139 research outputs found
Tail bounds via generic chaining
We modify Talagrand's generic chaining method to obtain upper bounds for all
p-th moments of the supremum of a stochastic process. These bounds lead to an
estimate for the upper tail of the supremum with optimal deviation parameters.
We apply our procedure to improve and extend some known deviation inequalities
for suprema of unbounded empirical processes and chaos processes. As an
application we give a significantly simplified proof of the restricted isometry
property of the subsampled discrete Fourier transform.Comment: Added detailed proof of Theorem 3.5; Application to dimensionality
reduction expanded and moved to separate note arXiv:1402.397
Modulated Unit-Norm Tight Frames for Compressed Sensing
In this paper, we propose a compressed sensing (CS) framework that consists
of three parts: a unit-norm tight frame (UTF), a random diagonal matrix and a
column-wise orthonormal matrix. We prove that this structure satisfies the
restricted isometry property (RIP) with high probability if the number of
measurements for -sparse signals of length
and if the column-wise orthonormal matrix is bounded. Some existing structured
sensing models can be studied under this framework, which then gives tighter
bounds on the required number of measurements to satisfy the RIP. More
importantly, we propose several structured sensing models by appealing to this
unified framework, such as a general sensing model with arbitrary/determinisic
subsamplers, a fast and efficient block compressed sensing scheme, and
structured sensing matrices with deterministic phase modulations, all of which
can lead to improvements on practical applications. In particular, one of the
constructions is applied to simplify the transceiver design of CS-based channel
estimation for orthogonal frequency division multiplexing (OFDM) systems.Comment: submitted to IEEE Transactions on Signal Processin
Quantized Compressed Sensing for Partial Random Circulant Matrices
We provide the first analysis of a non-trivial quantization scheme for
compressed sensing measurements arising from structured measurements.
Specifically, our analysis studies compressed sensing matrices consisting of
rows selected at random, without replacement, from a circulant matrix generated
by a random subgaussian vector. We quantize the measurements using stable,
possibly one-bit, Sigma-Delta schemes, and use a reconstruction method based on
convex optimization. We show that the part of the reconstruction error due to
quantization decays polynomially in the number of measurements. This is in line
with analogous results on Sigma-Delta quantization associated with random
Gaussian or subgaussian matrices, and significantly better than results
associated with the widely assumed memoryless scalar quantization. Moreover, we
prove that our approach is stable and robust; i.e., the reconstruction error
degrades gracefully in the presence of non-quantization noise and when the
underlying signal is not strictly sparse. The analysis relies on results
concerning subgaussian chaos processes as well as a variation of McDiarmid's
inequality.Comment: 15 page
Uniform Recovery from Subgaussian Multi-Sensor Measurements
Parallel acquisition systems are employed successfully in a variety of
different sensing applications when a single sensor cannot provide enough
measurements for a high-quality reconstruction. In this paper, we consider
compressed sensing (CS) for parallel acquisition systems when the individual
sensors use subgaussian random sampling. Our main results are a series of
uniform recovery guarantees which relate the number of measurements required to
the basis in which the solution is sparse and certain characteristics of the
multi-sensor system, known as sensor profile matrices. In particular, we derive
sufficient conditions for optimal recovery, in the sense that the number of
measurements required per sensor decreases linearly with the total number of
sensors, and demonstrate explicit examples of multi-sensor systems for which
this holds. We establish these results by proving the so-called Asymmetric
Restricted Isometry Property (ARIP) for the sensing system and use this to
derive both nonuniversal and universal recovery guarantees. Compared to
existing work, our results not only lead to better stability and robustness
estimates but also provide simpler and sharper constants in the measurement
conditions. Finally, we show how the problem of CS with block-diagonal sensing
matrices can be viewed as a particular case of our multi-sensor framework.
Specializing our results to this setting leads to a recovery guarantee that is
at least as good as existing results.Comment: 37 pages, 5 figure
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