The Trimmed Lasso: Sparse Recovery Guarantees and Practical Optimization by the Generalized Soft-Min Penalty

TL;DR

This paper introduces the trimmed lasso for sparse recovery, proves its theoretical guarantees, and proposes a practical, polynomial-time algorithm using the generalized soft-min surrogate, showing competitive empirical results.

Contribution

It develops a new non-convex penalty for sparse recovery, proves theoretical guarantees, and introduces a computationally feasible surrogate with strong empirical performance.

Findings

Theoretical sparse recovery guarantees for the trimmed lasso.

A polynomial-time algorithm for the generalized soft-min penalty.

Competitive empirical performance compared to state-of-the-art methods.

Abstract

We present a new approach to solve the sparse approximation or best subset selection problem, namely find a $k$-sparse vector ${\bf x}\in\mathbb{R}^d$ that minimizes the $\ell_2$ residual $\lVert A{\bf x}-{\bf y} \rVert_2$. We consider a regularized approach, whereby this residual is penalized by the non-convex $\textit{trimmed lasso}$, defined as the $\ell_1$-norm of ${\bf x}$ excluding its $k$ largest-magnitude entries. We prove that the trimmed lasso has several appealing theoretical properties, and in particular derive sparse recovery guarantees assuming successful optimization of the penalized objective. Next, we show empirically that directly optimizing this objective can be quite challenging. Instead, we propose a surrogate for the trimmed lasso, called the $\textit{generalized soft-min}$. This penalty smoothly interpolates between the classical lasso and the trimmed lasso, while taking into account all possible $k$-sparse patterns. The generalized soft-min penalty involves summation over $\binom{d}{k}$ terms, yet we derive a polynomial-time algorithm to compute it. This, in turn, yields a practical method for the original sparse approximation problem. Via simulations, we demonstrate its competitive performance compared to current state of the art.