1,111 research outputs found
Sparse Multivariate Factor Regression
We consider the problem of multivariate regression in a setting where the
relevant predictors could be shared among different responses. We propose an
algorithm which decomposes the coefficient matrix into the product of a long
matrix and a wide matrix, with an elastic net penalty on the former and an
penalty on the latter. The first matrix linearly transforms the
predictors to a set of latent factors, and the second one regresses the
responses on these factors. Our algorithm simultaneously performs dimension
reduction and coefficient estimation and automatically estimates the number of
latent factors from the data. Our formulation results in a non-convex
optimization problem, which despite its flexibility to impose effective
low-dimensional structure, is difficult, or even impossible, to solve exactly
in a reasonable time. We specify an optimization algorithm based on alternating
minimization with three different sets of updates to solve this non-convex
problem and provide theoretical results on its convergence and optimality.
Finally, we demonstrate the effectiveness of our algorithm via experiments on
simulated and real data
Optimization with Sparsity-Inducing Penalties
Sparse estimation methods are aimed at using or obtaining parsimonious
representations of data or models. They were first dedicated to linear variable
selection but numerous extensions have now emerged such as structured sparsity
or kernel selection. It turns out that many of the related estimation problems
can be cast as convex optimization problems by regularizing the empirical risk
with appropriate non-smooth norms. The goal of this paper is to present from a
general perspective optimization tools and techniques dedicated to such
sparsity-inducing penalties. We cover proximal methods, block-coordinate
descent, reweighted -penalized techniques, working-set and homotopy
methods, as well as non-convex formulations and extensions, and provide an
extensive set of experiments to compare various algorithms from a computational
point of view
Sparse Modeling for Image and Vision Processing
In recent years, a large amount of multi-disciplinary research has been
conducted on sparse models and their applications. In statistics and machine
learning, the sparsity principle is used to perform model selection---that is,
automatically selecting a simple model among a large collection of them. In
signal processing, sparse coding consists of representing data with linear
combinations of a few dictionary elements. Subsequently, the corresponding
tools have been widely adopted by several scientific communities such as
neuroscience, bioinformatics, or computer vision. The goal of this monograph is
to offer a self-contained view of sparse modeling for visual recognition and
image processing. More specifically, we focus on applications where the
dictionary is learned and adapted to data, yielding a compact representation
that has been successful in various contexts.Comment: 205 pages, to appear in Foundations and Trends in Computer Graphics
and Visio
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