Much of the existing literature in sparse recovery is concerned with the following question: given a sparsity pattern and a corresponding regularizer, derive conditions on the dictionary under which exact recovery is possible. In this paper, we study the opposite question: given a dictionary and the $\ell _{1}$ -norm regularizer, find the largest sparsity pattern that can be recovered. We show that such a pattern is described by a mathematical object called a “maximum abstract simplicial complex,” and provide two different characterizations of this object: one based on extreme points and the other based on vectors of minimal support. In addition, we show how this new framework is useful in the study of sparse recovery problems when the dictionary takes the form of a graph incidence matrix or a partial discrete Fourier transform. In case of incidence matrices, we show that the largest sparsity pattern that can be recovered is determined by the set of simple cycles of the graph. As a byproduct, we show that standard sparse recovery can be certified in polynomial time, although this is known to be NP-hard for general matrices. In the case of the partial discrete Fourier transform, our characterization of the largest sparsity pattern that can be recovered requires the unknown signal to be real and its dimension to be a prime number.

中文翻译：

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一范式最小化可以恢复的最大稀疏模式是什么？

稀疏恢复方面的许多现有文献都与以下问题有关：给定稀疏模式和相应的正则化函数，在字典上导出条件，在该条件下可以进行精确恢复。在本文中，我们研究了相反的问题：给定字典和 $ \ ell _ {1} $ -norm正则化器，找到可以恢复的最大稀疏模式。我们证明了这种模式是由称为“最大抽象简单复数”的数学对象描述的，并提供了该对象的两种不同特征：一种基于极端点，另一种基于最小支持向量。此外，当字典采用图入射矩阵或部分离散傅立叶变换的形式时，我们将展示此新框架在稀疏恢复问题的研究中如何有用。在入射矩阵的情况下，我们表明可以恢复的最大稀疏模式是由图形的简单循环集确定的。作为副产品，我们表明可以在多项式时间内验证标准的稀疏恢复，尽管这对于一般矩阵来说是NP难解的。