# A Dictionary Based Generalization of Robust PCA

## TL;DR

This paper extends Robust PCA to handle data decompositions where the sparse component is represented in a known dictionary, providing theoretical guarantees and empirical validation for successful recovery.

## Contribution

It introduces a unified convex demixing framework for low-rank plus dictionary-sparse components, covering both undercomplete and overcomplete dictionaries.

## Key findings

- Successful recovery under mild assumptions up to a certain sparsity level.
- Empirical phase transition results validate theoretical guarantees.
- Framework applicable to various dictionary sizes.

## Abstract

We analyze the decomposition of a data matrix, assumed to be a superposition of a low-rank component and a component which is sparse in a known dictionary, using a convex demixing method. We provide a unified analysis, encompassing both undercomplete and overcomplete dictionary cases, and show that the constituent components can be successfully recovered under some relatively mild assumptions up to a certain $\textit{global}$ sparsity level. Further, we corroborate our theoretical results by presenting empirical evaluations in terms of phase transitions in rank and sparsity for various dictionary sizes.