4,858 research outputs found
Signal Estimation with Additive Error Metrics in Compressed Sensing
Compressed sensing typically deals with the estimation of a system input from
its noise-corrupted linear measurements, where the number of measurements is
smaller than the number of input components. The performance of the estimation
process is usually quantified by some standard error metric such as squared
error or support set error. In this correspondence, we consider a noisy
compressed sensing problem with any arbitrary error metric. We propose a
simple, fast, and highly general algorithm that estimates the original signal
by minimizing the error metric defined by the user. We verify that our
algorithm is optimal owing to the decoupling principle, and we describe a
general method to compute the fundamental information-theoretic performance
limit for any error metric. We provide two example metrics --- minimum mean
absolute error and minimum mean support error --- and give the theoretical
performance limits for these two cases. Experimental results show that our
algorithm outperforms methods such as relaxed belief propagation (relaxed BP)
and compressive sampling matching pursuit (CoSaMP), and reaches the suggested
theoretical limits for our two example metrics.Comment: to appear in IEEE Trans. Inf. Theor
Sketching for Large-Scale Learning of Mixture Models
Learning parameters from voluminous data can be prohibitive in terms of
memory and computational requirements. We propose a "compressive learning"
framework where we estimate model parameters from a sketch of the training
data. This sketch is a collection of generalized moments of the underlying
probability distribution of the data. It can be computed in a single pass on
the training set, and is easily computable on streams or distributed datasets.
The proposed framework shares similarities with compressive sensing, which aims
at drastically reducing the dimension of high-dimensional signals while
preserving the ability to reconstruct them. To perform the estimation task, we
derive an iterative algorithm analogous to sparse reconstruction algorithms in
the context of linear inverse problems. We exemplify our framework with the
compressive estimation of a Gaussian Mixture Model (GMM), providing heuristics
on the choice of the sketching procedure and theoretical guarantees of
reconstruction. We experimentally show on synthetic data that the proposed
algorithm yields results comparable to the classical Expectation-Maximization
(EM) technique while requiring significantly less memory and fewer computations
when the number of database elements is large. We further demonstrate the
potential of the approach on real large-scale data (over 10 8 training samples)
for the task of model-based speaker verification. Finally, we draw some
connections between the proposed framework and approximate Hilbert space
embedding of probability distributions using random features. We show that the
proposed sketching operator can be seen as an innovative method to design
translation-invariant kernels adapted to the analysis of GMMs. We also use this
theoretical framework to derive information preservation guarantees, in the
spirit of infinite-dimensional compressive sensing
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
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