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Matrices And Tensors

 

Linear Algebra Review

Vector Basics

For {$m$}-dimensional vectors {$\mathbf{v}$} and {$\mathbf{u}$} that form angle {$\theta$} with each other:

Property Name	Denotation	Formula
Length of {$\mathbf{v}$}	{$|\mathbf{v}|$} or {$||\mathbf{v}||_2$}	{$\sqrt{\sum_{i = 1}^{m} v^2_i}$}
Inner (dot) product of {$\mathbf{v}$} and {$\mathbf{u}$}	{$\mathbf{v} \cdot \mathbf{u}$} or {$\mathbf{v}^T \mathbf{u}$} or {$<\mathbf{v}, \mathbf{u}>$}	{$\sum_{i = 1}^m v_i u_i$} or {$|v||u|\cos(\theta)$}
Projection of {$\mathbf{v}$} onto {$\mathbf{u}$}	{$\textrm{proj}_{\mathbf{u}}(\mathbf{v})$}	{$<\mathbf{v},\mathbf{u}> \frac{\mathbf{u}}{<\mathbf{u},\mathbf{u}>}$}

Vectors of unit length ({$|\mathbf{v}| = 1$}) are called normal vectors. Two vectors {$\mathbf{v}$} and {$\mathbf{u}$} are said to be orthogonal, {$\mathbf{v} \perp \mathbf{u}$}, if {$\mathbf{v} \cdot \mathbf{u} = 0$}. In {$2D$} this corresponds to them being at a {$90^\circ$} angle. Two normal vectors that are orthogonal are called orthonormal. You can get a geometric intuition for the dot product and orthogonality by playing with this dot product applet. Note that for the purposes of multiplication, it’s convention to assume vectors are vertical, and that their transposes are horizontal as shown below.

{$\mathbf{v} = \left[\begin{array}{c} v_1 \\ \vdots \\ v_m \end{array}\right] \;\;\;\;\;\;\textrm{and}\;\;\;\;\;\; \mathbf{v}^T = \left[\begin{array}{ccc} v_1 & \ldots & v_m \\ \end{array}\right]$}

Geometrically, a vector can be viewed as a point in a space. Multiplying by a scalar lengthens (or shortens) the vector, changing its norm (length). A vector can also be thought of as a mapping (using a dot product) from a vector to a scalar.

Matrix Basics

An {$n*m$} matrix {$\mathbf{A}$} is a linear mapping from a length {$m$} vector {$\mathbf{x}$} to a length {$n$} vector {$\mathbf{y}$}:

{$ \mathbf{y} = \mathbf{A}\mathbf{x} $}

or, equivalently, a mapping from two vectors to a scalar {$z = f(x,y) = \mathbf{y}^T \mathbf{A}\mathbf{x}$}

• If matrix {$A$} has {$m$} rows and {$n$} columns, it is called an “{$m$} by {$n$}” matrix, denoted {$m \times n$}.
• Two matrices {$A$} and {$B$} can be multiplied if their inner dimensions agree: {$A \in \mathbb{R}^{m \times n}$} and {$B \in \mathbb{R}^{n \times k}$} means {$AB \in \mathbb{R}^{m \times k}$}. {$AB$} can be thought of as the dot product of rows of A with columns of B.
• We will often deal with matrices that are symmetric: {$\mathbf{A}^T =\mathbf{A}$}.
• The identity matrix, denoted {$I$}, has all 1’s on the diagonal and zeros elsewhere. Thus, for any matrix {$A$}, {$AI = A$}.
• The matrix A is called orthogonal if {$A^TA = AA^T = I$}.
• A matrix is called diagonal if its only non-zero values are on the diagonal.
• For any square matrix (a matrix where the number of rows equals the number of columns) a quantity called the determinant is defined. It is denoted {$\det(A)$}, or {$|A|$} since it is the matrix analogue of vector length; the determinant of a matrix is the product of its eigenvalues, and measures the magnitude of its linear transformation power (how much the matrix would scale the unit box). Mathematicians look at characteristic equations, but we will use SVD to find eigenvalues, and if we use determinants it will be to note that the determinant is zero iff there is a least one zero eigenvalue, in which case the matrix is singular and has no inverse (only a generalized inverse).
• There are a number of different measures of ‘how big’ a matrix is, all functions of its eigenvalues
  • the sum of the eigenvalues
  • the product of the eigenvalues
  • the L2 norm of the eigenvalues
• The inverse of a matrix {$A$} is denoted {$A^{-1}$}, which is the matrix such that {$AA^{-1} = A^{-1}A = I$}. Math books talk about methods using co-factors and such, but we will use SVD methods to find inverses in this course.

Geometrically, multiplying by a matrix can be viewed as moving a point using a combination of a rotation and a dilation. The simplest matrix (apart from the zero) is the identity matrix {$\mathbf{I}$}, which has no effect on a point. The next simplest is a diagonal matrix {$\mathbf{\Delta}$}, which rescales each element (dimension) of a vector. Rotation matrices move a point without changing its length. Multiplying two matrices just does the two operations sequentially, but note that the order matters.

Tensor Basics

An {$n*n*n$} tensor {$\mathbf{\Gamma}$} is a bilinear mapping from two length {$n$} vectors to a length {$n$} vector:

{$\mathbf{z} = \mathbf{\Gamma}\mathbf{x} \mathbf{y} = f(\mathbf{x},\mathbf{y})$}

If one applies the tensor to a single vector, the result can be viewed as a matrix:

{$\mathbf{A} = \mathbf{\Gamma}\mathbf{x} $}

alternatively, you can view a tensor as a mapping from three vectors to a scalar.

Eigen-thingys

Consider the equation {$A\mathbf{v} = \lambda\mathbf{v}$} with {$A$} symmetric. This states that the result of multiplying matrix {$A$} with a vector {$\mathbf{v}$} is the same as multiplying that vector by the scalar {$\lambda$}. A graphical representation of this is shown in the illustration below.

All non-zero vectors {$\mathbf{v}$} that satisfy such an equation are called eigenvectors of {$A$}, and their respective {$\lambda$} are called eigenvalues. “Eigen” is a German term, meaning “own” (as in “my own” or “self”) or characteristic” or “peculiar to”, which is appropriate for eigenvectors and eigenvalues because to a great extent they describe the characteristics of the transformation that a matrix represents. In fact, for every real-valued, symmetric matrix we can even write the matrix entirely in terms of its eigenvalues and eigenvectors. Specifically, for an {$m \times m$} matrix {$A$} we can state {$A = V \Lambda V^T$}, where each column of {$V$} is an eigenvector of {$A$} and {$\Lambda$} is a diagonal matrix with the corresponding eigenvalues along its diagonal. This is called the eigendecomposition of the matrix.

If you want to find some function {$f(A)$} of a matrix {$A$}, e.g {$e^A$}, how would you do it?

Some functions are easy {$A^2$}. others not so. All are easy if {$A$} is a diagonal matrix — just apply the function to each element of the diagonal. SImilarly, to ask is a matrix is “positive”, we could ask if all the elements of its diagonal are positive.

How do we do this for a general (or for now a general symmetric) matrix?

Answer: do a eigendecomposition. Find the solutions to {$A\mathbf{v_j} = \lambda_j\mathbf{v_j}$}, then assemble the eigenvectors {$\mathbf{v_j}$} into a matrix {$V$} and put the eigenvalues {$\lambda_j$} as elements on the diagonal of a matrix {$\Lambda$}, and then {$A = V \Lambda V^T$}.

Now we can apply all of our operations {$f()$} to the eigenvalues as if they were scalars!

In practice, most eigenvalue solvers use methods such as power iteration to compute eigenvalues. Let’s see how that works by multiplying some arbitrary vector {$x$} repeatedly by a matrix {$A$}. What will {$A A A A A A A x$} approach? Answer: a constant times the eigenvalue corresponding to the largest eigenvalue {$\lambda_1$} How do we know? Answer: write {$x$} in terms of the basis vectors {$v_i$}, by projecting it on to each of them. {$x = \sum_i <x,v_i> v_i$}

How fast will it become “pure”? Each multiplication will increase the component of {$v_1$} relative to {$v_2$} by a factor of {$\lambda_1 / \lambda_2$}

SVD (singular value decomposition)

One can generalize eigenvalues/vectors to non-square matrices, in which case they are called singular vectors and singular values. The equations remain the same, but there are now both left and right singular values and vectors. {$A^TA\mathbf{v} = \lambda\mathbf{v}$}

{$ AA^T \mathbf{u} = \lambda\mathbf{u}$}

{$A = U \Lambda V^T$}.

where {$A$} is {$n*p$},{$U$} is {$n*n$}, {$\Lambda$} is {$n*p$}, and {$V^T$} is {$p*p$}, but {$U$} and {$V$} can be at most rank {$min(n,p)$}, so one of them will have {$n-p$} or {$p-n$} zero eigenvalues.

Exercise: show that the nonzero left and right singular values are identical.

Any matrix can be decomposed into a weighted sum of it’s singular vectors (giving a singular value decomposition).

