💡 Colab Link

<aside> ❓ Goal: Improve understanding of the Singular Value Decomposition (SVD) using low-rank approximations to compress images.

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Summary

The Eckart-Young Theorem states that the best rank $r$ approximation of a matrix can be found using the Singular Value Decomposition and computing the following:

$$ \begin{align*}\hat{A} = \sum_{i=1}^r \sigma_i u_i v_i^\top\end{align*} $$

So, we can perform basic image compression by computing the SVD of the image and specifying our desired rank $r$ approximation. More detail on the Singular Value Decomposition is included in this writeup.

The linked Colab Notebook demonstrates this.