Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Simultaneously Structured Models with Application to Sparse and Low-rank
  Matrices   The topic of recovery of a structured model given a small number of linear
observations has been well-studied in recent years. Examples include recovering
sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and
low-rank matrices, among others. In various applications in signal processing
and machine learning, the model of interest is known to be structured in
several ways at the same time, for example, a matrix that is simultaneously
sparse and low-rank.
  Often norms that promote each individual structure are known, and allow for
recovery using an order-wise optimal number of measurements (e.g., $\ell_1$
norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to
minimize a combination of such norms. We show that, surprisingly, if we use
multi-objective optimization with these norms, then we can do no better,
order-wise, than an algorithm that exploits only one of the present structures.
This result suggests that to fully exploit the multiple structures, we need an
entirely new convex relaxation, i.e. not one that is a function of the convex
relaxations used for each structure. We then specialize our results to the case
of sparse and low-rank matrices. We show that a nonconvex formulation of the
problem can recover the model from very few measurements, which is on the order
of the degrees of freedom of the matrix, whereas the convex problem obtained
from a combination of the $\ell_1$ and nuclear norms requires many more
measurements. This proves an order-wise gap between the performance of the
convex and nonconvex recovery problems in this case. Our framework applies to
arbitrary structure-inducing norms as well as to a wide range of measurement
ensembles. This allows us to give performance bounds for problems such as
sparse phase retrieval and low-rank tensor completion.

3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
13
The categories of this paper are: quant-ph
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
26
The categories of this paper are: bayes-an
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The categories of this paper are: q-fin
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph