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There are a lot of empirical results about that truncated SVD (TSVD) can help denoise the noises of images, but I wonder what is the theoretical support behind that?

We know that TSVD is the best low-rank matrix approximation method in terms of Frobenius norm and spectral norm, but why does this have anything to do with denoising?

Formally, say we have an image $A$, and a matrix of noises $E$ where each element $e$ in $E$ follows $e \sim N(0, \sigma^2)$, and we have a corrupted image $B = A + E$, we apply TSVD to denoise, the denoising performance must have something to do with $A$ and $\sigma^2$, I wonder where I can find some theoretical study about it.

So, overall, I hope I can find the answer to these two questions:

  • Why small errors in terms of Frobenius norm or spectral norm are related to denoising performance?
  • What is the relationship between denoising performance and $A$ and $\sigma^2$?
kjetil b halvorsen
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    Not a duplicate, but my answer at SVD? give some information, and indeed an example where the "noise" removed can be quite important/interesting – kjetil b halvorsen Oct 10 '18 at 22:38
  • See this question (https://mathoverflow.net/questions/373616/perturbation-bound-for-svd-denoising-for-a-low-rank-matrix) for a formal statement of how TSVD de-noises a low-rank matrix plus noise. – Holden Lee Oct 12 '20 at 21:15

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