg(u) & \mbox{if } u =v \end{cases}$

Looking at the above and recalling the definition for the adjacency matrix $A$, it is easy to see that $L_G = D - A$, where $D$ is the diagonal matrix with $D_{u,u} = deg(u)$. This is just the familiar definition that I mentioned at the start.

PS: From the above it is also clear that the Laplacian is positive semidefinite. Indeed, consider $f ^TL f$, which is written as $f^T \nabla^T \nabla f =\|\nabla f\|^2 = \sum_{(u,v) \in E} (f_u - f_v)^2$

## From Eros to Gaia

I had been re-reading “From Eros to Gaia” by Freeman Dyson after some years. I have a bad habit of never reading prefaces to books, however I am glad I read it this time around because of this sobering passage that appears in it:

My mother used to say that life begins at forty. That was her age when she had her first baby. I say, on the contrary, that life begins at fifty-five, the age when I published my first book. So long as you have courage and a sense of humour, it is never too late to start life afresh. A book is in many ways like a baby. While you are writing, it is curled up in your belly. You cannot get a clear view of it. As soon as it is born, it goes out into the world and develops a character of its own. Like a daughter coming home from school, it surprises you with unexpected flashes of wisdom. The same thing happens with scientific theories. You sit quietly gestating them, for nine months or whatever the required time may be, and then one day they are out on their own, not belonging to you anymore but to the whole community of scientists. Whatever it is that you produce– a baby, a book, or a theory– it is a piece of the magic of creation. You are producing something that you do not fully understand. As you watch it grow, it becomes part of a larger world, and fits itself into a larger design than you imagined. You belong to the company of those medieval craftsmen who added a carved stone here or a piece of scaffolding there, and together built Chartres Cathedral.

## Neighbourhood Gerrymandering: An Approach to Discriminative Metric Learning via Latent Structured Prediction

An informal summary of a recent project I had some involvement in.

Motivation: Why care about Metric Learning?

In many machine learning algorithms, such as k-means, Support Vector Machines, k-Nearest Neighbour based classification, kernel regression, methods based on Gaussian Processes etc etc – there is a fundamental reliance, that is to be able to measure dissimilarity between two examples. Usually this is done by using the Euclidean distance between points (i.e. points that are closer in this sense are considered more similar), which is usually suboptimal in the sense that will be explained below. Being able to compare examples and decide if they are similar or dissimilar or return a measure of similarity is one of the most fundamental problems in machine learning. Ofcourse a related question is: What does mean by “similar” afterall?

To illustrate the above let us work with k-Nearest Neighbour classification. Before starting, let us just illustrate the really simple idea (of kNN classification) by an example: Consider the following points in $\mathbb{R}^2$, with the classes marked by different colours.

Now suppose we have a new point – marked with black – whose class is unknown. We assign it a class by looking at the nearest neighbors and taking the majority vote:

Some notes on kNN:

A brief digression first before moving on the problem in the above (what is nearest?). kNN classifiers are very simple and yet in many cases they can give excellent performance. For example, consider the performance on the MNIST dataset, it is clear that kNN can give competitive performance as compared to other more complicated models.

Moreover, they are simple to implement, use local information and hence are inherently nonlinear. The biggest advantage in my opinion is that it is easy to add new classes (since no retraining from scratch is required) and since we average across points, kNN is also relatively robust to label noise. It also has some attractive theoretical properties: for example kNN is universally consistent (as the number of points approaches infinity, with appropriate choice of k, the kNN error will approach the Bayes Risk).

Notion of “Nearest”:

At the same time, kNN classifiers also have their disadvantages. One is related to the notion of “nearest” (which falls back on what was talked about at the start) i.e. how does one decide what points are “nearest”. Usually such points are decided on the basis of the Euclidean distance on the native feature space which usually has shortfalls. Why? Because nearness in the Euclidean space may not correspond to nearness in the label space i.e. points that might be far off in the Euclidean space may have similar labels. In such cases, clearly the notion of “near” using the euclidean distance is suboptimal. This is illustrated by a set of figures below (adapted from slides by Kilian Weinberger):

An Illustration:

Consider the image of this lady – now how do we decide what is more similar to it?

