1,016 research outputs found
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
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
The Sparsity Gap: Uncertainty Principles Proportional to Dimension
In an incoherent dictionary, most signals that admit a sparse representation
admit a unique sparse representation. In other words, there is no way to
express the signal without using strictly more atoms. This work demonstrates
that sparse signals typically enjoy a higher privilege: each nonoptimal
representation of the signal requires far more atoms than the sparsest
representation-unless it contains many of the same atoms as the sparsest
representation. One impact of this finding is to confer a certain degree of
legitimacy on the particular atoms that appear in a sparse representation. This
result can also be viewed as an uncertainty principle for random sparse signals
over an incoherent dictionary.Comment: 6 pages. To appear in the Proceedings of the 44th Ann. IEEE Conf. on
Information Sciences and System
Channel Protection: Random Coding Meets Sparse Channels
Multipath interference is an ubiquitous phenomenon in modern communication
systems. The conventional way to compensate for this effect is to equalize the
channel by estimating its impulse response by transmitting a set of training
symbols. The primary drawback to this type of approach is that it can be
unreliable if the channel is changing rapidly. In this paper, we show that
randomly encoding the signal can protect it against channel uncertainty when
the channel is sparse. Before transmission, the signal is mapped into a
slightly longer codeword using a random matrix. From the received signal, we
are able to simultaneously estimate the channel and recover the transmitted
signal. We discuss two schemes for the recovery. Both of them exploit the
sparsity of the underlying channel. We show that if the channel impulse
response is sufficiently sparse, the transmitted signal can be recovered
reliably.Comment: To appear in the proceedings of the 2009 IEEE Information Theory
Workshop (Taormina
Uncertainty Relations and Sparse Signal Recovery for Pairs of General Signal Sets
We present an uncertainty relation for the representation of signals in two
different general (possibly redundant or incomplete) signal sets. This
uncertainty relation is relevant for the analysis of signals containing two
distinct features each of which can be described sparsely in a suitable general
signal set. Furthermore, the new uncertainty relation is shown to lead to
improved sparsity thresholds for recovery of signals that are sparse in general
dictionaries. Specifically, our results improve on the well-known
-threshold for dictionaries with coherence by up to a factor of
two. Furthermore, we provide probabilistic recovery guarantees for pairs of
general dictionaries that also allow us to understand which parts of a general
dictionary one needs to randomize over to "weed out" the sparsity patterns that
prohibit breaking the square-root bottleneck.Comment: submitted to IEEE Trans. Inf. Theor
Decoding by Linear Programming
This paper considers the classical error correcting problem which is
frequently discussed in coding theory. We wish to recover an input vector from corrupted measurements . Here, is an by
(coding) matrix and is an arbitrary and unknown vector of errors. Is it
possible to recover exactly from the data ? We prove that under suitable
conditions on the coding matrix , the input is the unique solution to
the -minimization problem () provided that the support of the vector of
errors is not too large, for some . In short, can be recovered exactly by solving a
simple convex optimization problem (which one can recast as a linear program).
In addition, numerical experiments suggest that this recovery procedure works
unreasonably well; is recovered exactly even in situations where a
significant fraction of the output is corrupted.Comment: 22 pages, 4 figures, submitte
Optimally Sparse Frames
Frames have established themselves as a means to derive redundant, yet stable
decompositions of a signal for analysis or transmission, while also promoting
sparse expansions. However, when the signal dimension is large, the computation
of the frame measurements of a signal typically requires a large number of
additions and multiplications, and this makes a frame decomposition intractable
in applications with limited computing budget. To address this problem, in this
paper, we focus on frames in finite-dimensional Hilbert spaces and introduce
sparsity for such frames as a new paradigm. In our terminology, a sparse frame
is a frame whose elements have a sparse representation in an orthonormal basis,
thereby enabling low-complexity frame decompositions. To introduce a precise
meaning of optimality, we take the sum of the numbers of vectors needed of this
orthonormal basis when expanding each frame vector as sparsity measure. We then
analyze the recently introduced algorithm Spectral Tetris for construction of
unit norm tight frames and prove that the tight frames generated by this
algorithm are in fact optimally sparse with respect to the standard unit vector
basis. Finally, we show that even the generalization of Spectral Tetris for the
construction of unit norm frames associated with a given frame operator
produces optimally sparse frames
Geometric approach to error correcting codes and reconstruction of signals
We develop an approach through geometric functional analysis to error
correcting codes and to reconstruction of signals from few linear measurements.
An error correcting code encodes an n-letter word x into an m-letter word y in
such a way that x can be decoded correctly when any r letters of y are
corrupted. We prove that most linear orthogonal transformations Q from R^n into
R^m form efficient and robust robust error correcting codes over reals. The
decoder (which corrects the corrupted components of y) is the metric projection
onto the range of Q in the L_1 norm. An equivalent problem arises in signal
processing: how to reconstruct a signal that belongs to a small class from few
linear measurements? We prove that for most sets of Gaussian measurements, all
signals of small support can be exactly reconstructed by the L_1 norm
minimization. This is a substantial improvement of recent results of Donoho and
of Candes and Tao. An equivalent problem in combinatorial geometry is the
existence of a polytope with fixed number of facets and maximal number of
lower-dimensional facets. We prove that most sections of the cube form such
polytopes.Comment: 17 pages, 3 figure
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