4,209 research outputs found
An algebraic perspective on integer sparse recovery
Compressed sensing is a relatively new mathematical paradigm that shows a
small number of linear measurements are enough to efficiently reconstruct a
large dimensional signal under the assumption the signal is sparse.
Applications for this technology are ubiquitous, ranging from wireless
communications to medical imaging, and there is now a solid foundation of
mathematical theory and algorithms to robustly and efficiently reconstruct such
signals. However, in many of these applications, the signals of interest do not
only have a sparse representation, but have other structure such as
lattice-valued coefficients. While there has been a small amount of work in
this setting, it is still not very well understood how such extra information
can be utilized during sampling and reconstruction. Here, we explore the
problem of integer sparse reconstruction, lending insight into when this
knowledge can be useful, and what types of sampling designs lead to robust
reconstruction guarantees. We use a combination of combinatorial, probabilistic
and number-theoretic methods to discuss existence and some constructions of
such sensing matrices with concrete examples. We also prove sparse versions of
Minkowski's Convex Body and Linear Forms theorems that exhibit some limitations
of this framework
An extremal problem for integer sparse recovery
Motivated by the problem of integer sparse recovery we study the following
question. Let be an integer matrix whose entries are in
absolute value at most . How large can be if all
submatrices of are non-degenerate? We obtain new upper and lower bounds on
and answer a special case of the problem by Brass, Moser and Pach on
covering -dimensional grid by linear subspaces
Rank Minimization over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations
This paper establishes information-theoretic limits in estimating a finite
field low-rank matrix given random linear measurements of it. These linear
measurements are obtained by taking inner products of the low-rank matrix with
random sensing matrices. Necessary and sufficient conditions on the number of
measurements required are provided. It is shown that these conditions are sharp
and the minimum-rank decoder is asymptotically optimal. The reliability
function of this decoder is also derived by appealing to de Caen's lower bound
on the probability of a union. The sufficient condition also holds when the
sensing matrices are sparse - a scenario that may be amenable to efficient
decoding. More precisely, it is shown that if the n\times n-sensing matrices
contain, on average, \Omega(nlog n) entries, the number of measurements
required is the same as that when the sensing matrices are dense and contain
entries drawn uniformly at random from the field. Analogies are drawn between
the above results and rank-metric codes in the coding theory literature. In
fact, we are also strongly motivated by understanding when minimum rank
distance decoding of random rank-metric codes succeeds. To this end, we derive
distance properties of equiprobable and sparse rank-metric codes. These
distance properties provide a precise geometric interpretation of the fact that
the sparse ensemble requires as few measurements as the dense one. Finally, we
provide a non-exhaustive procedure to search for the unknown low-rank matrix.Comment: Accepted to the IEEE Transactions on Information Theory; Presented at
IEEE International Symposium on Information Theory (ISIT) 201
On Sparse Representation in Fourier and Local Bases
We consider the classical problem of finding the sparse representation of a
signal in a pair of bases. When both bases are orthogonal, it is known that the
sparse representation is unique when the sparsity of the signal satisfies
, where is the mutual coherence of the dictionary.
Furthermore, the sparse representation can be obtained in polynomial time by
Basis Pursuit (BP), when . Therefore, there is a gap between the
unicity condition and the one required to use the polynomial-complexity BP
formulation. For the case of general dictionaries, it is also well known that
finding the sparse representation under the only constraint of unicity is
NP-hard.
In this paper, we introduce, for the case of Fourier and canonical bases, a
polynomial complexity algorithm that finds all the possible -sparse
representations of a signal under the weaker condition that . Consequently, when , the proposed algorithm solves the
unique sparse representation problem for this structured dictionary in
polynomial time. We further show that the same method can be extended to many
other pairs of bases, one of which must have local atoms. Examples include the
union of Fourier and local Fourier bases, the union of discrete cosine
transform and canonical bases, and the union of random Gaussian and canonical
bases
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