10.1. Lecture 25¶
We’ll do a brief introduction to singular value decomposition via a Jupyter notebook: Linear algebra games including SVD for PCA. We won’t be able to look at everything (or prove everything) in detail, so we will start with the highlights and fill in what we can.
Preliminary exercises¶
Linear algebra using the index form. (This is duplicated from Lecture 13.) Recall that
\[ (AB)^\intercal = B^\intercal A^\intercal \quad\mbox{and}\quad (A^\intercal)_{ji} = A_{ij} \]\[ \chi^2 = [Y - A\thetavec]^\intercal\, \Sigmavec^{-1}\, [Y - A\thetavec] = (Y_i - A_{ij}\thetavec_j)(\Sigma^{-1})_{ii'}(Y_{i'}- A_{i'j'}\thetavec_{j'}) , \]where \(i,i'\) run from \(1\) to \(N\) and \(j,j'\) run from one to \(p\), where the highest power term is \(x^{p-1}\).
Summed over \(i,i',j,j'\) because they each appear (exactly) twice.
\((\Sigma^{-1})_{ii'} \neq (\Sigma_{i,i'})^{-1}\)
Be sure you understand the indices on the leftmost term, remembering that the matrix expression has this term transposed.
We know \(\chi^2\) is a scalar because there are no free indices.
We find the MLE from \(\partial\chi^2/\partial\thetavec_k = 0\) for \(k = 1,\ldots p\).
\[\begin{align} \left.\frac{\partial\chi^2}{\partial\thetavec_k}\right|_ {\thetavec=\thetavechat} &= -A_{ij}\delta_{jk}(\Sigma^{-1})_{i,i'}(Y_{i'} - A_ {i'j'}\thetavechat_{j'}) + (Y_{i} - A_{ij}\thetavechat_{j})(\Sigma^{-1})_{i,i'}(-A_ {i'j'}\delta_{j'k}) = 0 \end{align}\]Isolate the \(\thetavec\) terms on one side and show the doubled terms are equal:
\[ A_{ik}(\Sigmavec^{-1})_{i,i'}Y_{i'} + Y_i (\Sigmavec^{-1})_{i,i'} A_{i'k} = A_{ik}(\Sigmavec^{-1})_{i,i'}A_{i'j'}\thetavechat_{j'} + A_{ij}\thetavechat_{j}(\Sigmavec^{-1})_{i,i'}A_{i'k} \]In the second term on the left, switch \(i\leftrightarrow i'\) and use \((\Sigmavec^{-1})_{i',i} = (\Sigmavec^{-1})_{i,i'}\) because it is symmetric. This is then the same as the first term.
In the first term on the right, we switch \(j\leftrightarrow j'\) and use the symmetry of \(\Sigmavec\) again to show the two terms are the same.
Writing \(A_{ik} = (A^\intercal)_{ki}\), we get
\[\begin{align} 2(A^\intercal)_{ki} (\Sigmavec^{-1})_{i,i'} Y_i = 2 (A^{\intercal})_{ki} (\Sigmavec^{-1})_{i,i'} A_ {i'j}\thetavechat_j \end{align}\]or, removing the indices,
\[ (A^{\intercal}\Sigmavec^{-1} Y) = (A^{\intercal}\Sigmavec^{-1}A) \thetavechat \]and then inverting (which is possible because the expression in parentheses on the right is a square, invertible matrix), we finally obtain:
\[ \thetavechat = [A^\intercal \Sigmavec^{-1} A]^{-1} [A^\intercal \Sigmavec^{-1} Y] . \]Q.E.D.
Diagonalization of symmetric matrix¶
If \(A\) is a symmetric, real \(n \times n\) matrix, then we diagonalize it by finding an orthogonal matrix \(V\) and a diagonal matrix \(D\) such that \(A = V D V^\intercal\).
The diagonal elements of \(D\) are the eigenvalues of \(A\).
The columns of \(V\) are the eigenvectors of \(A\). They are orthonormal. (Can you prove this?)
Following Kalman, to compare to the singular value decomposition of \(A\), we might call this the eigenvalue decomposition (EVD) of \(A\).
Singular value decomposition (SVD)¶
The key feature of SVD is the decomposition of a matrix into three other matrices.
We can do full SVD or reduced SVD. The former is usually the starting point but the latter is used in practice.
Reduced form is \( A = U S V^\intercal\), with \(S\) diagonal.
\(A\) is an \(m\times n\) matrix, while \(U\) is \(m \times n\), \(S\) is \(n\times n\), and \(V\) is \(n\times n\).
Elements of \(S\), names \(S_{kk'} \equiv S_k \delta_{kk'}\) are singular values. For a square matrix \(A\), they would be the eigenvalues.
Key feature:
\[ A_{ij} \approx \sum_{k=1}^p U_{ik} S_k V_{jk} \ \mbox{with } p < n \]is a truncated representation of \(A\) with most of the content if the “dominant” \(p\) vectors kept, which are identified by the largest singular values.
Applications of SVD¶
Solving matrix equations with ill-conditioned matrices, which means that the smallest eigenvalue is zero or close to zero. (If zero, then the matrix is singular.)
The condition number of a matrix is the ratio of the largest to the smallest eigenvalue.
If solving \(Ax = h\) \(\Lra\) \(x = A^{-1}b\), then an error in \(h\) is magnified by the condition number to give the error in \(x\).
So finite precision in \(b\) leads to nonsense for \(x\) if the the condition number is larger than the inverse of the machine or calculational precision. If the condition number \(\kappa(A) = 10^k\), then up to \(k\) digits of accuracy is lost (roughly).
Data reduction application: identify the important basis elements \(\Lra\) e.g., what linear combinations of parameters are most important.
Note the Python syntax for matrix multiplication with
@, transpose withT.Step through the Python example for image compression. You are identifying the most important eigenvalues for the image.
If we keep 1 in 100 of the singulare values, then the numbers to be stored are significantly reduced.
Note the spectrum of eigenvalues.
Covariance, PCA, and SVD
Consider the covariance matrix and find its eigenvalues and eigenvectors. ``Center’’ the data first (subtract the mean so that you deal with data having mean zero).
PCA means to rank these. This is typically done via SVD (which is better numerically).
Then one uses fewer singular values (which are the eigenvalues if it is a square matrix) \(\Lra\) we have reduced the number of parameters in a model (for example).