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--- private:writeups:svd [2020/10/18 19:18]
admin
+++ private:writeups:svd [2020/10/18 21:25] (current)
admin
@@ Line 1: / Line 1: @@
+====== Math ======
-$$X=\begin{bmatrix}
+$$X_{(m,n)}=\begin{bmatrix}
 | & | &  & | \\
 x_1 & x_2 & \dots & x_m  \\
@@ Line 18: / Line 19: @@
 \end{bmatrix}
 \begin{bmatrix}
-\text{---}  & v_1^T & \text{---}  \\
+| & | &  & | \\
-\text{---}  & v_2^T & \text{---}  \\
+v_1 & v_2 & \dots & v_m  \\
-\text{---}  & \vdots & \text{---}  \\
+| & | &  & |
-\text{---}  & v_m^T & \text{---}
+\end{bmatrix}^T
-\end{bmatrix}$$
+$$
-where $n$ is the number of points in each $x$, and $m$ is the number of $x$ values. We're also assuming $n>>m$.
+where $n$ is the number of points in each $x$, and $m$ is the number of $x$ values.
 This can be further expanded into
-$$ X = x_1 \sigma_1 v_1^T + x_2 \sigma_2 v_2^T + \ldots + x_m \sigma_m v_m^T$$
+$$ X = \sigma_1 u_1  v_1^T +  \sigma_2 u_2  v_2^T + \ldots + \sigma_m u_m  v_m^T$$ (assuming $n>>m$).
+We can therefore select the first $r$ terms of this expansion to get a reduced-order approximation of X.  I.e.
+$$ X \approx \tilde{U}\tilde{\Sigma}\tilde{V}^T$$.
+====== Interpretation ======
+Correlation between the rows and columns of X.
+U and V are the eigenvectors of the correlation matrices,$X X^T$ and $X^T X$, respectively.  Because calculating $U$ and $V$ this way is slow, more optimized methods are available.    Which is the inner product of $x_i ^T x_j$ where $i$ and $j$ are the $i$th and $j$th column of $X$.  If the inner product is large, it means that $x_i$ and $x_j$ are similar. If small, it means that  $x_i$ and $x_j$  are different.
+====== Principal component analysis (PCA) ======
+This is the statistical interpretation of SVD.
+PCA finds elipsoids (depending on dimensionality) that account for the largest, second largest, etc. variance in your data.
+Note that X is defined differently as a matrix of horizontally aligned matrix (instead of vertical).
+https://www.youtube.com/watch?v=fkf4IBRSeEc
+Steps.
+.  Compute mean row, X'
+.  Subtract mean from X.  $B=X-X'$
+.  Calc covariance matrix, $C=B^T B$
+.  Computer eigs of C.  Eigenvectors, V, and eigvenvalues, D.
+.  Principal components, T=BV.  (V also called "loadings")
+If $B=U \Sigma V^T$, then $T=U \Sigma$.
+Takeaway, PCA can be found by saving the SVD of (X minus its row-mean).
+====== Least squares, regression, and pseudo-inverse ======
-We can therefore select the first "number" of these terms to
+SVD allows us to generalized Ax=b to non-square A matrices.  (pseudo-inverse)

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