304 research outputs found
The decimation process in random k-SAT
Let F be a uniformly distributed random k-SAT formula with n variables and m
clauses. Non-rigorous statistical mechanics ideas have inspired a message
passing algorithm called Belief Propagation Guided Decimation for finding
satisfying assignments of F. This algorithm can be viewed as an attempt at
implementing a certain thought experiment that we call the Decimation Process.
In this paper we identify a variety of phase transitions in the decimation
process and link these phase transitions to the performance of the algorithm
Approaching the ground states of the random maximum two-satisfiability problem by a greedy single-spin flipping process
In this brief report we explore the energy landscapes of two spin glass
models using a greedy single-spin flipping process, {\tt Gmax}. The
ground-state energy density of the random maximum two-satisfiability problem is
efficiently approached by {\tt Gmax}. The achieved energy density
decreases with the evolution time as
with a small prefactor and a scaling coefficient , indicating an
energy landscape with deep and rugged funnel-shape regions. For the
Viana-Bray spin glass model, however, the greedy single-spin dynamics quickly
gets trapped to a local minimal region of the energy landscape.Comment: 5 pages with 4 figures included. Accepted for publication in Physical
Review E as a brief repor
Spatial Mixing of Coloring Random Graphs
We study the strong spatial mixing (decay of correlation) property of proper
-colorings of random graph with a fixed . The strong spatial
mixing of coloring and related models have been extensively studied on graphs
with bounded maximum degree. However, for typical classes of graphs with
bounded average degree, such as , an easy counterexample shows that
colorings do not exhibit strong spatial mixing with high probability.
Nevertheless, we show that for with and
sufficiently large , with high probability proper -colorings of
random graph exhibit strong spatial mixing with respect to an
arbitrarily fixed vertex. This is the first strong spatial mixing result for
colorings of graphs with unbounded maximum degree. Our analysis of strong
spatial mixing establishes a block-wise correlation decay instead of the
standard point-wise decay, which may be of interest by itself, especially for
graphs with unbounded degree
Harnessing the Bethe Free Energy
Gibbs measures induced by random factor graphs play a prominent role in computer science, combinatorics and physics. A key problem is to calculate the typical value of the partition function. According to the "replica symmetric cavity method", a heuristic that rests on non-rigorous considerations from statistical mechanics, in many cases this problem can be tackled by way of maximising a functional called the "Bethe free energy". In this paper we prove that the Bethe free energy upper-bounds the partition function in a broad class of models. Additionally, we provide a sufficient condition for this upper bound to be tight
From one solution of a 3-satisfiability formula to a solution cluster: Frozen variables and entropy
A solution to a 3-satisfiability (3-SAT) formula can be expanded into a
cluster, all other solutions of which are reachable from this one through a
sequence of single-spin flips. Some variables in the solution cluster are
frozen to the same spin values by one of two different mechanisms: frozen-core
formation and long-range frustrations. While frozen cores are identified by a
local whitening algorithm, long-range frustrations are very difficult to trace,
and they make an entropic belief-propagation (BP) algorithm fail to converge.
For BP to reach a fixed point the spin values of a tiny fraction of variables
(chosen according to the whitening algorithm) are externally fixed during the
iteration. From the calculated entropy values, we infer that, for a large
random 3-SAT formula with constraint density close to the satisfiability
threshold, the solutions obtained by the survey-propagation or the walksat
algorithm belong neither to the most dominating clusters of the formula nor to
the most abundant clusters. This work indicates that a single solution cluster
of a random 3-SAT formula may have further community structures.Comment: 13 pages, 6 figures. Final version as published in PR
Random Projections For Large-Scale Regression
Fitting linear regression models can be computationally very expensive in
large-scale data analysis tasks if the sample size and the number of variables
are very large. Random projections are extensively used as a dimension
reduction tool in machine learning and statistics. We discuss the applications
of random projections in linear regression problems, developed to decrease
computational costs, and give an overview of the theoretical guarantees of the
generalization error. It can be shown that the combination of random
projections with least squares regression leads to similar recovery as ridge
regression and principal component regression. We also discuss possible
improvements when averaging over multiple random projections, an approach that
lends itself easily to parallel implementation.Comment: 13 pages, 3 Figure
Random forests with random projections of the output space for high dimensional multi-label classification
We adapt the idea of random projections applied to the output space, so as to
enhance tree-based ensemble methods in the context of multi-label
classification. We show how learning time complexity can be reduced without
affecting computational complexity and accuracy of predictions. We also show
that random output space projections may be used in order to reach different
bias-variance tradeoffs, over a broad panel of benchmark problems, and that
this may lead to improved accuracy while reducing significantly the
computational burden of the learning stage
Large Scale Spectral Clustering Using Approximate Commute Time Embedding
Spectral clustering is a novel clustering method which can detect complex
shapes of data clusters. However, it requires the eigen decomposition of the
graph Laplacian matrix, which is proportion to and thus is not
suitable for large scale systems. Recently, many methods have been proposed to
accelerate the computational time of spectral clustering. These approximate
methods usually involve sampling techniques by which a lot information of the
original data may be lost. In this work, we propose a fast and accurate
spectral clustering approach using an approximate commute time embedding, which
is similar to the spectral embedding. The method does not require using any
sampling technique and computing any eigenvector at all. Instead it uses random
projection and a linear time solver to find the approximate embedding. The
experiments in several synthetic and real datasets show that the proposed
approach has better clustering quality and is faster than the state-of-the-art
approximate spectral clustering methods
Clustering of solutions in the random satisfiability problem
Using elementary rigorous methods we prove the existence of a clustered phase
in the random -SAT problem, for . In this phase the solutions are
grouped into clusters which are far away from each other. The results are in
agreement with previous predictions of the cavity method and give a rigorous
confirmation to one of its main building blocks. It can be generalized to other
systems of both physical and computational interest.Comment: 4 pages, 1 figur
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