133 research outputs found
Dynamical Functional Theory for Compressed Sensing
We introduce a theoretical approach for designing generalizations of the
approximate message passing (AMP) algorithm for compressed sensing which are
valid for large observation matrices that are drawn from an invariant random
matrix ensemble. By design, the fixed points of the algorithm obey the
Thouless-Anderson-Palmer (TAP) equations corresponding to the ensemble. Using a
dynamical functional approach we are able to derive an effective stochastic
process for the marginal statistics of a single component of the dynamics. This
allows us to design memory terms in the algorithm in such a way that the
resulting fields become Gaussian random variables allowing for an explicit
analysis. The asymptotic statistics of these fields are consistent with the
replica ansatz of the compressed sensing problem.Comment: 5 pages, accepted for ISIT 201
On Capacity Optimality of OAMP: Beyond IID Sensing Matrices and Gaussian Signaling
This paper investigates a large unitarily invariant system (LUIS) involving a
unitarily invariant sensing matrix, an arbitrarily fixed signal distribution,
and forward error control (FEC) coding. A universal Gram-Schmidt
orthogonalization is considered for the construction of orthogonal approximate
message passing (OAMP), which renders the results applicable to general
prototypes without the differentiability restriction. For OAMP with Lipschitz
continuous local estimators, we develop two variational
single-input-single-output transfer functions, based on which we analyze the
achievable rate of OAMP. Furthermore, when the state evolution of OAMP has a
unique fixed point, we reveal that OAMP reaches the constrained capacity
predicted by the replica method of the LUIS with an arbitrary signal
distribution based on matched FEC coding. The replica method is rigorous for
LUIS with Gaussian signaling and for certain sub-classes of LUIS with arbitrary
signal distributions. Several area properties are established based on the
variational transfer functions of OAMP. Meanwhile, we elaborate a replica
constrained capacity-achieving coding principle for LUIS, based on which
irregular low-density parity-check (LDPC) codes are optimized for binary
signaling in the simulation results. We show that OAMP with the optimized codes
has significant performance improvement over the un-optimized ones and the
well-known Turbo linear MMSE algorithm. For quadrature phase-shift keying
(QPSK) modulation, replica constrained capacity-approaching bit error rate
(BER) performances are observed under various channel conditions.Comment: Single column, 34 pages, 9 figure
Expectation Propagation for Approximate Inference: Free Probability Framework
We study asymptotic properties of expectation propagation (EP) -- a method
for approximate inference originally developed in the field of machine
learning. Applied to generalized linear models, EP iteratively computes a
multivariate Gaussian approximation to the exact posterior distribution. The
computational complexity of the repeated update of covariance matrices severely
limits the application of EP to large problem sizes. In this study, we present
a rigorous analysis by means of free probability theory that allows us to
overcome this computational bottleneck if specific data matrices in the problem
fulfill certain properties of asymptotic freeness. We demonstrate the relevance
of our approach on the gene selection problem of a microarray dataset.Comment: Both authors are co-first authors. The main body of this paper is
accepted for publication in the proceedings of the 2018 IEEE International
Symposium on Information Theory (ISIT
Compressed Sensing with Upscaled Vector Approximate Message Passing
Recently proposed Vector Approximate Message Passing (VAMP) demonstrates a
great reconstruction potential at solving compressed sensing related linear
inverse problems. VAMP provides high per-iteration improvement, can utilize
powerful denoisers like BM3D, has rigorously defined dynamics and is able to
recover signals sampled by highly undersampled and ill-conditioned linear
operators. Yet, its applicability is limited to relatively small problem sizes
due to necessity to compute the expensive LMMSE estimator at each iteration. In
this work we consider the problem of upscaling VAMP by utilizing Conjugate
Gradient (CG) to approximate the intractable LMMSE estimator and propose a
CG-VAMP algorithm that can efficiently recover large-scale data. We derive
evolution models of certain key parameters of CG-VAMP and use the theoretical
results to develop fast and practical tools for correcting, tuning and
accelerating the CG algorithm within CG-VAMP to preserve all the main
advantages of VAMP, while maintaining reasonable and controllable computational
cost of the algorithm
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