362 research outputs found
On Sparse Vector Recovery Performance in Structurally Orthogonal Matrices via LASSO
In this paper, we consider the compressed sensing problem of reconstructing a sparse signal from an undersampled set of noisy linear measurements. The regularized least squares or least absolute shrinkage and selection operator (LASSO) formulation is used for signal estimation. The measurement matrix is assumed to be constructed by concatenating several randomly orthogonal bases, which we refer to as structurally orthogonal matrices. Such measurement matrix is highly relevant to large-scale compressive sensing applications because it facilitates rapid computation and parallel processing. Using the replica method in statistical physics, we derive the mean-squared-error (MSE) formula of reconstruction over the structurally orthogonal matrix in the large-system regime. Extensive numerical experiments are provided to verify the analytical result. We then consider the analytical result to investigate the MSE behaviors of the LASSO over the structurally orthogonal matrix, with an emphasis on performance comparisons with matrices with independent and identically distributed (i.i.d.) Gaussian entries. We find that structurally orthogonal matrices are at least as good as their i.i.d. Gaussian counterparts. Thus, the use of structurally orthogonal matrices is attractive in practical applications
Tensor decomposition with generalized lasso penalties
We present an approach for penalized tensor decomposition (PTD) that
estimates smoothly varying latent factors in multi-way data. This generalizes
existing work on sparse tensor decomposition and penalized matrix
decompositions, in a manner parallel to the generalized lasso for regression
and smoothing problems. Our approach presents many nontrivial challenges at the
intersection of modeling and computation, which are studied in detail. An
efficient coordinate-wise optimization algorithm for (PTD) is presented, and
its convergence properties are characterized. The method is applied both to
simulated data and real data on flu hospitalizations in Texas. These results
show that our penalized tensor decomposition can offer major improvements on
existing methods for analyzing multi-way data that exhibit smooth spatial or
temporal features
Projection-Based and Look Ahead Strategies for Atom Selection
In this paper, we improve iterative greedy search algorithms in which atoms
are selected serially over iterations, i.e., one-by-one over iterations. For
serial atom selection, we devise two new schemes to select an atom from a set
of potential atoms in each iteration. The two new schemes lead to two new
algorithms. For both the algorithms, in each iteration, the set of potential
atoms is found using a standard matched filter. In case of the first scheme, we
propose an orthogonal projection strategy that selects an atom from the set of
potential atoms. Then, for the second scheme, we propose a look ahead strategy
such that the selection of an atom in the current iteration has an effect on
the future iterations. The use of look ahead strategy requires a higher
computational resource. To achieve a trade-off between performance and
complexity, we use the two new schemes in cascade and develop a third new
algorithm. Through experimental evaluations, we compare the proposed algorithms
with existing greedy search and convex relaxation algorithms.Comment: sparsity, compressive sensing; IEEE Trans on Signal Processing 201
Isotropically Random Orthogonal Matrices: Performance of LASSO and Minimum Conic Singular Values
Recently, the precise performance of the Generalized LASSO algorithm for
recovering structured signals from compressed noisy measurements, obtained via
i.i.d. Gaussian matrices, has been characterized. The analysis is based on a
framework introduced by Stojnic and heavily relies on the use of Gordon's
Gaussian min-max theorem (GMT), a comparison principle on Gaussian processes.
As a result, corresponding characterizations for other ensembles of measurement
matrices have not been developed. In this work, we analyze the corresponding
performance of the ensemble of isotropically random orthogonal (i.r.o.)
measurements. We consider the constrained version of the Generalized LASSO and
derive a sharp characterization of its normalized squared error in the
large-system limit. When compared to its Gaussian counterpart, our result
analytically confirms the superiority in performance of the i.r.o. ensemble.
Our second result, derives an asymptotic lower bound on the minimum conic
singular values of i.r.o. matrices. This bound is larger than the corresponding
bound on Gaussian matrices. To prove our results we express i.r.o. matrices in
terms of Gaussians and show that, with some modifications, the GMT framework is
still applicable
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