12,135 research outputs found
Greedy Algorithms for Cone Constrained Optimization with Convergence Guarantees
Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe
(FW) algorithms regained popularity in recent years due to their simplicity,
effectiveness and theoretical guarantees. MP and FW address optimization over
the linear span and the convex hull of a set of atoms, respectively. In this
paper, we consider the intermediate case of optimization over the convex cone,
parametrized as the conic hull of a generic atom set, leading to the first
principled definitions of non-negative MP algorithms for which we give explicit
convergence rates and demonstrate excellent empirical performance. In
particular, we derive sublinear () convergence on general
smooth and convex objectives, and linear convergence () on
strongly convex objectives, in both cases for general sets of atoms.
Furthermore, we establish a clear correspondence of our algorithms to known
algorithms from the MP and FW literature. Our novel algorithms and analyses
target general atom sets and general objective functions, and hence are
directly applicable to a large variety of learning settings.Comment: NIPS 201
Computational Methods for Sparse Solution of Linear Inverse Problems
The goal of the sparse approximation problem is to approximate a target signal using a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, to the circumstances in which individual methods tend to perform well, and to the theoretical guarantees available. Many fundamental questions in electrical engineering, statistics, and applied mathematics can be posed as sparse approximation problems, making these algorithms versatile and relevant to a plethora of applications
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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