Dimensionality Reduction

Throughout this course, one of the principles we have adhered to can be summarized as “don’t show too much in a single plot”. In other words, instead of creating a visualization where every single record is mapped to a different visual channel, we tend to use visual channels sparingly. We choose only a few variables to display at a time, and use something like Small Multiples displays to portray variable differences.

In this module, however, we will break this rule is a very specific way. “Dimensionality reduction” techniques, in essence, compute a position on the plane based on every variable of the dataset. Specifically, we will think of every one of these methods as a procedure that, given a set of input points $s_i \in S$, determines a position in $x$ and $y$ coordinates for each $s_i$.

As we will see, there is sometimes good reasons to use dimensionality reduction (and often there isn’t).

From records to vectors

In the rest of the course, we have assumed that our data is represented by a set of records (concretely, Javascript objects), such as

{ "GPA": 3.5, "SATV": 780, "SATM": 800, "ACT": 23 }


or

{ "Sepal Length": 3,
"Petal Length": 4,
"Sepal Width": 2,
"Petal Length": 3.5 }


In this module, we will require our data to be in a different format. To start, we will convert every data point into a vector of finite dimension, and the dimension will need to be decided ahead of time. In the two examples above, the conversion is quite obvious: we choose some order for the fields, and simply list all of the numbers: [3.5, 780, 800, 23], or [3, 4, 2, 3.5].

Sometimes, the conversion is not that straightforward. For example, if one of the data fields store strings, how should we represent those as vectors? A full discussion of this is well beyond the scope of this module and course, Still, whenever you take use dimensionality reduction on complicated datasets, remember that one important question to consider is how to encode the data into vectors.

Linear dimensionality reduction

The first class of dimensionality reduction techniques we’ll consider are linear dimensionality reduction techniques. They’re called this because they are, well, linear methods. Specifically, they involve computing some matrix $M$ in one way or another, and then the result of the method is the multiplication of each input vector $d_i$ by $M$.

Linear dimensionality reduction methods are a good object of study because they are quite simple, very elegant, and often surprisingly powerful.

Scatterplots

When we earlier discussed scatterplots, we didn’t call them dimensionality reduction methods. But it’s easy to see how they are. Consider the iris dataset, from which the second above example is drawn. For every field in the dataset, there exists a vector $f$ such that $\langle f, d \rangle = \texttt{record}[\texttt{field}]$. For example, for the vector representation and example record we chose above, $\langle [3, 4, 2, 3.5], [1, 0, 0, 0] \rangle = 3$, the value of the $\texttt{Sepal Length}$ field. Similarly, the vector $[0, 1, 0, 0]$ extracts the field $\texttt{Petal Length}$ from the vectors. So consider the matrix

$M = \left ( \begin{array}{cccc} 1&0&0&0 \\ 0&1&0&0 \end{array} \right )$

This matrix $M$ creates a vector whose two coordinates are the values of the fields $\texttt{Sepal Length}$ and $\texttt{Petal Length}$. So if we simply plot those 2-dimensional vectors in a plane, we get precisely the scatterplot of those two variables:

However, there’s nothing stopping us from choosing other vectors for the $x$ and $y$ coordinates. Sometimes, these come naturally from the data. For example, if we wanted to plot the average of the petal measurements against the average of the sepal measurements, we would use the matrix

$M' = \left (\begin{array}{cccc} 0.5 & 0 & 0.5 & 0 \\ 0 & 0.5 & 0 & 0.5 \end{array}\right )$

Notice how this is a slightly different plot from the first one; we can pick weirder ones, too:

$M'' = \left (\begin{array}{cccc} 0.2 & -0.4 & -0.3 & 0.5 \\ -0.7 & 0.3 & -0.3 & 0.8 \end{array}\right )$

Now it’s hard to come up with a name for the axes, and it’s also become hard to see the clustered structure that we had before. This makes us think: is there a notion of a best matrix? How do we know which matrix is good?

