469 research outputs found
Sparsity-Cognizant Total Least-Squares for Perturbed Compressive Sampling
Solving linear regression problems based on the total least-squares (TLS)
criterion has well-documented merits in various applications, where
perturbations appear both in the data vector as well as in the regression
matrix. However, existing TLS approaches do not account for sparsity possibly
present in the unknown vector of regression coefficients. On the other hand,
sparsity is the key attribute exploited by modern compressive sampling and
variable selection approaches to linear regression, which include noise in the
data, but do not account for perturbations in the regression matrix. The
present paper fills this gap by formulating and solving TLS optimization
problems under sparsity constraints. Near-optimum and reduced-complexity
suboptimum sparse (S-) TLS algorithms are developed to address the perturbed
compressive sampling (and the related dictionary learning) challenge, when
there is a mismatch between the true and adopted bases over which the unknown
vector is sparse. The novel S-TLS schemes also allow for perturbations in the
regression matrix of the least-absolute selection and shrinkage selection
operator (Lasso), and endow TLS approaches with ability to cope with sparse,
under-determined "errors-in-variables" models. Interesting generalizations can
further exploit prior knowledge on the perturbations to obtain novel weighted
and structured S-TLS solvers. Analysis and simulations demonstrate the
practical impact of S-TLS in calibrating the mismatch effects of contemporary
grid-based approaches to cognitive radio sensing, and robust
direction-of-arrival estimation using antenna arrays.Comment: 30 pages, 10 figures, submitted to IEEE Transactions on Signal
Processin
Approximate message passing for nonconvex sparse regularization with stability and asymptotic analysis
We analyse a linear regression problem with nonconvex regularization called
smoothly clipped absolute deviation (SCAD) under an overcomplete Gaussian basis
for Gaussian random data. We propose an approximate message passing (AMP)
algorithm considering nonconvex regularization, namely SCAD-AMP, and
analytically show that the stability condition corresponds to the de
Almeida--Thouless condition in spin glass literature. Through asymptotic
analysis, we show the correspondence between the density evolution of SCAD-AMP
and the replica symmetric solution. Numerical experiments confirm that for a
sufficiently large system size, SCAD-AMP achieves the optimal performance
predicted by the replica method. Through replica analysis, a phase transition
between replica symmetric (RS) and replica symmetry breaking (RSB) region is
found in the parameter space of SCAD. The appearance of the RS region for a
nonconvex penalty is a significant advantage that indicates the region of
smooth landscape of the optimization problem. Furthermore, we analytically show
that the statistical representation performance of the SCAD penalty is better
than that of L1-based methods, and the minimum representation error under RS
assumption is obtained at the edge of the RS/RSB phase. The correspondence
between the convergence of the existing coordinate descent algorithm and RS/RSB
transition is also indicated
Regularized Gradient Descent: A Nonconvex Recipe for Fast Joint Blind Deconvolution and Demixing
We study the question of extracting a sequence of functions
from observing only the sum of
their convolutions, i.e., from . While convex optimization techniques
are able to solve this joint blind deconvolution-demixing problem provably and
robustly under certain conditions, for medium-size or large-size problems we
need computationally faster methods without sacrificing the benefits of
mathematical rigor that come with convex methods. In this paper, we present a
non-convex algorithm which guarantees exact recovery under conditions that are
competitive with convex optimization methods, with the additional advantage of
being computationally much more efficient. Our two-step algorithm converges to
the global minimum linearly and is also robust in the presence of additive
noise. While the derived performance bounds are suboptimal in terms of the
information-theoretic limit, numerical simulations show remarkable performance
even if the number of measurements is close to the number of degrees of
freedom. We discuss an application of the proposed framework in wireless
communications in connection with the Internet-of-Things.Comment: Accepted to Information and Inference: a Journal of the IM
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