Learning Trees of $\ell_0$-Minimization Problems

TL;DR

This paper explores a learning-based approach to solve $ ext{NP}$-hard $ ext{l}_0$-minimization problems by training on curricula of samples, aiming to mimic human mathematician learning processes.

Contribution

It introduces adaptable classes for $ ext{l}_0$-minimization that become tractable after training on increasingly difficult samples, modeling human learning.

Findings

Proposes a curriculum-based training framework for $ ext{l}_0$-minimization.

Demonstrates potential for adaptive algorithms to solve complex sparse recovery problems.

Bridges the gap between theoretical hardness and practical solvability through learning.

Abstract

The problem of computing minimally sparse solutions of under-determined linear systems is $NP$ hard in general. Subsets with extra properties, may allow efficient algorithms, most notably problems with the restricted isometry property (RIP) can be solved by convex $\ell_1$-minimization. While these classes have been very successful, they leave out many practical applications. In this paper, we consider adaptable classes that are tractable after training on a curriculum of increasingly difficult samples. The setup is intended as a candidate model for a human mathematician, who may not be able to tackle an arbitrary proof right away, but may be successful in relatively flexible subclasses, or areas of expertise, after training on a suitable curriculum.