1,936 research outputs found
Proximal Methods for Hierarchical Sparse Coding
Sparse coding consists in representing signals as sparse linear combinations
of atoms selected from a dictionary. We consider an extension of this framework
where the atoms are further assumed to be embedded in a tree. This is achieved
using a recently introduced tree-structured sparse regularization norm, which
has proven useful in several applications. This norm leads to regularized
problems that are difficult to optimize, and we propose in this paper efficient
algorithms for solving them. More precisely, we show that the proximal operator
associated with this norm is computable exactly via a dual approach that can be
viewed as the composition of elementary proximal operators. Our procedure has a
complexity linear, or close to linear, in the number of atoms, and allows the
use of accelerated gradient techniques to solve the tree-structured sparse
approximation problem at the same computational cost as traditional ones using
the L1-norm. Our method is efficient and scales gracefully to millions of
variables, which we illustrate in two types of applications: first, we consider
fixed hierarchical dictionaries of wavelets to denoise natural images. Then, we
apply our optimization tools in the context of dictionary learning, where
learned dictionary elements naturally organize in a prespecified arborescent
structure, leading to a better performance in reconstruction of natural image
patches. When applied to text documents, our method learns hierarchies of
topics, thus providing a competitive alternative to probabilistic topic models
Block-proximal methods with spatially adapted acceleration
We study and develop (stochastic) primal--dual block-coordinate descent
methods for convex problems based on the method due to Chambolle and Pock. Our
methods have known convergence rates for the iterates and the ergodic gap:
if each block is strongly convex, if no convexity is
present, and more generally a mixed rate for strongly convex
blocks, if only some blocks are strongly convex. Additional novelties of our
methods include blockwise-adapted step lengths and acceleration, as well as the
ability to update both the primal and dual variables randomly in blocks under a
very light compatibility condition. In other words, these variants of our
methods are doubly-stochastic. We test the proposed methods on various image
processing problems, where we employ pixelwise-adapted acceleration
Parallel extragradient-proximal methods for split equilibrium problems
In this paper, we introduce two parallel extragradient-proximal methods for
solving split equilibrium problems. The algorithms combine the extragradient
method, the proximal method and the hybrid (outer approximation) method. The
weak and strong convergence theorems for iterative sequences generated by the
algorithms are established under widely used assumptions for equilibrium
bifunctions.Comment: 13 pages, submitte
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