Someone might mean similar on the basis of the gender:

Or on the basis of age:

Or on the basis of the hairstyle!

Similarity depends on the context! Something that the euclidean distance in the native feature space would fail to capture. This context is provided by labels.

Distance Metric Learning:

The goal of Metric Learning is to learn a distance metric, so that the above label information is incorporated in the notion of distance i.e. points that are semantically similar are now closer in the new space. The idea is to take the original or native feature space, use the label information and then amplify directions that are more informative and squish directions that are not. This is illustrated in this figure – notice that the point marked in black would be incorrectly classified in the native feature space, however under the learnt metric it would be correctly classified.

It is worthwhile to have a brief look at what this means. The Euclidean distance (with $x_i \in \mathbb{R}^d$) is defined by

$\sqrt{(x_i - x_j)^T (x_i - x_j)}$

as also was evident in the above figure, this corresponds to the following euclidean ball in 2-D

A family of distance measure may be defined using an inner product matrix. These are called the Mahalanobis metrics.

$\sqrt{(x_i - x_j)^T \mathbf{W}(x_i - x_j)}$

The learnt metric affects a rescaling and rotation of the original space. The goal is now to learn this $\mathbf{W} \succeq 0$ using the label information so that the new distances correspond better to the semantic context. It is easy to see that when $\mathbf{W} \succeq 0$, the above is still a distance metric.

Learning $\mathbf{W}$:

Usually the real motivation for metric learning is to optimize for the kNN objective i.e. learn the matrix $\mathbf{W} \succeq 0$ so that the kNN error is reduced. But note that directly optimizing for the kNN loss is intractable because of the combinatorial nature of the optimization (we’ll see this in a bit), so instead, $\mathbf{W}$ is learnt as follows:

1. Define a set of “good” neighbors for each point. The definition of “good” is usually some combination of proximity to the query point and label agreement between the points.

2. Define a set of “bad” neighbours for each point. This might be a set of points that are “close” to the query point but disagree on the label (i.e. inspite of being close to the query point they might give a wrong classification if they were chosen to classify the query point).

3. Set up the optimization problem for $\mathbf{W}$ such that for each query point, “good” neighbours are pulled closer to it while “bad” neighbours are pushed farther away, and thus learn $\mathbf{W}$ so as to minimize the leave one out kNN error.

The exact formulation of “good” and “bad” varies from method to method. Here are some examples:

In one of the earliest papers on distance metric learning by Xing, Ng, Jordan and Russell (2002) – good neighbors are similarly labeled k points. The optimization is done so that each class is mapped into a ball of fixed radius. However no separation is enforced between the classes. This is illustrated in the following figure (the query point is marked with an X, similarly labeled k points are moved into a ball of a fixed radius):

One problem with the above is that the kNN objective does not really require that similarly labeled points are clustered together, hence in a way it optimizes for a harder objective. This is remedied by the LMNN described briefly below.

One of the more famous Metric Learning papers is the Large Margin Nearest Neighbors by Weinberger and Saul (2006). Here good neighbors are similarly labeled k points (and the circle around x is the distance of the farthest of the good neighbours) and “worst offenders” or “bad” neighbours are points that are of a different class but still in the nearest neighbors of the query point. The optimization is basically a semidefinite program that works to pull the good neighbours towards the query point and a margin is enforced by pushing the offending points out of this circle. Thus in a way, the goal in LMNN is to deform the metric in such a way that the neighbourhood for each point is “pure.

There are many approaches to the metric learning problem, however a few more notable ones are:

1. Neighbourhood Components Analysis (Goldberger, Roweis, Hinton and Salakhutdinov, 2004): Here the piecewise constant error of the kNN rule is replaced by a soft version. This leads to a non-convex objective that can be optimized by gradient descent. Basically, NCA tries to optimize for the choice  of neighbour at the price of losing convexity.

