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arXiv · 2210.02738

Sparse Approximation Over the Cube

Sabrina Bruckmeier·Christoph Hunkenschröder·Robert Weismantel
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Abstract

This paper presents an anlysis of the NP-hard minimization problem $\min \{\|b - Ax\|_2: \ x \in [0,1]^n, | \text{supp}(x) | \leq \sigma\}$, where $\text{supp}(x) = \{i \in [n]: x_i \neq 0\}$ and $\sigma$ is a positive integer. The object of investigation is a natural relaxation where we replace $| \text{supp}(x) | \leq \sigma$ by $\sum_i x_i \leq \sigma$. Our analysis includes a probabilistic view on when the relaxation is exact. We also consider the problem from a deterministic point of view and provide a bound on the distance between the images of optimal solutions of the original problem and its relaxation under $A$. This leads to an algorithm for generic matrices $A \in \mathbb{Z}^{m \times n}$ and achieves a polynomial running time provided that $m$ and $\|A\|_{\infty}$ are fixed.

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Sabrina Bruckmeier, Christoph Hunkenschröder, Robert Weismantel. 2022-10-06. Sparse Approximation Over the Cube. https://arxiv.org/abs/2210.02738

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