485 research outputs found
Nonmonotone Barzilai-Borwein Gradient Algorithm for -Regularized Nonsmooth Minimization in Compressive Sensing
This paper is devoted to minimizing the sum of a smooth function and a
nonsmooth -regularized term. This problem as a special cases includes
the -regularized convex minimization problem in signal processing,
compressive sensing, machine learning, data mining, etc. However, the
non-differentiability of the -norm causes more challenging especially
in large problems encountered in many practical applications. This paper
proposes, analyzes, and tests a Barzilai-Borwein gradient algorithm. At each
iteration, the generated search direction enjoys descent property and can be
easily derived by minimizing a local approximal quadratic model and
simultaneously taking the favorable structure of the -norm. Moreover, a
nonmonotone line search technique is incorporated to find a suitable stepsize
along this direction. The algorithm is easily performed, where the values of
the objective function and the gradient of the smooth term are required at
per-iteration. Under some conditions, the proposed algorithm is shown to be
globally convergent. The limited experiments by using some nonconvex
unconstrained problems from CUTEr library with additive -regularization
illustrate that the proposed algorithm performs quite well. Extensive
experiments for -regularized least squares problems in compressive
sensing verify that our algorithm compares favorably with several
state-of-the-art algorithms which are specifically designed in recent years.Comment: 20 page
Structured Sparsity: Discrete and Convex approaches
Compressive sensing (CS) exploits sparsity to recover sparse or compressible
signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity
is also used to enhance interpretability in machine learning and statistics
applications: While the ambient dimension is vast in modern data analysis
problems, the relevant information therein typically resides in a much lower
dimensional space. However, many solutions proposed nowadays do not leverage
the true underlying structure. Recent results in CS extend the simple sparsity
idea to more sophisticated {\em structured} sparsity models, which describe the
interdependency between the nonzero components of a signal, allowing to
increase the interpretability of the results and lead to better recovery
performance. In order to better understand the impact of structured sparsity,
in this chapter we analyze the connections between the discrete models and
their convex relaxations, highlighting their relative advantages. We start with
the general group sparse model and then elaborate on two important special
cases: the dispersive and the hierarchical models. For each, we present the
models in their discrete nature, discuss how to solve the ensuing discrete
problems and then describe convex relaxations. We also consider more general
structures as defined by set functions and present their convex proxies.
Further, we discuss efficient optimization solutions for structured sparsity
problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure
Recovery of Low-Rank Plus Compressed Sparse Matrices with Application to Unveiling Traffic Anomalies
Given the superposition of a low-rank matrix plus the product of a known fat
compression matrix times a sparse matrix, the goal of this paper is to
establish deterministic conditions under which exact recovery of the low-rank
and sparse components becomes possible. This fundamental identifiability issue
arises with traffic anomaly detection in backbone networks, and subsumes
compressed sensing as well as the timely low-rank plus sparse matrix recovery
tasks encountered in matrix decomposition problems. Leveraging the ability of
- and nuclear norms to recover sparse and low-rank matrices, a convex
program is formulated to estimate the unknowns. Analysis and simulations
confirm that the said convex program can recover the unknowns for sufficiently
low-rank and sparse enough components, along with a compression matrix
possessing an isometry property when restricted to operate on sparse vectors.
When the low-rank, sparse, and compression matrices are drawn from certain
random ensembles, it is established that exact recovery is possible with high
probability. First-order algorithms are developed to solve the nonsmooth convex
optimization problem with provable iteration complexity guarantees. Insightful
tests with synthetic and real network data corroborate the effectiveness of the
novel approach in unveiling traffic anomalies across flows and time, and its
ability to outperform existing alternatives.Comment: 38 pages, submitted to the IEEE Transactions on Information Theor
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