Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Analysis of sparse recovery for Legendre expansions using envelope bound   We provide novel sufficient conditions for the uniform recovery of sparse
Legendre expansions using $\ell_1$ minimization, where the sampling points are
drawn according to orthogonalization (uniform) measure. So far, conditions of
the form $m \gtrsim \Theta^2 s \times \textit{log factors}$ have been relied on
to determine the minimum number of samples $m$ that guarantees successful
reconstruction of $s$-sparse vectors when the measurement matrix is associated
to an orthonormal system. However, in case of sparse Legendre expansions, the
uniform bound $\Theta$ of Legendre systems is so high that these conditions are
unable to provide meaningful guarantees. In this paper, we present an analysis
which employs the envelop bound of all Legendre polynomials instead, and prove
a new recovery guarantee for $s$-sparse Legendre expansions, $$ m \gtrsim {s^2}
\times \textit{log factors}, $$ which is independent of $\Theta$. Arguably,
this is the first recovery condition established for orthonormal systems
without assuming the uniform boundedness of the sampling matrix. The key
ingredient of our analysis is an extension of chaining arguments, recently
developed in [Bou14,CDTW15], to handle the envelope bound. Furthermore, our
recovery condition is proved via restricted eigenvalue property, a less
demanding replacement of restricted isometry property which is perfectly suited
to the considered scenario. Along the way, we derive simple criteria to detect
good sample sets. Our numerical tests show that sets of uniformly sampled
points that meet these criteria will perform better recovery on average.

24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
12
The categories of this paper are: nucl-th
21
The categories of this paper are: adap-org
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol