1,226 research outputs found
An asymptotic existence result on compressed sensing matrices
For any rational number and all sufficiently large we give a
deterministic construction for an compressed
sensing matrix with -recoverability where . Our
method uses pairwise balanced designs and complex Hadamard matrices in the
construction of -equiangular frames, which we introduce as a
generalisation of equiangular tight frames. The method is general and produces
good compressed sensing matrices from any appropriately chosen pairwise
balanced design. The -recoverability performance is specified as a
simple function of the parameters of the design. To obtain our asymptotic
existence result we prove new results on the existence of pairwise balanced
designs in which the numbers of blocks of each size are specified.Comment: 15 pages, no figures. Minor improvements and updates in February 201
Explicit constructions of RIP matrices and related problems
We give a new explicit construction of matrices satisfying the
Restricted Isometry Property (RIP). Namely, for some c>0, large N and any n
satisfying N^{1-c} < n < N, we construct RIP matrices of order k^{1/2+c}. This
overcomes the natural barrier k=O(n^{1/2}) for proofs based on small coherence,
which are used in all previous explicit constructions of RIP matrices. Key
ingredients in our proof are new estimates for sumsets in product sets and for
exponential sums with the products of sets possessing special additive
structure. We also give a construction of sets of n complex numbers whose k-th
moments are uniformly small for 1\le k\le N (Turan's power sum problem), which
improves upon known explicit constructions when (\log N)^{1+o(1)} \le n\le
(\log N)^{4+o(1)}. This latter construction produces elementary explicit
examples of n by N matrices that satisfy RIP and whose columns constitute a new
spherical code; for those problems the parameters closely match those of
existing constructions in the range (\log N)^{1+o(1)} \le n\le (\log
N)^{5/2+o(1)}.Comment: v3. Minor correction
Universal Spatiotemporal Sampling Sets for Discrete Spatially Invariant Evolution Systems
Let be a finite abelian group and be a circular
convolution operator on . The problem under consideration is how to
construct minimal and such that is
a frame for , where is the canonical
basis of . This problem is motivated by the spatiotemporal sampling
problem in discrete spatially invariant evolution systems. We will show that
the cardinality of should be at least equal to the largest geometric
multiplicity of eigenvalues of , and we consider the universal
spatiotemporal sampling sets for convolution operators
with eigenvalues subject to the same largest geometric
multiplicity. We will give an algebraic characterization for such sampling sets
and show how this problem is linked with sparse signal processing theory and
polynomial interpolation theory
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