# A general theory of singular values with applications to signal denoising

## TL;DR

This paper develops a general theory extending singular values to arbitrary finite-dimensional spaces with dual norms, with applications in various signal denoising techniques.

## Contribution

It introduces a unified framework for singular values in general normed spaces, generalizing matrix and tensor decompositions for improved denoising methods.

## Key findings

- Unified theory of singular values for dual norm spaces
- Applications to diverse denoising problems like TV and LASSO
- Enhanced understanding of signal decomposition and noise separation

## Abstract

We study the Pareto frontier for two competing norms $\|\cdot\|_X$ and $\|\cdot\|_Y$ on a vector space. For a given vector $c$, the pareto frontier describes the possible values of $(\|a\|_X,\|b\|_Y)$ for a decomposition $c=a+b$. The singular value decomposition of a matrix is closely related to the Pareto frontier for the spectral and nuclear norm. We will develop a general theory that extends the notion of singular values of a matrix to arbitrary finite dimensional euclidean vector spaces equipped with dual norms. This also generalizes the diagonal singular value decompositions for tensors introduced by the author in previous work. We can apply the results to denoising, where $c$ is a noisy signal, $a$ is a sparse signal and $b$ is noise. Applications include 1D total variation denoising, 2D total variation Rudin-Osher-Fatemi image denoising, LASSO, basis pursuit denoising and tensor decompositions.