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Guohua Zhang - One of the best experts on this subject based on the ideXlab platform.
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Local Entropy theory of a random dynamical system
2015Co-Authors: A H Dooley, Guohua ZhangAbstract:Introduction Preliminaries: Infinite countable discrete amenable groups Measurable dynamical systems Continuous bundle random dynamical systems A Local Variational Principle for Fiber Topological Pressure: Local fiber topological pressure Factor excellent and good covers A variational principle for Local fiber topological pressure Proof of main result Theorem 7.1 Assumption i?½ on the family D The Local variational principle for amenable groups admitting a tiling Folner sequence Another version of the Local variational principle Applications of the Local Variational Principle Entropy tuples for a continuous bundle random dynamical system Bibliography
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relativization of dynamical properties
Science China-mathematics, 2012Co-Authors: Guohua ZhangAbstract:In the past twenty years, great achievements have been made by many researchers in the studies of chaotic behavior and Local Entropy theory of dynamical systems. Most of the results have been generalized to the relative case in the sense of a given factor map. In this survey we offer an overview of these developments.
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Local Entropy theory for a countable discrete amenable group action
Journal of Functional Analysis, 2011Co-Authors: Wen Huang, Guohua ZhangAbstract:In the paper we throw the first light on studying systematically the Local Entropy theory for a countable discrete amenable group action. For such an action, we introduce Entropy tuples in both topological and measure-theoretic settings and build the variational relation be- tween these two kinds of Entropy tuples by establishing a Local variational principle for a given finite open cover. Moreover, based the idea of topological Entropy pairs, we introduce and study two special classes of such an action: uniformly positive Entropy and completely positive Entropy. Note that in the building of the Local variational principle, following Romagnoli's ideas two kinds of measure-theoretic Entropy are introduced for finite Borel covers. These two kinds of Entropy turn out to be the same, where Danilenko's orbital approach becomes an inevitable tool.
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Local Entropy theory of a random dynamical system
arXiv: Dynamical Systems, 2011Co-Authors: A H Dooley, Guohua ZhangAbstract:In this paper we extend the notion of a continuous bundle random dynamical system to the setting where the action of $\R$ or $\N$ is replaced by the action of an infinite countable discrete amenable group. Given such a system, and a monotone sub-additive invariant family of random continuous functions, we introduce the concept of Local fiber topological pressure and establish an associated variational principle, relating it to measure-theoretic Entropy. We also discuss some variants of this variational principle. We introduce both topological and measure-theoretic Entropy tuples for continuous bundle random dynamical systems, and apply our variational principles to obtain a relationship between these of Entropy tuples. Finally, we give applications of these results to general topological dynamical systems, recovering and extending many recent results in Local Entropy theory.
Riccardo Zecchina - One of the best experts on this subject based on the ideXlab platform.
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Entropy sgd biasing gradient descent into wide valleys
International Conference on Learning Representations, 2017Co-Authors: Pratik Chaudhari, Carlo Baldassi, Anna Choromanska, Stefano Soatto, Yann Lecun, Christian Borgs, Jennifer Chayes, Levent Sagun, Riccardo ZecchinaAbstract:This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the Local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenvalues. We leverage upon this observation to construct a Local-Entropy-based objective function that favors well-generalizable solutions lying in large flat regions of the energy landscape, while avoiding poorly-generalizable solutions located in the sharp valleys. Conceptually, our algorithm resembles two nested loops of SGD where we use Langevin dynamics in the inner loop to compute the gradient of the Local Entropy before each update of the weights. We show that the new objective has a smoother energy landscape and show improved generalization over SGD using uniform stability, under certain assumptions. Our experiments on convolutional and recurrent neural networks demonstrate that Entropy-SGD compares favorably to state-of-the-art techniques in terms of generalization error and training time.
