3,645 research outputs found
Emergence of Compositional Representations in Restricted Boltzmann Machines
Extracting automatically the complex set of features composing real
high-dimensional data is crucial for achieving high performance in
machine--learning tasks. Restricted Boltzmann Machines (RBM) are empirically
known to be efficient for this purpose, and to be able to generate distributed
and graded representations of the data. We characterize the structural
conditions (sparsity of the weights, low effective temperature, nonlinearities
in the activation functions of hidden units, and adaptation of fields
maintaining the activity in the visible layer) allowing RBM to operate in such
a compositional phase. Evidence is provided by the replica analysis of an
adequate statistical ensemble of random RBMs and by RBM trained on the
handwritten digits dataset MNIST.Comment: Supplementary material available at the authors' webpag
Probing the entanglement and locating knots in ring polymers: a comparative study of different arc closure schemes
The interplay between the topological and geometrical properties of a polymer
ring can be clarified by establishing the entanglement trapped in any portion
(arc) of the ring. The task requires to close the open arcs into a ring, and
the resulting topological state may depend on the specific closure scheme that
is followed. To understand the impact of this ambiguity in contexts of
practical interest, such as knot localization in a ring with non trivial
topology, we apply various closure schemes to model ring polymers. The rings
have the same length and topological state (a trefoil knot) but have different
degree of compactness. The comparison suggests that a novel method, termed the
minimally-interfering closure, can be profitably used to characterize the arc
entanglement in a robust and computationally-efficient way. This closure method
is finally applied to the knot localization problem which is tackled using two
different localization schemes based on top-down or bottom-up searches.Comment: 9 pages, 7 figures. Submitted to Progress of Theoretical Physic
L'homme et l'animal dans le bassin du lac Tchad
Le mythe de l'antilope (oryx, damalisque ou bubale selon les populations concernĂ©es) fait partie de la cultures des "Arabes Zaghawa" appelĂ©s parfois aussi Djumbo Dirong, des Mimi Ab-tetel, des Dadjo. Un despote, dont les faits et gestes abusifs sont minutieusement dĂ©crits, est invitĂ© par ceux qui souhaitent s'affranchir de sa tyrannie, Ă prendre pour monture une antilope. Alors qu'il galope, ligotĂ© sur le dos de l'animal, Ă l'intĂ©rieur d'un cercle composĂ© de ses parents et alliĂ©s, le cercle s'ouvre et le tyran est emportĂ© vers l'ouest. Sa course se termine dans un groupe Ă©tranger et son mariage avec une fille de ce groupe est Ă l'origine d'un nouveau groupe ethnique ; dans une variante ses membres dĂ©chiquetĂ©s jalonnent sa fuite et donnent naissance Ă autant de groupes humains. Des deux schĂ©mas correspondent Ă l'histoire du peuplement que permettent de retracer les enquĂȘtes de terrain. On retrouve ce mythe dans l'histoire de divers groupes sara. Il est alors "rĂ©duite" Ă la seule partie fondation d'un groupe humain et le hĂ©ros n'apparaĂźt plus comme un tyran mais comme un homme sage et gĂ©nĂ©reux. (RĂ©sumĂ© d'auteur
Equilibrium and kinetic properties of knotted ring polymers: a computational approach
We provide hereafter a summary of the Thesis organization. Chapter 1 contains a short
introduction to the mathematical theory of knots. Starting from the mathematical
definition of knotting, we introduce the fundamental concepts and knot properties used
throughout this Thesis.
In chapter 2 we tackle the problem of measuring the degree of localization of a knot.
This is in general a very challenging task, involving the assignment of a topological state
to open arcs of the ring. To assign a topological state to an open arc, one must first close it
into a ring whose topological state can be assessed using the tools introduced in chapter 1.
Consequently, the resulting topological state may depend on the specific closure scheme
that is followed. To reduce this ambiguity we introduce a novel closure scheme, the
minimally-interfering closure. We prove the robustness of the minimally-interfering
closure by comparing its results against several standard closure schemes.
We further show that the identified knotted portion depends also on the search
algorithm adopted to find it. The knot search algorithms adopted in literature can be
divided in two general categories: bottom-up searches and top-down searches. We show
that bottom-up and knot-down searches give in general different results for the length of
a knot, the difference increasing with increasing length of the polymer rings. We suggest
that this systematic difference can explain the discrepancies between previous numerical
results on the scaling behaviour of the knot length with increasing length of polymer
rings in good solvent.
In chapter 3 we investigate the mutual entanglement between multiple prime knots
tied on the same ring. Knots like these, which can be decomposed into simpler ones, are
called composite knots and dominate the knot spectrum of sufficiently long polymers [131].
Since prime knots are expected to localize to point-like decorations for asymptotically
large chain lengths, it is expected that composite knots should factorize into separate
prime components [101, 82, 43, 11]. Therefore the asymptotic properties of composite
knots should merely depend on the number of prime components (factor knots) by which
they are formed [101, 82, 43, 11] and the properties of the single prime components should be largely independent from the presence of other knots on the ring. We show that this
factorization into separate prime components is only partial for composite knots which
are dominant in an equilibrium population of Freely Jointed Rings. As a consequence
the properties of those prime knots which are found as separate along the chain depend
on the number of knots tied on it. We further show that these results can be explained
using a transparent one-dimensional model in which prime knots are substituted with
paraknots.
