38,503 research outputs found
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
Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions
We develop a robust uncertainty principle for finite signals in C^N which
states that for almost all subsets T,W of {0,...,N-1} such that |T|+|W| ~ (log
N)^(-1/2) N, there is no sigal f supported on T whose discrete Fourier
transform is supported on W. In fact, we can make the above uncertainty
principle quantitative in the sense that if f is supported on T, then only a
small percentage of the energy (less than half, say) of its Fourier transform
is concentrated on W.
As an application of this robust uncertainty principle (QRUP), we consider
the problem of decomposing a signal into a sparse superposition of spikes and
complex sinusoids. We show that if a generic signal f has a decomposition using
spike and frequency locations in T and W respectively, and obeying |T| + |W| <=
C (\log N)^{-1/2} N, then this is the unique sparsest possible decomposition
(all other decompositions have more non-zero terms). In addition, if |T| + |W|
<= C (\log N)^{-1} N, then this sparsest decomposition can be found by solving
a convex optimization problem.Comment: 25 pages, 9 figure
Audio Source Separation Using Sparse Representations
This is the author's final version of the article, first published as A. Nesbit, M. G. Jafari, E. Vincent and M. D. Plumbley. Audio Source Separation Using Sparse Representations. In W. Wang (Ed), Machine Audition: Principles, Algorithms and Systems. Chapter 10, pp. 246-264. IGI Global, 2011. ISBN 978-1-61520-919-4. DOI: 10.4018/978-1-61520-919-4.ch010file: NesbitJafariVincentP11-audio.pdf:n\NesbitJafariVincentP11-audio.pdf:PDF owner: markp timestamp: 2011.02.04file: NesbitJafariVincentP11-audio.pdf:n\NesbitJafariVincentP11-audio.pdf:PDF owner: markp timestamp: 2011.02.04The authors address the problem of audio source separation, namely, the recovery of audio signals from recordings of mixtures of those signals. The sparse component analysis framework is a powerful method for achieving this. Sparse orthogonal transforms, in which only few transform coefficients differ significantly from zero, are developed; once the signal has been transformed, energy is apportioned from each transform coefficient to each estimated source, and, finally, the signal is reconstructed using the inverse transform. The overriding aim of this chapter is to demonstrate how this framework, as exemplified here by two different decomposition methods which adapt to the signal to represent it sparsely, can be used to solve different problems in different mixing scenarios. To address the instantaneous (neither delays nor echoes) and underdetermined (more sources than mixtures) mixing model, a lapped orthogonal transform is adapted to the signal by selecting a basis from a library of predetermined bases. This method is highly related to the windowing methods used in the MPEG audio coding framework. In considering the anechoic (delays but no echoes) and determined (equal number of sources and mixtures) mixing case, a greedy adaptive transform is used based on orthogonal basis functions that are learned from the observed data, instead of being selected from a predetermined library of bases. This is found to encode the signal characteristics, by introducing a feedback system between the bases and the observed data. Experiments on mixtures of speech and music signals demonstrate that these methods give good signal approximations and separation performance, and indicate promising directions for future research
Polar Polytopes and Recovery of Sparse Representations
Suppose we have a signal y which we wish to represent using a linear
combination of a number of basis atoms a_i, y=sum_i x_i a_i = Ax. The problem
of finding the minimum L0 norm representation for y is a hard problem. The
Basis Pursuit (BP) approach proposes to find the minimum L1 norm representation
instead, which corresponds to a linear program (LP) that can be solved using
modern LP techniques, and several recent authors have given conditions for the
BP (minimum L1 norm) and sparse (minimum L0 solutions) representations to be
identical. In this paper, we explore this sparse representation problem} using
the geometry of convex polytopes, as recently introduced into the field by
Donoho. By considering the dual LP we find that the so-called polar polytope P
of the centrally-symmetric polytope P whose vertices are the atom pairs +-a_i
is particularly helpful in providing us with geometrical insight into
optimality conditions given by Fuchs and Tropp for non-unit-norm atom sets. In
exploring this geometry we are able to tighten some of these earlier results,
showing for example that the Fuchs condition is both necessary and sufficient
for L1-unique-optimality, and that there are situations where Orthogonal
Matching Pursuit (OMP) can eventually find all L1-unique-optimal solutions with
m nonzeros even if ERC fails for m, if allowed to run for more than m steps
Uncertainty Relations for Shift-Invariant Analog Signals
The past several years have witnessed a surge of research investigating
various aspects of sparse representations and compressed sensing. Most of this
work has focused on the finite-dimensional setting in which the goal is to
decompose a finite-length vector into a given finite dictionary. Underlying
many of these results is the conceptual notion of an uncertainty principle: a
signal cannot be sparsely represented in two different bases. Here, we extend
these ideas and results to the analog, infinite-dimensional setting by
considering signals that lie in a finitely-generated shift-invariant (SI)
space. This class of signals is rich enough to include many interesting special
cases such as multiband signals and splines. By adapting the notion of
coherence defined for finite dictionaries to infinite SI representations, we
develop an uncertainty principle similar in spirit to its finite counterpart.
We demonstrate tightness of our bound by considering a bandlimited lowpass
train that achieves the uncertainty principle. Building upon these results and
similar work in the finite setting, we show how to find a sparse decomposition
in an overcomplete dictionary by solving a convex optimization problem. The
distinguishing feature of our approach is the fact that even though the problem
is defined over an infinite domain with infinitely many variables and
constraints, under certain conditions on the dictionary spectrum our algorithm
can find the sparsest representation by solving a finite-dimensional problem.Comment: Accepted to IEEE Trans. on Inform. Theor
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