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Entropy sgd biasing gradient descent into wide valleys
arXiv: Learning, 2016Co-Authors: Pratik Chaudhari, Carlo Baldassi, Anna Choromanska, Stefano Soatto, Yann Lecun, Christian Borgs, Jennifer Chayes, Levent Sagun, Riccardo ZecchinaAbstract:This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the Local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenvalues. We leverage upon this observation to construct a Local-Entropy-based objective function that favors well-generalizable solutions lying in large flat regions of the energy landscape, while avoiding poorly-generalizable solutions located in the sharp valleys. Conceptually, our algorithm resembles two nested loops of SGD where we use Langevin dynamics in the inner loop to compute the gradient of the Local Entropy before each update of the weights. We show that the new objective has a smoother energy landscape and show improved generalization over SGD using uniform stability, under certain assumptions. Our experiments on convolutional and recurrent networks demonstrate that Entropy-SGD compares favorably to state-of-the-art techniques in terms of generalization error and training time.
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Local Entropy as a measure for sampling solutions in constraint satisfaction problems
Journal of Statistical Mechanics: Theory and Experiment, 2016Co-Authors: Carlo Baldassi, Alessandro Ingrosso, Carlo Lucibello, Luca Saglietti, Riccardo ZecchinaAbstract:We introduce a novel Entropy-driven Monte Carlo (EdMC) strategy to efficiently sample solutions of random constraint satisfaction problems (CSPs). First, we extend a recent result that, using a large-deviation analysis, shows that the geometry of the space of solutions of the binary perceptron learning problem (a prototypical CSP), contains regions of very high-density of solutions. Despite being sub-dominant, these regions can be found by optimizing a Local Entropy measure. Building on these results, we construct a fast solver that relies exclusively on a Local Entropy estimate, and can be applied to general CSPs. We describe its performance not only for the perceptron learning problem but also for the random K-satisfiabilty problem (another prototypical CSP with a radically different structure), and show numerically that a simple zero-temperature Metropolis search in the smooth Local Entropy landscape can reach sub-dominant clusters of optimal solutions in a small number of steps, while standard Simulated Annealing either requires extremely long cooling procedures or just fails. We also discuss how the EdMC can heuristically be made even more efficient for the cases we studied.
C A Stafford - One of the best experts on this subject based on the ideXlab platform.
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Local Entropy of a nonequilibrium fermion system
arXiv: Mesoscale and Nanoscale Physics, 2016Co-Authors: C A Stafford, Abhay ShastryAbstract:The Local Entropy of a nonequilibrium system of independent fermions is investigated, and analyzed in the context of the laws of thermodynamics. It is shown that the Local temperature and chemical potential can only be expressed in terms of derivatives of the Local Entropy for linear deviations from Local equilibrium. The first law of thermodynamics is shown to lead to an inequality, not an equality, for the change in the Local Entropy as the nonequilibrium state of the system is changed. The maximum Entropy principle (second law of thermodynamics) is proven: a nonequilibrium distribution has a Local Entropy less than or equal to a Local equilibrium distribution satisfying the same constraints. It is shown that the Local Entropy of the system tends to zero when the Local temperature tends to zero, consistent with the third law of thermodynamics.
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cold spots in quantum systems far from equilibrium Local entropies and temperatures near absolute zero
Physical Review B, 2015Co-Authors: Abhay Shastry, C A StaffordAbstract:We consider a question motivated by the third law of thermodynamics: can there be a Local temperature arbitrarily close to absolute zero in a nonequilibrium quantum system? We consider nanoscale quantum conductors with the source reservoir held at finite temperature and the drain held at or near absolute zero, a problem outside the scope of linear response theory. We obtain Local temperatures close to absolute zero when electrons originating from the finite temperature reservoir undergo destructive quantum interference. The Local temperature is computed by numerically solving a nonlinear system of equations describing equilibration of a scanning thermoelectric probe with the system, and we obtain excellent agreement with analytic results derived using the Sommerfeld expansion. A Local Entropy for a nonequilibrium quantum system is introduced, and used as a metric quantifying the departure from Local equilibrium. It is shown that the Local Entropy of the system tends to zero when the probe temperature tends to zero, consistent with the third law of thermodynamics.
Sayantani Bhattacharyya - One of the best experts on this subject based on the ideXlab platform.
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Entropy current from partition function one example
arXiv: High Energy Physics - Theory, 2014Co-Authors: Sayantani BhattacharyyaAbstract:In hydrodynamics the existence of an Entropy current with non-negative divergence is related to the existence of a time-independent solution in a static background. Recently there has been a proposal for how to construct an Entropy current from the equilibrium partition function of the fluid system. In this note, we have applied this algorithm for the charged fluid at second order in derivative expansion. From the partition function we first constructed one example of Entropy current with non-negative divergence upto the required order. Finally we extended it to its most general form, consistent with the principle of Local Entropy production. As a by-product we got the constraints on the second order transport coefficients for a parity even charged fluid, but in some non-standard fluid frame.