2. Collapsing Classes (Globerson and Roweis, 2006): This method attempts to remedy the non-convexity above by optimizing a similar stochastic rule while attempting to collapse each class to one point, making the problem convex.

3. Metric Learning to Rank (McFee and Lankriet, 2010): This paper takes a different take on metric learning, treating it as a ranking problem. Note that given a fixed p.s.d matrix $\mathbf{W}$ a query point induces a permutation on the training set (in order of increasing distance). The idea thus is to optimize the metric for some ranking measure (such as precision@k). But note that this is not necessarily the same as requiring correct classification.

Neighbourhood Gerrymandering:

As a motivation we can look at the cartoon above for LMNN. Since we are looking to optimize for the kNN objective, the requirement to learn the metric should just be correct classification. Thus, we should need to push the points to ensure the same. Thus we can have the circle around x as simply the distance of the farthest point in the k nearest neighbours (irrespective of class). Now, we would like to deform the metric such that enough points are pulled in and pushed out of this circle so as to ensure correct classification. This is illustrated below.

This method is akin to the common practice of Gerrymandering, in drawing up borders of election districts so as to provide advantages to desired political parties. This is done by concentrating voters from a particular party and/or by spreading out voters from other parties. In the above, the “districts” are cells in the Voronoi diagram defined by the Mahalanobis metric and “parties” are class labels voted for by each neighbour.

Motivations and Intuition:

Now we can step back a little from the survey above, and think a bit about the kNN problem in somewhat more precise terms so that the above approach can be motivated better.

For kNN, given a query point and a fixed metric, there is an implicit latent variable: The choice of the k “neighbours”.

Given this latent variable – inference of the label for the query point is trivial – since it is just the majority vote. But notice that for any given query point, there can exist a very large number of  choices of k points that may correspond to correct classification (basically any set of points with majority of correct class will work). Now we basically want to learn a metric so that we prefer one of these sets over any set of k neighbours which would vote for a wrong class. In particular, from the sets that affects correct classification we would like to pick the set that is on average most similar to the query point.

We can write kNN prediction as an inference problem with a structured latent variable being the choice of k neighbours.

The learning then corresponds to  minimizing a sum of structured latent hinge loss and a regularizer. Computing the latent hinge loss involves loss-augmented inference which is basically looking for the worst offending k points (points that have high average similarity with the query point, yet correspond to a high loss). Given the combinatorial nature of the problem, efficient inference and loss-augmented inference is key. Optimization can basically be just gradient descent on the surrograte loss. To make this a bit more clear, the setup is described below:

Problem Setup:

Suppose we are given $N$ training examples that are represented by a “native” feature map, $\mathbf{X} = \{x_1, \dots, x_N\}$ with $x_i \in \mathbb{R}^d$ with class labels $\mathbf{y} = [y_1, \dots, y_N]^T$ with $y_i \in [\mathbf{R}]$, where $[\mathbf{R}]$ stands for the set $\{1, \dots, \mathbf{R}\}$.

Suppose are also provided with a loss matrix $\Lambda$ with $\Lambda(r,r')$ being the loss incurred by predicting $r'$ when the correct class is $r$. We assume that $\Lambda(r,r) = 0$ and $\forall (r,r'), \Lambda(r,r') \geq 0$.

Now let $h \subset \mathbf{X}$ be a set of examples in $\mathbf{X}$.

As stated earlier, we are interested in the Mahalanobis metrics:

$D_W(x,x_i) = (x-x_i)^T W (x-x_i)$

For a fixed $W$ we may define the distance of $h$ with respect to a point $x$ as:

$\displaystyle S_W(x,h) - \sum_{x_j \in h} D_W(x, x_j)$

Therefore, the set of k-Nearest Neighbours of $x$ in $\mathbf{X}$ is:

$h_W(x ) = \arg\max_{|h|=k} S_W(x,h)$

For any set $h$ of $k$ examples from $\mathbf{X}$ we can predict the label of $x$ by a simple majority vote.

