Low-rank Approximation with SVD on a Kernel Matrix

Your work is an example of low rank approximation.

Solution

Use a threshold. The singular value decomposition, by construction, sorts the singular values: $$ \sigma_{1} \ge \sigma_{2} \ge \dots \ge \sigma_{\rho} > 0 $$ Mathematically, you want the rank to be $k$, the numerical rank is $\rho$. The extraneous singular values are artifacts of finite precision arithmetic.

Take the first 65 or the first 75. Pick a threshold.

Seeing the low rank approximation

A sequence of low rank approximations follows. We see the input image, the low rank approximation, the error, and the threshold for the spectrum. The image matrix has rank 266. Using a rank of 25 produces a reasonable approximation.