The Grand Tour

Now, before we discuss how to think of the “best” possible 2D linear projection of a dataset, what if we could animate along all possible such views? The Grand Tour is one classic algorithm that achieves this. In the animation below, we are slowly—and smoothly—moving through all possible 2D views of the dataset:

As the animation progresses, notice how some of the points of view make a visualization that’s all bunched up in the middle. This happens because that particular point of view behaves like the shadow of a sheet of paper when we light it along its edge. So, in a sense, that point of view is not very informative of the dataset: all the interesting stuff is getting “squashed on the screen”. Conversely, the “best” possible point of view is the one that squashed the data the least. This brings us directly to the canonical dimensionality reduction method: Principal Components Analysis, or PCA.

Principal Components Analysis

(In what follows, we are going to assume that each feature in the dataset is centered, or has mean 0).

Let’s start considering a simpler case, that of one-dimensional projections. That is, instead of thinking of a matrix $M$, let’s consider a single vector $m$. Now if we think of the intuition of the Grand Tour visualization, what we want is a vector for which most of the values of $\langle m, d_i \rangle$ are large (so they don’t “bunch up” near zero). We have to be a little careful, though. Imagine that you somehow find a relatively good $m$. If that $m$ is good, then I could give you a new vector that’s twice as good, by simply multiplying it by $2$, since that would spread out the points twice as much. That means that we need to search for a good $m$, but we need to make sure we fix its length, to avoid this artificial situation in which we find a good vector, but it’s good only because its length is very large.

Let’s make this a bit more formal. We have a list of vectors $d_i$, which we will organize on a matrix $A$:

$A = \left ( \begin{array}{c} d_0 \\ d_1 \\ \vdots \\ d_{n-1} \end{array} \right )$

What we are looking for is a vector $m$, such that the length of $Am$ is maximized, but constrained to having length 1: $\arg \max _m\ \langle Am, Am \rangle$, subject to $\langle m, m \rangle - 1 = 0$.

We can solve this directly, but first you’ll need to remember your Lagrangian multipliers, though. The expression we need to maximize is:

$\begin{eqnarray*} L(m, \lambda) & = & \langle Am, Am \rangle - \lambda (\langle m, m \rangle - 1) \\ & = & m^T A^T A m - \lambda m^T m \end{eqnarray*}$

When we take the gradient of $L$ with respect to $m$, we end up with

$A^T A m = \lambda m$

In other words, the vector $m$ that maximizes the one-dimensional projection of the vectors $d_i$ is one of the eigenvectors of $A^T A$. Specifically, it is the eigenvector associated to the largest eigenvalue. (Since $A^T A$ is symmetric and positive-definite, its eigenvalues are all non-negative and real, which means we can choose the largest.) To get the first principal component of ${ d_i }$, we simply compute $A m$. And, although we won’t prove it here, it’s relatively easy to show that the $n$-th principal component of the dataset is the result of projecting the points using the eigenvector corresponding to the $n$-th largest eigenvalue. When we plot the PCA components on the screen, we usually plot the first principal component against the second principal component, since those are the two projection directions that minimize the degree to which the dataset is squashed:

It is important to consider how to interpret a PCA plot, especially when compared to a regular scatterplot. In a regular scatterplot, the axes are decided by someone, and so by definition they correspond to some phenomenon of interest of the analyst. The PCA, on the other hand, can have fairly arbitrary axes. So how are we supposed to read it? The main way in which we do this is to project the axes of interest:

Here, what we’ve done is we’ve projected the basis vectors of the dataset. Because PCA is a linear method, we can use those vector to make judgments of “where the points would go if those column values changed”, since adding or (subtracting) a certain amount along one coordinate changes the overall projection exactly by the amount given by the projection of the change:

$M(v + [1, 0, 0, 0]) = Mv + M[1,0,0,0]$

This is one of the main advantages of PCA: even though the axes of the scatterplot it generates are a potentially complicated combination of all the data features, it is still possible to interpret the plot somewhat effectively, since we can easily “work out” what the plot would look like, if the data values were different. (In a way, this is exactly how we interpret regular scatterplots: if the data were different along the axes we plot in a scatterplot, then the point would move horizontally or vertically)

But let’s get back to the math of the PCA. We defined it in terms of the eigenvectors of $A^T A$, but there’s a slightly more general way to think of what’s happening. We start by writing $A$ in terms of its singular value decomposition: $A = U \Sigma V^T$, where $U$ and $V$ are orthogonal matrices, and $\Sigma$ is diagonal with non-negative, non-increasing real entries along the diagonal (we index the diagonal values by $i$ and call them $\sigma_i$). The main trick we’ll use with singular value decompositions is that if $u_i$ is the $i$-th column of $U$, then $u_i^T A = \sigma_i v_i$, where $v_i$ is the $i$-th column of $V$ ($u_i$ and $v_i$ are known as left and right singular vectors, respectively).

