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private:writeups:svd
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$$X=\begin{bmatrix} | & | & & | \\ x_1 & x_2 & \dots & x_m \\ | & | & & | \end{bmatrix} =U\Sigma V^T =\begin{bmatrix} | & | & & | \\ u_1 & u_2 & \dots & u_n \\ | & | & & | \end{bmatrix} \begin{bmatrix} \sigma_1 & & \\ & \ddots & \\ & & \sigma_m \\ & & \end{bmatrix} \begin{bmatrix} \text{---} & v_1^T & \text{---} \\ \text{---} & v_2^T & \text{---} \\ \text{---} & \vdots & \text{---} \\ \text{---} & v_m^T & \text{---} \end{bmatrix}$$
where $n$ is the number of points in each $x$, and $m$ is the number of $x$ values. We're also assuming $n>>m$.
This can be further expanded into
$$ X = x_1 \sigma_1 v_1^T + x_2 \sigma_2 v_2^T + \ldots + x_m \sigma_m v_m^T$$
We can therefore select the first “number” of these terms to
private/writeups/svd.1603048719.txt.gz · Last modified: 2020/10/18 19:18 by admin
