6,472 research outputs found

### Sampling Sparse Signals on the Sphere: Algorithms and Applications

We propose a sampling scheme that can perfectly reconstruct a collection of
spikes on the sphere from samples of their lowpass-filtered observations.
Central to our algorithm is a generalization of the annihilating filter method,
a tool widely used in array signal processing and finite-rate-of-innovation
(FRI) sampling. The proposed algorithm can reconstruct $K$ spikes from
$(K+\sqrt{K})^2$ spatial samples. This sampling requirement improves over
previously known FRI sampling schemes on the sphere by a factor of four for
large $K$. We showcase the versatility of the proposed algorithm by applying it
to three different problems: 1) sampling diffusion processes induced by
localized sources on the sphere, 2) shot noise removal, and 3) sound source
localization (SSL) by a spherical microphone array. In particular, we show how
SSL can be reformulated as a spherical sparse sampling problem.Comment: 14 pages, 8 figures, submitted to IEEE Transactions on Signal
Processin

### 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 $K$ of the signal satisfies
$K<1/\mu(D)$, where $\mu(D)$ is the mutual coherence of the dictionary.
Furthermore, the sparse representation can be obtained in polynomial time by
Basis Pursuit (BP), when $K<0.91/\mu(D)$. 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 $K$-sparse
representations of a signal under the weaker condition that $K<\sqrt{2}
/\mu(D)$. Consequently, when $K<1/\mu(D)$, 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

### Generalized Approximate Survey Propagation for High-Dimensional Estimation

In Generalized Linear Estimation (GLE) problems, we seek to estimate a signal
that is observed through a linear transform followed by a component-wise,
possibly nonlinear and noisy, channel. In the Bayesian optimal setting,
Generalized Approximate Message Passing (GAMP) is known to achieve optimal
performance for GLE. However, its performance can significantly degrade
whenever there is a mismatch between the assumed and the true generative model,
a situation frequently encountered in practice. In this paper, we propose a new
algorithm, named Generalized Approximate Survey Propagation (GASP), for solving
GLE in the presence of prior or model mis-specifications. As a prototypical
example, we consider the phase retrieval problem, where we show that GASP
outperforms the corresponding GAMP, reducing the reconstruction threshold and,
for certain choices of its parameters, approaching Bayesian optimal
performance. Furthermore, we present a set of State Evolution equations that
exactly characterize the dynamics of GASP in the high-dimensional limit

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