In practice, we often don’t care about decomposing {$A$} exactly, but only approximating it. For example, we will often take {$A$} to be our “design matrix” of observations {$X$}, and approximate it by the thin SVD obtained when one only keeps the top {$k$} singular vectors and values.

{$A \sim U_k \Lambda_k V_k^T$} where {$A$} is {$n*p$}, {$U_k$} is {$n*k$}, {$\Lambda_k$} is {$k*k$}, and {$V_k^T$} is {$k*p$}

The thin SVD gives the best rank {$k$} approximation to a matrix, where “best” means minimum Frobenious norm, which is basically the {$L_2$} norm of the elements of the matrix (i.e., the square root of the sum of the squares of all the elements of the matrix, which magically equals the sum of the squares of the singular values).

This is often useful. Consider a matrix {$A$} where each student is a row, and each column is set of test result for that student (e.g. a score on an exam question, or to an IQ test, or anything. (More generally, rows are observations, and columns are features.) We can now approximate the student grades by {$\hat{A} \sim U_k \Lambda_k V_k^T$}. Here we have approximated each original {$p$}-dimensional vector by a {$k$}-dimensional one, called the scores. Each student’s grade vector {$x$} can be written in terms of a weighted combination of the basis vectors {$V_k$}. The weights are the “scores”, and the components of the basis vectors are called the loadings of the original features.

For example, in a battery of tests of intellectual abilities, the score on the “largest singular vector”—which means the singular vector corresponding to the largest singular value — after all, all the singular vectors have been normalized to be of length 1—is often called g, for “general intelligence”.

Why is all this useful? We’ll see many examples over the next couple classes, but here are a few:

• insight and visualization; often you can characterize a feature vector by a couple of scores, and often the scores can be given names. For example the famous “Five factor” model of personality psychology
• dimension reduction — instead of working in the {$p$}-dimensional space, one works is a much smaller {$k$} dimensional space
• regularization — we’ll see that using the scores instead of the original features in linear regression (this is called Principle components regression (PCR) has a similar effect to Ridge regression,

but is sometimes a little better — or a little worse.

Generalized inverses

Linear regression requires estimating {$w$} in {$y = Xw$}, which can be viewed as computing a pseudo-inverse (or “generalized inverse”) {$X^+$} of {$X$}, so that

{$w = X^+y$}

Thus far, we have done this by

{$X^+ = (X^TX)^{-1} X^T$}

But one could also compute a generalized inverse using the SVD decomposition

{$X^+ = (U \Lambda^{-1} V^T)^T = V \Lambda^{-1} U^T$}

then {$X X^+= U \Lambda V^T V \Lambda^{-1} U^T= U U^T=I$} if {$U$} and {$V$} are orthonormal and {$\Lambda$} is invertible. similarly, {$ X^+ X = V \Lambda U^T U \Lambda^{-1} V^T= V V^T=I$}

Note that {$ U^T U =U U^T = I$} and similarly with {$V$}.

But {$\Lambda$} is often not invertible — since it is, in general, rectangular. So we use a “thin SVD” {$X^+ = V_k \Lambda_k^{-1} U_k^T$} In the thin SVD, we only keep {$k$} nonzero singular values, so that inverse {$\Lambda^{-1}$} is well-defined

Fast SVD (Randomized singular value decomposition)

This is an extremely useful algorithm that shows how one really does SVD on big data sets.

Input: matrix {$A$} of size {$n \times p$}, the desired hidden state dimension {$k$}, and the number of “extra” singular vectors, {$l$}

1. Generate a {$(k+l) \times n$} random matrix {$\Omega$}
2. Find the SVD {$U_1 D_1 V_1^T$} of {$\Omega A$}, and keep the {$k+l$} components of {$V_1$} with the largest singular values
3. Find the SVD {$U_2 D_2 V_2^T$} of {$A V_1$}, and keep the ‘largest’ {$k+l$} components of {$U_2$}
4. Find the SVD {$U_3 D_3 V_{final}^T$} of {$U_2^T A $}, and keep the ‘largest’ {$k$} components of {$V_{final}$}
5. Find the SVD {$U_{final} D_4 V_4^T$} of {$A V_{final}$} and keep the ‘largest’ {$k$} components of {$U_{final}$}

Output: The left and right singular vectors {$U_{final}$}, {$V_{final}^T$}

• Why does randomized PCA keep extra components?

Practice exercises

• Show how to compute {$e^A$}, and explain why this works.
• Prove that a matrix with only positive eigenvalues is positive definite in the sense that {$x^TAx > 0$} for all nonzero vectors {$x$}
• Given an {$n*p$} matrix with {$n$} much bigger that {$p$}, how would you efficiently compute the left singular vectors without using a randomized SVD algorithm?

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