Chapters 4, 5 and 6 are dedicated to investigate the interplay between topological
entanglement and geometrical entanglement produced either by surrounding rings in a
dense solution or spherical confinement.
In chapter 4 we investigate the equilibrium and kinetic properties of solutions of
model ring polymers, modulating the interplay of inter- and intra-chain entanglement by
varying both solution density (from infinite dilution up to 40% volume occupancy) and
ring topology (by considering unknotted and trefoil-knotted chains). The equilibrium
metric properties of rings with either topology are found to be only weakly affected
by the increase of solution density. Even at the highest density, the average ring size,
shape anisotropy and length of the knotted region differ at most by 40% from those
of isolated rings. Conversely, kinetics are strongly affected by the degree of inter-chain
entanglement: for both unknots and trefoils the characteristic times of ring size relaxation,
reorientation and diffusion change by one order of magnitude across the considered range
of concentrations. Yet, significant topology-dependent differences in kinetics are observed
only for very dilute solutions (much below the ring overlap threshold). For knotted rings,
the slowest kinetic process is found to correspond to the diffusion of the knotted region
along the ring backbone.
In chapter 5 we study the interplay of geometrical and topological entanglement in
semiflexible knotted polymer rings under spherical confinement. We first characterize
how the top-down knot length lk depends on the ring contour length, Lc and the radius of
the confining sphere, Rc. In the no- and strong-confinement cases we observe weak knot
localization and complete knot delocalization, respectively. We show that the complex
interplay of lk, Lc and Rc that seamlessly bridges these two limits can be encompassed
by a simple scaling argument based on deflection theory. We then move to study the
behaviour of the bottom-up knot length lsk under the same conditions and observe
that it follows a qualitatively different behaviour from lk, decreasing upon increasing
confinement. The behaviour of lsk is rationalized using the same argument based on deflection theory. The qualitative difference between the two knot lengths highlights a
multiscale character of the entanglement emerging upon increasing confinement.
Finally, in chapter 6 we adopt a complementary approach, using topological analysis
(the properties of the knot spectrum) to infer the physical properties of packaged
bacteriophage genome. With their m long dsDNA genome packaged inside capsids
whose diameter are in the 50 80 nm range, bacteriophages bring the highest level of
compactification and arguably the simplest example of genome organization in living
organisms [31, 40]. Cryo-em studies showed that DNA in bacteriophages epsilon-15 and
phi-29 is neatly ordered in concentric shells close to the capsid wall, while an increasing
level of disorder was measured when moving away from the capsid internal surface. On
the other hand the detected spectrum of knots formed by DNA that is circularised
inside the P4 viral capsid showed that DNA tends to be knotted with high probability,
with a knot spectrum characterized by complex knots and biased towards torus knots
and against achiral ones. Existing coarse-grain DNA models, while being capable of
reproducing the salient physical aspects of free, unconstrained DNA, are not able to
reproduce the experimentally observed features of packaged viral DNA. We show, using
stochastic simulation techniques, that both the shell ordering and the knot spectrum
can be reproduced quantitatively if one accounts for the preference of contacting DNA
strands to juxtapose at a small twist angle, as in cholesteric liquid crystals
Statistical Physics and Representations in Real and Artificial Neural Networks
This document presents the material of two lectures on statistical physics
and neural representations, delivered by one of us (R.M.) at the Fundamental
Problems in Statistical Physics XIV summer school in July 2017. In a first
part, we consider the neural representations of space (maps) in the
hippocampus. We introduce an extension of the Hopfield model, able to store
multiple spatial maps as continuous, finite-dimensional attractors. The phase
diagram and dynamical properties of the model are analyzed. We then show how
spatial representations can be dynamically decoded using an effective Ising
model capturing the correlation structure in the neural data, and compare
applications to data obtained from hippocampal multi-electrode recordings and
by (sub)sampling our attractor model. In a second part, we focus on the problem
of learning data representations in machine learning, in particular with
artificial neural networks. We start by introducing data representations
through some illustrations. We then analyze two important algorithms, Principal
Component Analysis and Restricted Boltzmann Machines, with tools from
statistical physics
Multiscale entanglement in ring polymers under spherical confinement
The interplay of geometrical and topological entanglement in semiflexible
knotted polymer rings confined inside a spherical cavity is investigated using
advanced numerical methods. By using stringent and robust algorithms for
locating knots, we characterize how the knot length lk depends on the ring
contour length, Lc and the radius of the confining sphere, Rc . In the no- and
strong- confinement cases we observe weak knot localization and complete knot
delocalization, respectively. We show that the complex interplay of lk, Lc and
Rc that seamlessly bridges these two limits can be encompassed by a simple
scaling argument based on deflection theory. The same argument is used to
rationalize the multiscale character of the entanglement that emerges with
increasing confinement.Comment: 9 pages 9 figure
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