Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: What is the Largest Sparsity Pattern that Can Be Recovered by 1-Norm
  Minimization?   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 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.

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The categories of this paper are: q-alg
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The categories of this paper are: nucl-th
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The categories of this paper are: solv-int
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The categories of this paper are: physics
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-ph
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn