Sparse Approximation Over the Cube
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 $|...
Gespeichert in:
Hauptverfasser: | , , |
---|---|
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext bestellen |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Zusammenfassung: | 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. |
---|---|
DOI: | 10.48550/arxiv.2210.02738 |