10,393 research outputs found
The mean field theory of spin glasses: the heuristic replica approach and recent rigorous results
The mathematically correct computation of the spin glasses free energy in the
infinite range limit crowns 25 years of mathematic efforts in solving this
model. The exact solution of the model was found many years ago by using a
heuristic approach; the results coming from the heuristic approach were crucial
in deriving the mathematical results. The mathematical tools used in the
rigorous approach are quite different from those of the heuristic approach. In
this note we will review the heuristic approach to spin glasses in the light of
the rigorous results; we will also discuss some conjectures that may be useful
to derive the solution of the model in an alternative way.Comment: 12 pages, 1 figure; lecture at the Flato Colloquia Day, Thursday 27
November, 200
Asymptotic Analysis of MAP Estimation via the Replica Method and Applications to Compressed Sensing
The replica method is a non-rigorous but well-known technique from
statistical physics used in the asymptotic analysis of large, random, nonlinear
problems. This paper applies the replica method, under the assumption of
replica symmetry, to study estimators that are maximum a posteriori (MAP) under
a postulated prior distribution. It is shown that with random linear
measurements and Gaussian noise, the replica-symmetric prediction of the
asymptotic behavior of the postulated MAP estimate of an n-dimensional vector
"decouples" as n scalar postulated MAP estimators. The result is based on
applying a hardening argument to the replica analysis of postulated posterior
mean estimators of Tanaka and of Guo and Verdu.
The replica-symmetric postulated MAP analysis can be readily applied to many
estimators used in compressed sensing, including basis pursuit, lasso, linear
estimation with thresholding, and zero norm-regularized estimation. In the case
of lasso estimation the scalar estimator reduces to a soft-thresholding
operator, and for zero norm-regularized estimation it reduces to a
hard-threshold. Among other benefits, the replica method provides a
computationally-tractable method for precisely predicting various performance
metrics including mean-squared error and sparsity pattern recovery probability.Comment: 22 pages; added details on the replica symmetry assumptio
Support Recovery with Sparsely Sampled Free Random Matrices
Consider a Bernoulli-Gaussian complex -vector whose components are , with X_i \sim \Cc\Nc(0,\Pc_x) and binary mutually independent
and iid across . This random -sparse vector is multiplied by a square
random matrix \Um, and a randomly chosen subset, of average size , , of the resulting vector components is then observed in additive
Gaussian noise. We extend the scope of conventional noisy compressive sampling
models where \Um is typically %A16 the identity or a matrix with iid
components, to allow \Um satisfying a certain freeness condition. This class
of matrices encompasses Haar matrices and other unitarily invariant matrices.
We use the replica method and the decoupling principle of Guo and Verd\'u, as
well as a number of information theoretic bounds, to study the input-output
mutual information and the support recovery error rate in the limit of . We also extend the scope of the large deviation approach of Rangan,
Fletcher and Goyal and characterize the performance of a class of estimators
encompassing thresholded linear MMSE and relaxation
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