# Compressive Sensing with a Multiple Convex Sets Domain

## TL;DR

This paper introduces a flexible framework for compressive sensing that leverages prior knowledge of a signal belonging to a union of convex sets, enhancing recovery guarantees and proposing an efficient algorithm.

## Contribution

It develops a general approach for CS with union-of-convex-sets priors, analyzes measurement bounds, and proposes a novel multiplicative weight update algorithm for signal recovery.

## Key findings

- The prior knowledge reduces the number of measurements needed for unique recovery.
- The proposed algorithm is computationally inexpensive and adaptable.
- Regularizers can further improve signal reconstruction quality.

## Abstract

In this paper, we study a general framework for compressive sensing assuming the existence of the prior knowledge that $\boldsymbol{x}^*$ belongs to the union of multiple convex sets, $\boldsymbol{x}^{*} \in \bigcup_i \mathcal{C}_i$. In fact, by proper choices of these convex sets in the above framework, the problem can be transformed to well known CS problems such as the phase retrieval, quantized compressive sensing, and model-based CS. First we analyze the impact of this prior knowledge on the minimum number of measurements $M$ to guarantee the uniqueness of the solution. Then we formulate a universal objective function for signal recovery, which is both computationally inexpensive and flexible. Then, an algorithm based on \emph{multiplicative weight update} and \emph{proximal gradient descent} is proposed and analyzed for computation and its properties are analyzed for signal reconstruction. Finally, we investigate as to how we can improve the signal recovery by introducing regularizers into the objective function.