25,471 research outputs found
Conditioning of Random Block Subdictionaries with Applications to Block-Sparse Recovery and Regression
The linear model, in which a set of observations is assumed to be given by a
linear combination of columns of a matrix, has long been the mainstay of the
statistics and signal processing literature. One particular challenge for
inference under linear models is understanding the conditions on the dictionary
under which reliable inference is possible. This challenge has attracted
renewed attention in recent years since many modern inference problems deal
with the "underdetermined" setting, in which the number of observations is much
smaller than the number of columns in the dictionary. This paper makes several
contributions for this setting when the set of observations is given by a
linear combination of a small number of groups of columns of the dictionary,
termed the "block-sparse" case. First, it specifies conditions on the
dictionary under which most block subdictionaries are well conditioned. This
result is fundamentally different from prior work on block-sparse inference
because (i) it provides conditions that can be explicitly computed in
polynomial time, (ii) the given conditions translate into near-optimal scaling
of the number of columns of the block subdictionaries as a function of the
number of observations for a large class of dictionaries, and (iii) it suggests
that the spectral norm and the quadratic-mean block coherence of the dictionary
(rather than the worst-case coherences) fundamentally limit the scaling of
dimensions of the well-conditioned block subdictionaries. Second, this paper
investigates the problems of block-sparse recovery and block-sparse regression
in underdetermined settings. Near-optimal block-sparse recovery and regression
are possible for certain dictionaries as long as the dictionary satisfies
easily computable conditions and the coefficients describing the linear
combination of groups of columns can be modeled through a mild statistical
prior.Comment: 39 pages, 3 figures. A revised and expanded version of the paper
published in IEEE Transactions on Information Theory (DOI:
10.1109/TIT.2015.2429632); this revision includes corrections in the proofs
of some of the result
Approximate cross-validation formula for Bayesian linear regression
Cross-validation (CV) is a technique for evaluating the ability of
statistical models/learning systems based on a given data set. Despite its wide
applicability, the rather heavy computational cost can prevent its use as the
system size grows. To resolve this difficulty in the case of Bayesian linear
regression, we develop a formula for evaluating the leave-one-out CV error
approximately without actually performing CV. The usefulness of the developed
formula is tested by statistical mechanical analysis for a synthetic model.
This is confirmed by application to a real-world supernova data set as well.Comment: 5 pages, 2 figures, invited paper for Allerton2016 conferenc
Rectified Gaussian Scale Mixtures and the Sparse Non-Negative Least Squares Problem
In this paper, we develop a Bayesian evidence maximization framework to solve
the sparse non-negative least squares (S-NNLS) problem. We introduce a family
of probability densities referred to as the Rectified Gaussian Scale Mixture
(R- GSM) to model the sparsity enforcing prior distribution for the solution.
The R-GSM prior encompasses a variety of heavy-tailed densities such as the
rectified Laplacian and rectified Student- t distributions with a proper choice
of the mixing density. We utilize the hierarchical representation induced by
the R-GSM prior and develop an evidence maximization framework based on the
Expectation-Maximization (EM) algorithm. Using the EM based method, we estimate
the hyper-parameters and obtain a point estimate for the solution. We refer to
the proposed method as rectified sparse Bayesian learning (R-SBL). We provide
four R- SBL variants that offer a range of options for computational complexity
and the quality of the E-step computation. These methods include the Markov
chain Monte Carlo EM, linear minimum mean-square-error estimation, approximate
message passing and a diagonal approximation. Using numerical experiments, we
show that the proposed R-SBL method outperforms existing S-NNLS solvers in
terms of both signal and support recovery performance, and is also very robust
against the structure of the design matrix.Comment: Under Review by IEEE Transactions on Signal Processin
- …