Time-Data Tradeoffs in Structured Signals Recovery via the Proximal-Gradient Homotopy Method

TL;DR

This paper analyzes the data-time tradeoffs of the proximal-gradient homotopy method for solving linear inverse problems with sub-Gaussian measurements, showing sharp results and linear convergence under certain conditions.

Contribution

It provides a sharp characterization of data-time tradeoffs and demonstrates linear convergence without strong convexity assumptions for the proximal-gradient homotopy method.

Findings

Achieves linear convergence rate with sufficient measurements

Characterizes data-time tradeoffs sharply

Verifies results through numerical simulations

Abstract

In this paper, we characterize data-time tradeoffs of the proximal-gradient homotopy method used for solving linear inverse problems under sub-Gaussian measurements. Our results are sharp up to an absolute constant factor. We demonstrate that, in the absence of the strong convexity assumption, the proximal-gradient homotopy update can achieve a linear rate of convergence when the number of measurements is sufficiently large. Numerical simulations are provided to verify our theoretical results.