# On the Phase Transition of Corrupted Sensing

## TL;DR

This paper provides a theoretical explanation for the sharp phase transition observed in corrupted sensing problems, identifying the threshold where convex recovery methods succeed or fail, supported by numerical validation.

## Contribution

It establishes the precise threshold for successful recovery in corrupted sensing, linking it to the Gaussian widths of tangent cones, thus explaining the phase transition phenomenon.

## Key findings

- Sharp phase transition occurs around the sum of Gaussian widths squared.
- Theoretical thresholds match numerical experiments.
- Convex procedures fail or succeed based on this threshold.

## Abstract

In \cite{FOY2014}, a sharp phase transition has been numerically observed when a constrained convex procedure is used to solve the corrupted sensing problem. In this paper, we present a theoretical analysis for this phenomenon. Specifically, we establish the threshold below which this convex procedure fails to recover signal and corruption with high probability. Together with the work in \cite{FOY2014}, we prove that a sharp phase transition occurs around the sum of the squares of spherical Gaussian widths of two tangent cones. Numerical experiments are provided to demonstrate the correctness and sharpness of our results.