37 research outputs found
Almost Lossless Analog Signal Separation
We propose an information-theoretic framework for analog signal separation.
Specifically, we consider the problem of recovering two analog signals from a
noiseless sum of linear measurements of the signals. Our framework is inspired
by the groundbreaking work of Wu and Verd\'u (2010) on almost lossless analog
compression. The main results of the present paper are a general achievability
bound for the compression rate in the analog signal separation problem, an
exact expression for the optimal compression rate in the case of signals that
have mixed discrete-continuous distributions, and a new technique for showing
that the intersection of generic subspaces with subsets of sufficiently small
Minkowski dimension is empty. This technique can also be applied to obtain a
simplified proof of a key result in Wu and Verd\'u (2010).Comment: To be presented at IEEE Int. Symp. Inf. Theory 2013, Istanbul, Turke
Compressed Sensing with Coherent and Redundant Dictionaries
This article presents novel results concerning the recovery of signals from
undersampled data in the common situation where such signals are not sparse in
an orthonormal basis or incoherent dictionary, but in a truly redundant
dictionary. This work thus bridges a gap in the literature and shows not only
that compressed sensing is viable in this context, but also that accurate
recovery is possible via an L1-analysis optimization problem. We introduce a
condition on the measurement/sensing matrix, which is a natural generalization
of the now well-known restricted isometry property, and which guarantees
accurate recovery of signals that are nearly sparse in (possibly) highly
overcomplete and coherent dictionaries. This condition imposes no incoherence
restriction on the dictionary and our results may be the first of this kind. We
discuss practical examples and the implications of our results on those
applications, and complement our study by demonstrating the potential of
L1-analysis for such problems
Efficient Resolution of Anisotropic Structures
We highlight some recent new delevelopments concerning the sparse
representation of possibly high-dimensional functions exhibiting strong
anisotropic features and low regularity in isotropic Sobolev or Besov scales.
Specifically, we focus on the solution of transport equations which exhibit
propagation of singularities where, additionally, high-dimensionality enters
when the convection field, and hence the solutions, depend on parameters
varying over some compact set. Important constituents of our approach are
directionally adaptive discretization concepts motivated by compactly supported
shearlet systems, and well-conditioned stable variational formulations that
support trial spaces with anisotropic refinements with arbitrary
directionalities. We prove that they provide tight error-residual relations
which are used to contrive rigorously founded adaptive refinement schemes which
converge in . Moreover, in the context of parameter dependent problems we
discuss two approaches serving different purposes and working under different
regularity assumptions. For frequent query problems, making essential use of
the novel well-conditioned variational formulations, a new Reduced Basis Method
is outlined which exhibits a certain rate-optimal performance for indefinite,
unsymmetric or singularly perturbed problems. For the radiative transfer
problem with scattering a sparse tensor method is presented which mitigates or
even overcomes the curse of dimensionality under suitable (so far still
isotropic) regularity assumptions. Numerical examples for both methods
illustrate the theoretical findings