$\hat{y}(h) = majority\{y_j: x_j \in h\}$

The kNN classifier therefore predicts $\hat{y}(h_W(x))$.

Thus, the classification loss incurred using the set $h$ can be defined as:

$\Delta(y,h) = \Lambda(y,\hat{y}(h))$

Learning and Inference:

One might want to learn $W$ so as to minimize the training loss:

$\displaystyle \sum_i \Delta(y_i, h_W(x_i))$

However as mentioned in passing above, this fails because of the intractable nature of  the classification loss $\Delta$. Thus we’d have to resort to the usual remedy: define a tractable surrograte loss.

It must be stressed again that the output of prediction is a structured object $h_W$. The loss in structured prediction penalizes the gap between score of the correct structured output and the score of the “worst offending” incorrect output. This leads to the following definition of the surrogate:

$L(x,y,W) = \max_h [S_W(x,h) + \Delta(y,h)] - \max_{h: \Delta(y,h) = 0} S_W(x,h)$

This corresponds to our earlier intuition on wanting to learn $W$ such that the gap between the “good neighbours” and “worst offenders” is increased.

So, although the loss above was arrived at by intuitive arguments, it turns out that our problem is an instance of a familiar type of problem: Latent Structured Prediction and hence the machinery for optimization there can be used here as well. The objective for us corresponds to:

$\displaystyle \min_W \| W\|^2_{F} + C \sum_i (L(x_i, y_i,W))$

Where $\| \cdot \|_F$ is the Frobenius norm.

Note that the regularizer is convex, but the loss is not convex to the subtraction of the max term i.e. now it is a difference of convex functions which means the concave convex procedure may be used for optimization (although we just use stochastic gradient descent). Also note that the optimization at each step needs an efficient subroutine to determine the correct structured output (inference of the best set of neighbours) and the worst offending incorrect structured output (loss augmented inference i.e. finding the worst set of neighbors). Turns out that for this problem this is possible (although not presented here).

It is interesting to think about how this approach extends to regression and to see how it works when the embeddings learnt are not linear.

## The Jacobian Inner Product

This post may be considered an extension of the previous post.

The setup and notation is the same as in the previous post (linked above). But to summarize: Earlier we had an unknown smooth regression function $f: \mathbb{R}^d \to \mathbb{R}$. The idea was to estimate at each training point, the gradient of this unknown function $f$, and then taking the sample expectation of the outerproduct of the gradient. This quantity has some interesting properties and applications.

However it has its limitations, for one, the mapping $f: \mathbb{R}^d \to \mathbb{R}$ restricts the Gradient Outer Product being helpful for only regression and binary classification (since for binary classification the problem can be thought of as regression). It is not clear if a similar operator can be constructed when one is dealing with classification, that is the unknown smooth function is a vector valued function $f: \mathbb{R}^d \to \mathbb{R}^c$ where $c$ is the number of classes (let us say for the purpose of this discussion, that for each data point we have a probability distribution over the classes, a $c$ dimensional vector).

In the case of the gradient outer product since we were working with a real valued function, it was possible to define the gradient at each point, which is simply:

$\displaystyle \Bigg[ \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \dots, \frac{\partial f}{\partial x_d} \Bigg]$

For a vector valued function $f: \mathbb{R}^d \to \mathbb{R}^c$, we can’t have the gradient, but instead can define the Jacobian at each point:

$\displaystyle \mathbf{J} = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \dots & \frac{\partial f_1}{\partial x_d} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_c}{\partial x_1} & \frac{\partial f_c}{\partial x_2} & \dots & \frac{\partial f_c}{\partial x_d}\end{bmatrix}$

Note that $\mathbf{J}$ may be estimated in a similar manner as estimating gradients as in the previous posts. Which leads us to define the quantity $\mathbb{E}_X G(X) = \mathbb{E}_X ( \mathbf{J}^T \mathbf{J})$.

