2,715 research outputs found
Compressive Sensing of Analog Signals Using Discrete Prolate Spheroidal Sequences
Compressive sensing (CS) has recently emerged as a framework for efficiently
capturing signals that are sparse or compressible in an appropriate basis.
While often motivated as an alternative to Nyquist-rate sampling, there remains
a gap between the discrete, finite-dimensional CS framework and the problem of
acquiring a continuous-time signal. In this paper, we attempt to bridge this
gap by exploiting the Discrete Prolate Spheroidal Sequences (DPSS's), a
collection of functions that trace back to the seminal work by Slepian, Landau,
and Pollack on the effects of time-limiting and bandlimiting operations. DPSS's
form a highly efficient basis for sampled bandlimited functions; by modulating
and merging DPSS bases, we obtain a dictionary that offers high-quality sparse
approximations for most sampled multiband signals. This multiband modulated
DPSS dictionary can be readily incorporated into the CS framework. We provide
theoretical guarantees and practical insight into the use of this dictionary
for recovery of sampled multiband signals from compressive measurements
User-Friendly Covariance Estimation for Heavy-Tailed Distributions
We offer a survey of recent results on covariance estimation for heavy-tailed
distributions. By unifying ideas scattered in the literature, we propose
user-friendly methods that facilitate practical implementation. Specifically,
we introduce element-wise and spectrum-wise truncation operators, as well as
their -estimator counterparts, to robustify the sample covariance matrix.
Different from the classical notion of robustness that is characterized by the
breakdown property, we focus on the tail robustness which is evidenced by the
connection between nonasymptotic deviation and confidence level. The key
observation is that the estimators needs to adapt to the sample size,
dimensionality of the data and the noise level to achieve optimal tradeoff
between bias and robustness. Furthermore, to facilitate their practical use, we
propose data-driven procedures that automatically calibrate the tuning
parameters. We demonstrate their applications to a series of structured models
in high dimensions, including the bandable and low-rank covariance matrices and
sparse precision matrices. Numerical studies lend strong support to the
proposed methods.Comment: 56 pages, 2 figure
Approximate Message Passing with a Colored Aliasing Model for Variable Density Fourier Sampled Images
The Approximate Message Passing (AMP) algorithm efficiently reconstructs
signals which have been sampled with large i.i.d. sub-Gaussian sensing
matrices. Central to AMP is its "state evolution", which guarantees that the
difference between the current estimate and ground truth (the "aliasing") at
every iteration obeys a Gaussian distribution that can be fully characterized
by a scalar. However, when Fourier coefficients of a signal with non-uniform
spectral density are sampled, such as in Magnetic Resonance Imaging (MRI), the
aliasing is intrinsically colored, AMP's scalar state evolution is no longer
accurate and the algorithm encounters convergence problems. In response, we
propose the Variable Density Approximate Message Passing (VDAMP) algorithm,
which uses the wavelet domain to model the colored aliasing. We present
empirical evidence that VDAMP obeys a "colored state evolution", where the
aliasing obeys a Gaussian distribution that can be fully characterized with one
scalar per wavelet subband. A benefit of state evolution is that Stein's
Unbiased Risk Estimate (SURE) can be effectively implemented, yielding an
algorithm with subband-dependent thresholding that has no free parameters. We
empirically evaluate the effectiveness of VDAMP on three variations of Fast
Iterative Shrinkage-Thresholding (FISTA) and find that it converges in around
10 times fewer iterations on average than the next-fastest method, and to a
comparable mean-squared-error.Comment: 13 pages, 7 figures, 3 tables. arXiv admin note: text overlap with
arXiv:1911.0123
Robust Principal Component Analysis?
This paper is about a curious phenomenon. Suppose we have a data matrix,
which is the superposition of a low-rank component and a sparse component. Can
we recover each component individually? We prove that under some suitable
assumptions, it is possible to recover both the low-rank and the sparse
components exactly by solving a very convenient convex program called Principal
Component Pursuit; among all feasible decompositions, simply minimize a
weighted combination of the nuclear norm and of the L1 norm. This suggests the
possibility of a principled approach to robust principal component analysis
since our methodology and results assert that one can recover the principal
components of a data matrix even though a positive fraction of its entries are
arbitrarily corrupted. This extends to the situation where a fraction of the
entries are missing as well. We discuss an algorithm for solving this
optimization problem, and present applications in the area of video
surveillance, where our methodology allows for the detection of objects in a
cluttered background, and in the area of face recognition, where it offers a
principled way of removing shadows and specularities in images of faces
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