207,522 research outputs found
Sufficient conditions for convergence of the Sum-Product Algorithm
We derive novel conditions that guarantee convergence of the Sum-Product
algorithm (also known as Loopy Belief Propagation or simply Belief Propagation)
to a unique fixed point, irrespective of the initial messages. The
computational complexity of the conditions is polynomial in the number of
variables. In contrast with previously existing conditions, our results are
directly applicable to arbitrary factor graphs (with discrete variables) and
are shown to be valid also in the case of factors containing zeros, under some
additional conditions. We compare our bounds with existing ones, numerically
and, if possible, analytically. For binary variables with pairwise
interactions, we derive sufficient conditions that take into account local
evidence (i.e., single variable factors) and the type of pair interactions
(attractive or repulsive). It is shown empirically that this bound outperforms
existing bounds.Comment: 15 pages, 5 figures. Major changes and new results in this revised
version. Submitted to IEEE Transactions on Information Theor
Inertial Douglas-Rachford splitting for monotone inclusion problems
We propose an inertial Douglas-Rachford splitting algorithm for finding the
set of zeros of the sum of two maximally monotone operators in Hilbert spaces
and investigate its convergence properties. To this end we formulate first the
inertial version of the Krasnosel'ski\u{\i}--Mann algorithm for approximating
the set of fixed points of a nonexpansive operator, for which we also provide
an exhaustive convergence analysis. By using a product space approach we employ
these results to the solving of monotone inclusion problems involving linearly
composed and parallel-sum type operators and provide in this way iterative
schemes where each of the maximally monotone mappings is accessed separately
via its resolvent. We consider also the special instance of solving a
primal-dual pair of nonsmooth convex optimization problems and illustrate the
theoretical results via some numerical experiments in clustering and location
theory.Comment: arXiv admin note: text overlap with arXiv:1402.529
Orthogonal Trace-Sum Maximization: Applications, Local Algorithms, and Global Optimality
This paper studies a problem of maximizing the sum of traces of matrix
quadratic forms on a product of Stiefel manifolds. This orthogonal trace-sum
maximization (OTSM) problem generalizes many interesting problems such as
generalized canonical correlation analysis (CCA), Procrustes analysis, and
cryo-electron microscopy of the Nobel prize fame. For these applications
finding global solutions is highly desirable but has been out of reach for a
long time. For example, generalizations of CCA do not possess obvious global
solutions unlike their classical counterpart to which a global solution is
readily obtained through singular value decomposition; it is also not clear how
to test global optimality. We provide a simple method to certify global
optimality of a given local solution. This method only requires testing the
sign of the smallest eigenvalue of a symmetric matrix, and does not rely on a
particular algorithm as long as it converges to a stationary point. Our
certificate result relies on a semidefinite programming (SDP) relaxation of
OTSM, but avoids solving an SDP of lifted dimensions. Surprisingly, a popular
algorithm for generalized CCA and Procrustes analysis may generate oscillating
iterates. We propose a simple modification of this standard algorithm and prove
that it reliably converges. Our notion of convergence is stronger than
conventional objective value convergence or subsequence convergence.The
convergence result utilizes the Kurdyka-Lojasiewicz property of the problem.Comment: 22 pages, 1 figur
A Novel Partitioning Method for Accelerating the Block Cimmino Algorithm
We propose a novel block-row partitioning method in order to improve the
convergence rate of the block Cimmino algorithm for solving general sparse
linear systems of equations. The convergence rate of the block Cimmino
algorithm depends on the orthogonality among the block rows obtained by the
partitioning method. The proposed method takes numerical orthogonality among
block rows into account by proposing a row inner-product graph model of the
coefficient matrix. In the graph partitioning formulation defined on this graph
model, the partitioning objective of minimizing the cutsize directly
corresponds to minimizing the sum of inter-block inner products between block
rows thus leading to an improvement in the eigenvalue spectrum of the iteration
matrix. This in turn leads to a significant reduction in the number of
iterations required for convergence. Extensive experiments conducted on a large
set of matrices confirm the validity of the proposed method against a
state-of-the-art method
- …