3 research outputs found
Statistical RIP and Semi-Circle Distribution of Incoherent Dictionaries
In this paper we formulate and prove a statistical version of the Candes-Tao
restricted isometry property (SRIP for short) which holds in general for any
incoherent dictionary which is a disjoint union of orthonormal bases. In
addition, we prove that, under appropriate normalization, the eigenvalues of
the associated Gram matrix fluctuate around 1 according to the Wigner
semicircle distribution. The result is then applied to various dictionaries
that arise naturally in the setting of finite harmonic analysis, giving, in
particular, a better understanding on a remark of
Applebaum-Howard-Searle-Calderbank concerning RIP for the Heisenberg dictionary
of chirp like functions.Comment: This version Includes all the proofs. Submitted for publication
(2009
Compressive neural representation of sparse, high-dimensional probabilities
This paper shows how sparse, high-dimensional probability distributions could
be represented by neurons with exponential compression. The representation is a
novel application of compressive sensing to sparse probability distributions
rather than to the usual sparse signals. The compressive measurements
correspond to expected values of nonlinear functions of the probabilistically
distributed variables. When these expected values are estimated by sampling,
the quality of the compressed representation is limited only by the quality of
sampling. Since the compression preserves the geometric structure of the space
of sparse probability distributions, probabilistic computation can be performed
in the compressed domain. Interestingly, functions satisfying the requirements
of compressive sensing can be implemented as simple perceptrons. If we use
perceptrons as a simple model of feedforward computation by neurons, these
results show that the mean activity of a relatively small number of neurons can
accurately represent a high-dimensional joint distribution implicitly, even
without accounting for any noise correlations. This comprises a novel
hypothesis for how neurons could encode probabilities in the brain.Comment: 9 pages, 4 figure
Random Subdictionaries and Coherence Conditions for Sparse Signal Recovery
The most frequently used condition for sampling matrices employed in
compressive sampling is the restricted isometry (RIP) property of the matrix
when restricted to sparse signals. At the same time, imposing this condition
makes it difficult to find explicit matrices that support recovery of signals
from sketches of the optimal (smallest possible)dimension. A number of attempts
have been made to relax or replace the RIP property in sparse recovery
algorithms. We focus on the relaxation under which the near-isometry property
holds for most rather than for all submatrices of the sampling matrix, known as
statistical RIP or StRIP condition. We show that sampling matrices of
dimensions with maximum coherence and
mean square coherence support stable recovery of
-sparse signals using Basis Pursuit. These assumptions are satisfied in many
examples. As a result, we are able to construct sampling matrices that support
recovery with low error for sparsity higher than which exceeds
the range of parameters of the known classes of RIP matrices