180 research outputs found
Properties of spatial coupling in compressed sensing
In this paper we address a series of open questions about the construction of
spatially coupled measurement matrices in compressed sensing. For hardware
implementations one is forced to depart from the limiting regime of parameters
in which the proofs of the so-called threshold saturation work. We investigate
quantitatively the behavior under finite coupling range, the dependence on the
shape of the coupling interaction, and optimization of the so-called seed to
minimize distance from optimality. Our analysis explains some of the properties
observed empirically in previous works and provides new insight on spatially
coupled compressed sensing.Comment: 5 pages, 6 figure
Following Gibbs States Adiabatically - The Energy Landscape of Mean Field Glassy Systems
We introduce a generalization of the cavity, or Bethe-Peierls, method that
allows to follow Gibbs states when an external parameter, e.g. the temperature,
is adiabatically changed. This allows to obtain new quantitative results on the
static and dynamic behavior of mean field disordered systems such as models of
glassy and amorphous materials or random constraint satisfaction problems. As a
first application, we discuss the residual energy after a very slow annealing,
the behavior of out-of-equilibrium states, and demonstrate the presence of
temperature chaos in equilibrium. We also explore the energy landscape, and
identify a new transition from an computationally easier canyons-dominated
region to a harder valleys-dominated one.Comment: 6 pages, 7 figure
Quiet Planting in the Locked Constraint Satisfaction Problems
We study the planted ensemble of locked constraint satisfaction problems. We
describe the connection between the random and planted ensembles. The use of
the cavity method is combined with arguments from reconstruction on trees and
first and second moment considerations; in particular the connection with the
reconstruction on trees appears to be crucial. Our main result is the location
of the hard region in the planted ensemble. In a part of that hard region
instances have with high probability a single satisfying assignment.Comment: 21 pages, revised versio
Spectral Clustering of Graphs with the Bethe Hessian
Spectral clustering is a standard approach to label nodes on a graph by
studying the (largest or lowest) eigenvalues of a symmetric real matrix such as
e.g. the adjacency or the Laplacian. Recently, it has been argued that using
instead a more complicated, non-symmetric and higher dimensional operator,
related to the non-backtracking walk on the graph, leads to improved
performance in detecting clusters, and even to optimal performance for the
stochastic block model. Here, we propose to use instead a simpler object, a
symmetric real matrix known as the Bethe Hessian operator, or deformed
Laplacian. We show that this approach combines the performances of the
non-backtracking operator, thus detecting clusters all the way down to the
theoretical limit in the stochastic block model, with the computational,
theoretical and memory advantages of real symmetric matrices.Comment: 8 pages, 2 figure
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