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constraints on the second order transport coefficients of an uncharged fluid
Journal of High Energy Physics, 2012Co-Authors: Sayantani BhattacharyyaAbstract:In this note we have tried to determine how the existence of a Local Entropy current with non-negative divergence constrains the second order transport coefficients of an uncharged fluid, following the procedure described in [1]. Just on symmetry ground the stress tensor of an uncharged fluid can have 15 transport coefficients at second order in derivative expansion. The condition of Entropy-increase gives five relations among these 15 coefficients. So finally the relativistic stress tensor of an uncharged fluid can have 10 independent transport coefficients at second order.
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constraints on the second order transport coefficients of an uncharged fluid
arXiv: High Energy Physics - Theory, 2012Co-Authors: Sayantani BhattacharyyaAbstract:In this note we have tried to determine how the existence of a Local Entropy current with non-negative divergence constrains the second order transport coefficients of an uncharged fluid, following the procedure described in \cite{Romatschke:2009kr}. Just on symmetry ground the stress tensor of an uncharged fluid can have 15 transport coefficients at second order in derivative expansion. The condition of Entropy-increase gives five relations among these 15 coefficients. So finally the relativistic stress tensor of an uncharged fluid can have 10 independent transport coefficients at second order.
Günter Mahler - One of the best experts on this subject based on the ideXlab platform.
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quantum thermodynamics emergence of thermodynamic behavior within composite quantum systems
2004Co-Authors: Jochen Gemmer, M Michel, Günter MahlerAbstract:Background.- Basics of Quantum Mechanics.- Basics of Thermodynamics and Statistics.- Brief Review of Pertinent Concepts.- Equilibrium.- The Program for the Foundation of Thermodynamics.- Outline of the Present Approach.- Dynamics and Averages in Hilbert Space.- Typicality of Observables and States.- System and Environment.- The Typical Reduced State of the System.- Entanglement, Correlations and Local Entropy.- Generic Spectra of Large Systems.- Temperature.- Pressure and Adiabatic Processes.- Quantum Mechanical and Classical State Densities.- Equilibration in Model Systems.- Non-Equilibrium.- Brief Review of Relaxation and Transport Theories.- Projection Operator Techniques and Hilbert Space Average Method.- Finite Systems as Thermostats.- Projective Approach to Dynamical Transport.- Open System Approach to Transport.- Applications and Models.- Purity and Local Entropy in Product Hilbert Space.- Observability of Intensive Variables.- Observability of Extensive Variables.- Quantum Thermodynamic Processes.
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distribution of Local Entropy in the hilbert space of bi partite quantum systems origin of jaynes principle
European Physical Journal B, 2003Co-Authors: Jochen Gemmer, Günter MahlerAbstract:For a closed bi-partite quantum system partitioned into system proper and environment we interpret the microcanonical and the canonical condition as constraints for the interaction between those two subsystems. In both cases the possible pure-state trajectories are confined to certain regions in Hilbert space. We show that in a properly defined thermodynamical limit almost all states within those accessible regions represent states of some maximum Local Entropy. For the microcanonical condition this dominant state still depends on the initial state; for the canonical condition it coincides with that defined by Jaynes' principle. It is these states which thermodynamical systems should generically evolve into.
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distribution of Local Entropy in the hilbert space of bi partite quantum systems origin of jaynes principle
arXiv: Quantum Physics, 2002Co-Authors: Jochen Gemmer, Günter MahlerAbstract:For a closed bi-partite quantum system partitioned into system proper and environment we interprete the microcanonical and the canonical condition as constraints for the interaction between those two subsystems. In both cases the possible pure-state trajectories are confined to certain regions in Hilbert space. We show that in a properly defined thermodynamical limit almost all states within those accessible regions represent states of some maximum Local Entropy. For the microcanonical condition this dominant state still depends on the initial state; for the canonical condition it coincides with that defined by Jaynes' principle. It is these states which thermodynamical systems should generically evolve into.