77 research outputs found
Joint Sparsity with Different Measurement Matrices
We consider a generalization of the multiple measurement vector (MMV)
problem, where the measurement matrices are allowed to differ across
measurements. This problem arises naturally when multiple measurements are
taken over time, e.g., and the measurement modality (matrix) is time-varying.
We derive probabilistic recovery guarantees showing that---under certain (mild)
conditions on the measurement matrices---l2/l1-norm minimization and a variant
of orthogonal matching pursuit fail with a probability that decays
exponentially in the number of measurements. This allows us to conclude that,
perhaps surprisingly, recovery performance does not suffer from the individual
measurements being taken through different measurement matrices. What is more,
recovery performance typically benefits (significantly) from diversity in the
measurement matrices; we specify conditions under which such improvements are
obtained. These results continue to hold when the measurements are subject to
(bounded) noise.Comment: Allerton 201
On Unlimited Sampling
Shannon's sampling theorem provides a link between the continuous and the
discrete realms stating that bandlimited signals are uniquely determined by its
values on a discrete set. This theorem is realized in practice using so called
analog--to--digital converters (ADCs). Unlike Shannon's sampling theorem, the
ADCs are limited in dynamic range. Whenever a signal exceeds some preset
threshold, the ADC saturates, resulting in aliasing due to clipping. The goal
of this paper is to analyze an alternative approach that does not suffer from
these problems. Our work is based on recent developments in ADC design, which
allow for ADCs that reset rather than to saturate, thus producing modulo
samples. An open problem that remains is: Given such modulo samples of a
bandlimited function as well as the dynamic range of the ADC, how can the
original signal be recovered and what are the sufficient conditions that
guarantee perfect recovery? In this paper, we prove such sufficiency conditions
and complement them with a stable recovery algorithm. Our results are not
limited to certain amplitude ranges, in fact even the same circuit architecture
allows for the recovery of arbitrary large amplitudes as long as some estimate
of the signal norm is available when recovering. Numerical experiments that
corroborate our theory indeed show that it is possible to perfectly recover
function that takes values that are orders of magnitude higher than the ADC's
threshold.Comment: 11 pages, 4 figures, copy of initial version to appear in Proceedings
of 12th International Conference on Sampling Theory and Applications (SampTA
A Dimension Reduction Scheme for the Computation of Optimal Unions of Subspaces
Given a set of points \F in a high dimensional space, the problem of finding
a union of subspaces \cup_i V_i\subset \R^N that best explains the data \F
increases dramatically with the dimension of \R^N. In this article, we study a
class of transformations that map the problem into another one in lower
dimension. We use the best model in the low dimensional space to approximate
the best solution in the original high dimensional space. We then estimate the
error produced between this solution and the optimal solution in the high
dimensional space.Comment: 15 pages. Some corrections were added, in particular the title was
changed. It will appear in "Sampling Theory in Signal and Image Processing
Nearness to Local Subspace Algorithm for Subspace and Motion Segmentation
There is a growing interest in computer science, engineering, and mathematics
for modeling signals in terms of union of subspaces and manifolds. Subspace
segmentation and clustering of high dimensional data drawn from a union of
subspaces are especially important with many practical applications in computer
vision, image and signal processing, communications, and information theory.
This paper presents a clustering algorithm for high dimensional data that comes
from a union of lower dimensional subspaces of equal and known dimensions. Such
cases occur in many data clustering problems, such as motion segmentation and
face recognition. The algorithm is reliable in the presence of noise, and
applied to the Hopkins 155 Dataset, it generates the best results to date for
motion segmentation. The two motion, three motion, and overall segmentation
rates for the video sequences are 99.43%, 98.69%, and 99.24%, respectively
On the Effective Measure of Dimension in the Analysis Cosparse Model
Many applications have benefited remarkably from low-dimensional models in
the recent decade. The fact that many signals, though high dimensional, are
intrinsically low dimensional has given the possibility to recover them stably
from a relatively small number of their measurements. For example, in
compressed sensing with the standard (synthesis) sparsity prior and in matrix
completion, the number of measurements needed is proportional (up to a
logarithmic factor) to the signal's manifold dimension.
Recently, a new natural low-dimensional signal model has been proposed: the
cosparse analysis prior. In the noiseless case, it is possible to recover
signals from this model, using a combinatorial search, from a number of
measurements proportional to the signal's manifold dimension. However, if we
ask for stability to noise or an efficient (polynomial complexity) solver, all
the existing results demand a number of measurements which is far removed from
the manifold dimension, sometimes far greater. Thus, it is natural to ask
whether this gap is a deficiency of the theory and the solvers, or if there
exists a real barrier in recovering the cosparse signals by relying only on
their manifold dimension. Is there an algorithm which, in the presence of
noise, can accurately recover a cosparse signal from a number of measurements
proportional to the manifold dimension? In this work, we prove that there is no
such algorithm. Further, we show through numerical simulations that even in the
noiseless case convex relaxations fail when the number of measurements is
comparable to the manifold dimension. This gives a practical counter-example to
the growing literature on compressed acquisition of signals based on manifold
dimension.Comment: 19 pages, 6 figure
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