# Research Report: Exact biconvex reformulation of the $\ell_2-\ell_0$ minimization problem

## TL;DR

This paper introduces an exact biconvex reformulation of the $ ext{l}_2- ext{l}_0$ minimization problem, enabling improved algorithms for sparse optimization with applications in microscopy.

## Contribution

It presents a novel exact biconvex reformulation of the $ ext{l}_0$ minimization problem, facilitating more effective optimization algorithms.

## Key findings

- The reformulation is exact and biconvex, preserving solutions.
- The proposed algorithm outperforms Iterative Hard Thresholding.
- Application to microscopy demonstrates improved results.

## Abstract

We focus on the minimization of the least square loss function either under a $k$-sparse constraint or with a sparse penalty term. Based on recent results, we reformulate the $\ell_0$ pseudo-norm exactly as a convex minimization problem by introducing an auxiliary variable. We then propose an exact biconvex reformulation of the $\ell_2-\ell_0$ constrained and penalized problems. We give correspondence results between minimizers of the initial function and the reformulated ones. The reformulation is biconvex and the non-convexity is due to a penalty term. These two properties are used to derive a minimization algorithm. We apply the algorithm to the problem of single-molecule localization microscopy and compare the results with the well-known Iterative Hard Thresholding algorithm. Visually and numerically the biconvex reformulations perform better.