The first thing to note is that $\mathbb{E}_X G(X) = \mathbb{E}_X ( \nabla f(X)\nabla f(X)^T)$ defined in the previous post is simply the quantity for the special case when $f: \mathbb{R}^d \to \mathbb{R}$. Another note is also in order: The reason why we suffixed that quantity with “outer product” (as opposed to “inner product” here) is simply because we considered the gradient to be a column vector, otherwise they are similar in spirit.

Another thing to note is that it is easy to see that the quantity $\mathbb{E}_X G(X) = \mathbb{E}_X ( \mathbf{J}^T \mathbf{J})$ is a positive semi-definite matrix and hence is a Reimannian Metric, which is defined below:

Definition: A Reimannian Metric $G$ on a manifold $\mathcal{M}$ is a symmetric and positive semi-definite matrix, which defines a smoothly varying inner product in the tangent space $\mathbf{T}_x \mathcal{M}$, for each point $x \in \mathcal{M}$ and $a, b \in \mathbf{T}_x \mathcal{M}$. This associated p.s.d matrix is called the metric tensor. In the above case, since $\mathbb{E}_X G(X) = \mathbb{E}_X ( \mathbf{J}^T \mathbf{J})$ is p.s.d it defines a Reimannian metric:

$\langle a, b \rangle_x = a^T \mathbb{E}_X ( \mathbf{J}^T \mathbf{J}) b$

Thus, $\mathbb{E}_X ( \mathbf{J}^T \mathbf{J})$ is a specific metric (more general metrics are dealt with in areas such as metric learning).

Properties: We saw some properties of $\mathbb{E}_X G(X) = \mathbb{E}_X ( \nabla f(X)\nabla f(X)^T)$ in the previous post. In the same vein, does $\mathbb{E}_X G(X) = \mathbb{E}_X ( \mathbf{J}^T \mathbf{J})$ have similar properties? i.e. does the first eigenvector also correspond to the direction of highest average variation? What about the $k$-dimensional subspace? What difference does it make that we are looking at a vector valued function? Also what about the cases when $d > c$ and otherwise?

These are questions that I need to think about and should be the topic for a future post to be made soon, hopefully.

Recently, in course of a project that I had some involvement in, I came across an interesting quadratic form. It is called in the literature as the Gradient Outer Product. This operator, which has applications in supervised dimensionality reduction, inverse regression and metric learning can be motivated in two (related) ways, but before doing so, the following is the set up:

Setup: Suppose we have the usual set up as for nonparametric regression and (binary) classification i.e. let $Y \approx f(X)$ for some unknown smooth $f$, the input $X$ is $d$ dimensional $X = (X^i)_{i=1}^d$

1. Supervised Dimensionality Reduction: It is often the case that $f$ varies most along only some relevant coordinates. This is the main motivation behind variable selection.

The idea in variable selection is the following: That $f(X)$ may be written as $f(PX)$ where $P \in \{0,1\}^{k \times d}$. $P$ projects down the data to only $k$ relevant coordinates (i.e. some features are selected by $P$ while others are discarded).

This idea is generalized in Multi-Index Regression, where the goal is to recover a subspace most relevant to prediction. That is, now suppose the data varies significantly along all coordinates but it still depends on some subspace of smaller dimensionality. This might be achieved by letting $P$ from the above to be $P \in \mathbb{R}^{k \times d}$. It is important to note that $P$ is not any subspace, but rather the $k$-dimensional subspace to which if the data is projected, the regression error would be the least. This idea might be further generalized by means of mapping $X$ to some $P$ non-linearly, but for now we only stick to the relevant subspace.

How can we recover such a subspace?

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2. Average Variation of $f$: Another way to motivate this quantity is the following: Suppose we want to find the direction in which $f$ varies the most on average, or the direction in which $f$ varies the second fastest on average and so on. Or more generally, given any direction, we want to find the variation of $f$ along it. How can we recover these?

