# On the linear convergence rates of exchange and continuous methods for total variation minimization

## TL;DR

This paper studies the linear convergence of exchange and continuous methods for total variation minimization, showing under certain conditions that both approaches converge linearly and proposing a combined alternating method.

## Contribution

It provides the first analysis of linear convergence rates for exchange and continuous algorithms in total variation regularized inverse problems.

## Key findings

- Exchange algorithm converges linearly under regularity conditions.
- Continuous amplitude optimization achieves linear convergence with good initialization.
- Combining both methods offers advantages in convergence and performance.

## Abstract

We analyze an exchange algorithm for the numerical solution total-variation regularized inverse problems over the space M($\Omega$) of Radon measures on a subset $\Omega$ of R d. Our main result states that under some regularity conditions, the method eventually converges linearly. Additionally, we prove that continuously optimizing the amplitudes of positions of the target measure will succeed at a linear rate with a good initialization. Finally, we propose to combine the two approaches into an alternating method and discuss the comparative advantages of this approach.