This helps because we can see that if we have the SVD of A, then $m$ (the eigenvector of $A^T A$) is necessarily a right singular vector of $a$. And, because of that, the result of projecting $A$ along $m$ comes out directly from the definition of SVD, and it is the corresponding left singular vector, scaled by its singular value. If $m = v_i$, then $A m = u_i \sigma_i$. As a result, the SVD not only gives us the projection vectors, but also the coordinates for the projected vectors themselves. So far, it’s mostly a notational convenience. Hold on, though, because it’s about to get really cool.

Classical MDS

In the beginning of this lecture, we started by assuming that we needed a vector representation for each value in our dataset. Sometimes, this is not really feasible. Often, it’s much more convenient to think of a black-box subroutine that reports the distance between two points in the dataset, without resorting to any vector representation. For example, consider the situation of using edit distance to compare strings. It’s entirely unclear how to create a vector representation for strings such that the vector distance between them corresponds to edit distances. Miraculously, though, it’s often possible to recover vector representations direct from the distance subroutines. This is what we’re going to do now.

Imagine, to start, that we have a square matrix where each entry records the squared distance between point $i$ and $j$. We informally proceed by the method of hope: we pretend that there exist vector representations $d_i$ and $d_j$ for the data points, and we just don’t know what they are. Instead, we have the matrix $D$ with entries $D_{ij} = || d_i - d_j ||^2$. The expression $||d_i - d_j||^2$ can be massaged a bit:

$\begin{eqnarray*} ||d_i - d_j||^2 & = & \langle d_i - d_j , d_i - d_j \rangle \\ & = & \langle d_i - d_j, d_i \rangle - \langle d_i - d_j, d_j \rangle \\ & = & \langle d_i, d_i \rangle - 2 \langle d_i, d_j \rangle + \langle d_j, d_j \rangle \end{eqnarray*}$

We now write $D$ as $\alpha - 2 \beta + \gamma$. $\alpha$ is a matrix where the $i$-th row has repeated values $\langle d_i, d_i \rangle$. $\beta_{ij}$ simply holds the inner products: $\beta_{ij} = \langle d_i, d_j \rangle$. $\gamma$, finally, is a matrix where the $j$-th column has repeated values $\langle d_j, d_j \rangle$. The next trick is that we subtract row averages from $D$, as well as column averages. Because the average is a linear operator, this is equivalent to subtracting row and column averages from $\alpha$, $\beta$, and $\gamma$.

But notice what subtracting row averages does to $\alpha$: since each row is made of a single value, the row average is exactly $\langle d_i, d_i \rangle$, if we subtract that, we end up canceling $\alpha$ exactly. In addition, subtracting column averages cancels $\gamma$ in turn. And, finally, if we assume that our $d_i$ representation is already centered around the origin (that is, if we assume that $\sum_i d_i = 0$), then subtracting row and column averages from $\beta$ does precisely nothing1! As a result, subtracting row and column averages from $D$ leaves us precisely with $-2 \beta$. So to actually get the inner products, divide each entry by $-1/2$, and call the resulting matrix $B$.

What now? Well, let’s pretend that we had a matrix $A$ like the one we used for PCA. If we had that matrix $A$, then it would be the case that $A A^T = B$ (because the $(i,j)$-th entry of $B$ is the inner product of the $i$-th row and $j$-th columns of $A$ and $A^T$ respectively). Notice how this is pretty similar to the PCA case (where we dealt with $A^T A$). We come to the next leap of faith: maybe the same eigenvalue and SVD tricks could work.

