555 research outputs found
Encoding for the Blackwell Channel with Reinforced Belief Propagation
A key idea in coding for the broadcast channel (BC) is binning, in which the
transmitter encode information by selecting a codeword from an appropriate bin
(the messages are thus the bin indexes). This selection is normally done by
solving an appropriate (possibly difficult) combinatorial problem. Recently it
has been shown that binning for the Blackwell channel --a particular BC-- can
be done by iterative schemes based on Survey Propagation (SP). This method uses
decimation for SP and suffers a complexity of O(n^2). In this paper we propose
a new variation of the Belief Propagation (BP) algorithm, named Reinforced BP
algorithm, that turns BP into a solver. Our simulations show that this new
algorithm has complexity O(n log n). Using this new algorithm together with a
non-linear coding scheme, we can efficiently achieve rates close to the border
of the capacity region of the Blackwell channel.Comment: 5 pages, 8 figures, submitted to ISIT 200
Structure of the space of folding protein sequences defined by large language models
Proteins populate a manifold in the high-dimensional sequence space whose
geometrical structure guides their natural evolution. Leveraging
recently-developed structure prediction tools based on transformer models, we
first examine the protein sequence landscape as defined by the folding score
function. This landscape shares characteristics with optimization challenges
encountered in machine learning and constraint satisfaction problems. Our
analysis reveals that natural proteins predominantly reside in wide, flat
minima within this energy landscape. To investigate further, we employ
statistical mechanics algorithms specifically designed to explore regions with
high local entropy in relatively flat landscapes. Our findings indicate that
these specialized algorithms can identify valleys with higher entropy compared
to those found using traditional methods such as Monte Carlo Markov Chains. In
a proof-of-concept case, we find that these highly entropic minima exhibit
significant similarities to natural sequences, especially in critical key sites
and local entropy. Additionally, evaluations through Molecular Dynamics
suggests that the stability of these sequences closely resembles that of
natural proteins. Our tool combines advancements in machine learning and
statistical physics, providing new insights into the exploration of sequence
landscapes where wide, flat minima coexist alongside a majority of narrower
minima
Native state of natural proteins optimizes local entropy
The differing ability of polypeptide conformations to act as the native state of proteins has long been rationalized in terms of differing kinetic accessibility or thermodynamic stability. Building on the successful applications of physical concepts and sampling algorithms recently introduced in the study of disordered systems, in particular artificial neural networks, we quantitatively explore how well a quantity known as the local entropy describes the native state of model proteins. In lattice models and all-atom representations of proteins, we are able to efficiently sample high local entropy states and to provide a proof of concept of enhanced stability and folding rate. Our methods are based on simple and general statistical-mechanics arguments, and thus we expect that they are of very general use
Survey Propagation as local equilibrium equations
It has been shown experimentally that a decimation algorithm based on Survey
Propagation (SP) equations allows to solve efficiently some combinatorial
problems over random graphs. We show that these equations can be derived as
sum-product equations for the computation of marginals in an extended space
where the variables are allowed to take an additional value -- -- when they
are not forced by the combinatorial constraints. An appropriate ``local
equilibrium condition'' cost/energy function is introduced and its entropy is
shown to coincide with the expected logarithm of the number of clusters of
solutions as computed by SP. These results may help to clarify the geometrical
notion of clusters assumed by SP for the random K-SAT or random graph coloring
(where it is conjectured to be exact) and helps to explain which kind of
clustering operation or approximation is enforced in general/small sized models
in which it is known to be inexact.Comment: 13 pages, 3 figure
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
Bicoloring Random Hypergraphs
We study the problem of bicoloring random hypergraphs, both numerically and
analytically. We apply the zero-temperature cavity method to find analytical
results for the phase transitions (dynamic and static) in the 1RSB
approximation. These points appear to be in agreement with the results of the
numerical algorithm. In the second part, we implement and test the Survey
Propagation algorithm for specific bicoloring instances in the so called
HARD-SAT phase.Comment: 14 pages, 10 figure
Statistical Mechanics of Steiner trees
The Minimum Weight Steiner Tree (MST) is an important combinatorial
optimization problem over networks that has applications in a wide range of
fields. Here we discuss a general technique to translate the imposed global
connectivity constrain into many local ones that can be analyzed with cavity
equation techniques. This approach leads to a new optimization algorithm for
MST and allows to analyze the statistical mechanics properties of MST on random
graphs of various types
Lossy data compression with random gates
We introduce a new protocol for a lossy data compression algorithm which is
based on constraint satisfaction gates. We show that the theoretical capacity
of algorithms built from standard parity-check gates converges exponentially
fast to the Shannon's bound when the number of variables seen by each gate
increases. We then generalize this approach by introducing random gates. They
have theoretical performances nearly as good as parity checks, but they offer
the great advantage that the encoding can be done in linear time using the
Survey Inspired Decimation algorithm, a powerful algorithm for constraint
satisfaction problems derived from statistical physics
Entropy landscape and non-Gibbs solutions in constraint satisfaction problems
We study the entropy landscape of solutions for the bicoloring problem in
random graphs, a representative difficult constraint satisfaction problem. Our
goal is to classify which type of clusters of solutions are addressed by
different algorithms. In the first part of the study we use the cavity method
to obtain the number of clusters with a given internal entropy and determine
the phase diagram of the problem, e.g. dynamical, rigidity and SAT-UNSAT
transitions. In the second part of the paper we analyze different algorithms
and locate their behavior in the entropy landscape of the problem. For instance
we show that a smoothed version of a decimation strategy based on Belief
Propagation is able to find solutions belonging to sub-dominant clusters even
beyond the so called rigidity transition where the thermodynamically relevant
clusters become frozen. These non-equilibrium solutions belong to the most
probable unfrozen clusters.Comment: 38 pages, 10 figure
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