257 research outputs found
New Analysis of Manifold Embeddings and Signal Recovery from Compressive Measurements
Compressive Sensing (CS) exploits the surprising fact that the information
contained in a sparse signal can be preserved in a small number of compressive,
often random linear measurements of that signal. Strong theoretical guarantees
have been established concerning the embedding of a sparse signal family under
a random measurement operator and on the accuracy to which sparse signals can
be recovered from noisy compressive measurements. In this paper, we address
similar questions in the context of a different modeling framework. Instead of
sparse models, we focus on the broad class of manifold models, which can arise
in both parametric and non-parametric signal families. Using tools from the
theory of empirical processes, we improve upon previous results concerning the
embedding of low-dimensional manifolds under random measurement operators. We
also establish both deterministic and probabilistic instance-optimal bounds in
for manifold-based signal recovery and parameter estimation from noisy
compressive measurements. In line with analogous results for sparsity-based CS,
we conclude that much stronger bounds are possible in the probabilistic
setting. Our work supports the growing evidence that manifold-based models can
be used with high accuracy in compressive signal processing.Comment: arXiv admin note: substantial text overlap with arXiv:1002.124
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
Dimensionality reduction with subgaussian matrices: a unified theory
We present a theory for Euclidean dimensionality reduction with subgaussian
matrices which unifies several restricted isometry property and
Johnson-Lindenstrauss type results obtained earlier for specific data sets. In
particular, we recover and, in several cases, improve results for sets of
sparse and structured sparse vectors, low-rank matrices and tensors, and smooth
manifolds. In addition, we establish a new Johnson-Lindenstrauss embedding for
data sets taking the form of an infinite union of subspaces of a Hilbert space
The generalized Lasso with non-linear observations
We study the problem of signal estimation from non-linear observations when
the signal belongs to a low-dimensional set buried in a high-dimensional space.
A rough heuristic often used in practice postulates that non-linear
observations may be treated as noisy linear observations, and thus the signal
may be estimated using the generalized Lasso. This is appealing because of the
abundance of efficient, specialized solvers for this program. Just as noise may
be diminished by projecting onto the lower dimensional space, the error from
modeling non-linear observations with linear observations will be greatly
reduced when using the signal structure in the reconstruction. We allow general
signal structure, only assuming that the signal belongs to some set K in R^n.
We consider the single-index model of non-linearity. Our theory allows the
non-linearity to be discontinuous, not one-to-one and even unknown. We assume a
random Gaussian model for the measurement matrix, but allow the rows to have an
unknown covariance matrix. As special cases of our results, we recover
near-optimal theory for noisy linear observations, and also give the first
theoretical accuracy guarantee for 1-bit compressed sensing with unknown
covariance matrix of the measurement vectors.Comment: 21 page
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