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We present a new approach to solve the sparse approximation or best subsetselection problem, namely find a $k$-sparse vector ${\bf x}\in\mathbb{R}^d$that minimizes the $\ell_2$ residual $\lVert A{\bf x}-{\bf y} \rVert_2$. Weconsider a regularized approach, whereby this residual is penalized by thenon-convex $\textit{trimmed lasso}$, defined as the $\ell_1$-norm of ${\bf x}$excluding its $k$ largest-magnitude entries. We prove that the trimmed lassohas several appealing theoretical properties, and in particular derive sparserecovery guarantees assuming successful optimization of the penalizedobjective. Next, we show empirically that directly optimizing this objectivecan be quite challenging. Instead, we propose a surrogate for the trimmedlasso, called the $\textit{generalized soft-min}$. This penalty smoothlyinterpolates between the classical lasso and the trimmed lasso, while takinginto account all possible $k$-sparse patterns. The generalized soft-min penaltyinvolves summation over $\binom{d}{k}$ terms, yet we derive a polynomial-timealgorithm to compute it. This, in turn, yields a practical method for theoriginal sparse approximation problem. Via simulations, we demonstrate itscompetitive performance compared to current state of the art.

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