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The Expected Gradient Outer Product:  The expected gradient outer product of the unknown classification or regression function is the quantity: $\mathbb{E}_X G(X) = \mathbb{E}_X ( \nabla f(X)\nabla f(X)^T)$

The expected gradient outer product recovers the average variation of $f$ in all directions. This can be seen as follows: The directional derivative at $x$ along $v \in \mathbb{R}^d$ is given by $\displaystyle {f'}_v(x) = \nabla f(x)^T v$ or $\mathbb{E}_X |{f'}_v(X)|^2 = \mathbb{E}_X (v^T G(X) v) = v^T (\mathbb{E}_X G(X))v$.

From the above it follows that if $f$ does not vary along $v$ then $v$ must be in the null space of $\mathbb{E}_X (G(X))$.

Infact it is not hard to show that the relevant subspace $P$ as defined earlier can also be recovered from $\mathbb{E}_X (G(X))$. This fact is given in the following lemma.

Lemma: Under the assumed model i.e. $Y \approx f(PX)$, the gradient outer product matrix $\mathbb{E}_X (G(X))$ is of rank at most $k$. Let $\{v_1, v_2, \dots, v_k \}$ be the eigenvectors of $\mathbb{E}_X (G(X))$ corresponding to the top $k$ eigenvalues of $\mathbb{E}_X (G(X))$. Then the following is true:

$span(P) = span(v_1, v_2, \dots, v_k)$

This means that a spectral decomposition of $\mathbb{E}_X (G(X))$ recovers the relevant subspace. Also note that the Gradient Outer Product corresponds to a kind of a supervised version of Principal Component Analysis.

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Estimation: Ofcourse in real settings the function is unknown and we are only given points sampled from it. There are various estimators for $\mathbb{E}_X (G(X))$, which usually involve estimation of the derivatives. In one of them the idea is to estimate, at each point $x$ a linear approximation to $f$. The slope of this approximation approximates the gradient at that point. Repeating this at the $n$ sample points, gives a sample gradient outer product. There is some work that shows that some of these estimators are statistically consistent.

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Related: Gradient Based Diffusion Maps: The gradient outer product can not isolate local information or geometry and its spectral decomposition, as seen above, gives only a linear embedding. One way to obtain a non-linear dimensionality reduction would be to borrow from and extend the idea of diffusion maps, which are well established tools in semi supervised learning. The central quantity of interest for diffusion maps is the graph laplacian $L = I - D^{-\frac{1}{2}} W D^{-\frac{1}{2}}$, where $D$ is the degree matrix and $W$ the adjacency matrix of the nearest neighbor graph constructed on the data points. The non linear embedding is obtained by a spectral decomposition of the operator $L$ or its powers $L^t$.

As above, a similar diffusion operator may be constructed by using local gradient information. One such possible operator could be:

$\displaystyle W_{ij} = W_{f}(x_i, x_j) = exp \Big( - \frac{ \| x_i - x_j \| ^2}{\sigma_1} - \frac{ | \frac{1}{2} (\nabla f(x_i) + \nabla f(x_j)) (x_i - x_j) |^2 }{\sigma_2}\Big)$

Note that the first term is the same that is used in unsupervised dimension reduction techniques such as laplacian eigenmaps and diffusion maps. The second term can be interpreted as a diffusion on function values. This operator gives a way for non linear supervised dimension reduction using gradient information.

The above operator was defined here, however no consistency results for the same are provided.

Also see: The Jacobian Inner Product.

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## Implementation and Abstraction in Mathematics

I recently noticed on arXiv that the following manuscript “Implementation and Abstraction in Mathematics” by David McAllester. A couple of years ago, I had taken a graduate course taught by David that had a similar flavour (the material in the manuscript is more advanced, in particular the main results, not to mention it is better organized and the presentation more polished), presenting a type theoretic foundation of mathematics. Although I can’t say I did very well in the course, I certainly enjoyed the ideas in it very much, and thus the above manuscript might be worth a look. Perhaps it might be a good idea to audit that course again, just to make sure I understand the main ideas better this time. :)