Let’s see how far we can get. Since we started with distances between vectors, talking about projection directions doesn’t really make sense. What we’re looking for are the positions of the projected vectors themselves. Using the notation from the PCA above, what we are looking for is the equivalent of $A m = u_i \sigma_i$, that is, the left singular values of $A$. But expand $A A^T$ in terms of the SVD of $A$: $A A^T = U \Sigma V^T V \Sigma U^T = U \Sigma^2 U^T$ 2. If you remember that the eigendecomposition of a matrix $M$ gives $M = Q \Lambda Q^T$, where Q is orthogonal and $\Lambda$ is diagonal, then you notice that the eigenvectors of $B$ (the matrix we are pretending is equal to $B = A A^T$) would be equal to the left singular vectors of $A$ (the matrix we don’t even have!). And, in that case, $A m = u_i \sigma_i$, and we can recover $\sigma$ values as the square roots of the eigenvalues of $B$, and so we can pretend that $Q \Lambda^{1/2}$ are the principal components of the matrix $A$ we don’t actually have. And it actually works! This piece of linear algebra magic is called Classical Multidimensional Scaling, or “CMDS”. You can do some really very cool things with it, and here’s the actual pseudocode for the algorithm.

D[i,j] = distance(i, j) ** 2
D = subtract-row-averages(D)
D = subtract-column-averages(D)
B = -D/2
Compute eigenvectors Q and eigenvalues Lambda of B
return Q * sqrt(Lambda) as the coordinates of your vectors


Let’s start with a basic example. Imagine you have a table with highway distances between major cities:

     ATL  BOS  ORD  DCA  DEN  LAX  MIA  JFK  SEA  SFO  MSY
ATL    0  934  585  542 1209 1942  605  751 2181 2139  424
BOS  934    0  853  392 1769 2601 1252  183 2492 2700 1356
ORD  585  853    0  598  918 1748 1187  720 1736 1857  830
DCA  542  392  598    0 1493 2305  922  209 2328 2442  964
DEN 1209 1769  918 1493    0  836 1723 1636 1023  951 1079
LAX 1942 2601 1748 2305  836    0 2345 2461  957  341 1679
MIA  605 1252 1187  922 1723 2345    0 1092 2733 2594  669
JFK  751  183  720  209 1636 2461 1092    0 2412 2577 1173
SEA 2181 2492 1736 2328 1023  957 2733 2412    0  681 2101
SFO 2139 2700 1857 2442  951  341 2594 2577  681    0 1925
MSY  424 1356  830  964 1079 1679  669 1173 2101 1925    0


You can use this data to recover an approximate map of the US:

(Full disclosure, I had to mildly cheat here; the map as returned by the algorithm doesn’t know east from west or north from south, and so it’s flipped upside down. This is because CMDS only knows distances, and not our notions of what to expect from a map.)

This is already neat. But we can do much better. Notice that the gist of the CMDS trick above is to extract a matrix of inner products between the vectors that we don’t actually have3. But remember that one way to think of inner products is as a kind of similarity between objects. So if you have a way to roughly measure how similar two objects are, then you can extract vector representations, and you can even visualize them. One of the coolest instances of this trick is a way to visualize the way that people make Morse code mistakes. In a classic experiment, Rothkopf asked subjects to judge whether two Morse code sounds were perceived to be the same or different. With this setup, one can record how often the Morse code for a given letter (or number) is perceived to be the Morse code of a different letter. The more often people confuse two Morse codes, the more we can think of those codes as being “humanly” similar. By performing CMDS on that matrix, then, we can get, quite literally, a picture of human auditory perception of Morse code:

Notice how this is even somewhat interpretable: vertically, we go from codes that are predominantly dots to codes that are predominantly dashes; horizontally, we go from short codes to long codes. The multi-dimensional scaling principle, this idea that we can convert similarities and distances to positions, is quite striking and general.

Nonlinear dimensionality reduction

TBF.

TBF.

t-SNE, UMAP, oh my

TBF.

1. As an exercise, see if you can prove why this is the case.

2. Try to prove that this holds as well.

3. This is an instance of a much more general idea, namely that if we have access to operations that behave like inner products, we can do pretty much anything we could do with the vectors themselves. This is known in machine learning as the kernel trick, and it’s a central, celebrated tool in the field.