Poster Session (with Coffee & Tea)

Nonequilibrium properties of memristive circuits: connection to Hopfield models

Memristors are nonlinear passive circuit elements which can be thought as time varying resistances. Circuits with memristors have been shown both experimentally and numerically to be of use for various problems of machine learning. In this talk we provide further theoretical background for this statement. We show that the dynamics for memristive circuits is such that a specific class of quadratic functional is being minimized. This shows that this class of optimization problems, in general hard to solve, can be thus approximately quickly solved using memristive circuits. We analyze this statement in the case of random circuits, showing in a certain approximation what is the behavior of the number of stationary points as a function of the topological parameters of the circuit. This provides a connection between spin glasses and the Hopfield model.

Towards a Canonical Divergence within Information Geometry

We propose a new canonical divergence and prove that it is a potential function for the dual structure of a statistical manifold. We recover the canonical divergence proposed by Ay and Amari [1] in the case of: (a) dual flatness [2], (b) conjugate symmetry [3], and (3) statistical symmetry as studied by Henmi and Kobayashi [4]. The new divergence coincides with the well-known canonical divergence of dually flat manifolds [2] and is proportional to the squared Riemannian distance in the self-dual case.

1. Ay, N., Amari, S.I.: A novel approach to canonical divergences within information geometry. Entropy 17, 8111-8129 (2015)
2. Amari, S.I., Nagaoka, H.: Methods of Informatin Geometry, Translations of Mathematical monographs, vol. 191. Oxford University Press (2000)
3. Lauritzen, S.L.: Differential geometry in statistical inference. Lecture Notes-Monograph Series 10, 163-218 (1987)
4. Henmi,M., Kobayashi, R.: Hooke's law in statistical manifolds. Nagoya Math. J. 159, 1-24 (2000)

Agent-based modeling of housing market: can methods of statistical physics predict human behavior?

Joint work: Kirill Glavatskiy, Mikhail Prokopenko, Michael Harre.

Following the crashes of global market in 2008 questions were asked of the ability of economics to address the vulnerability of such an important system. It became evident that conventional tools are not able to address systemic risks in markets with heterogeneous structure, where humans behave according to multiple behavioral patterns. This is the case, in particular, for housing market. Over the recent years there has been observed a steady increase of real estate prices in major metropolitan cities in Australia, while some other cities do not show these trends. This indicates that there might be a possibility of “housing bubble” which will eventually burst.

In this work we employ methods of statistical physics to investigate the evolution of the housing market in Australia and possibilities for existence of the housing bubble and its burst. The ideological background for this approach is the fact that when it comes to macroscopic phenomena, individual human choices do not play a significant role. Rather, their collective behavior determines the overall evolution, and statistical interactions between individuals become decisive. Employing the analogy between humans in economic description and molecules in physical description, we investigate the conditions for human self-organization from the perspective of statistical physics.

We address the problem using three approaches, which complement each other on different levels of description: agent-based numerical simulations, maximum entropy principle, and bifurcations at the phase transitions.

Within agent based modeling we introduce heterogeneous society with agents who follow one of the several behavioral patterns. This corresponds to a multicomponent fluid, which self-organize in the mean-field fashion. Unlike a fluid, the agents are allowed to change their identity, switching between behavioral patterns. Depending on the particular state the current community (city), this evolution may or may not lead to a bifurcation and hence phase transition. Coming close to the bifurcation point would indicate that the particular community is close to a “bubble” state.

Within maximum entropy principle we investigate the macroscopic structure of interacting communities. Introducing the analogue of the entropy for the system of our agents, we identify the set of economic constraints, within which they realize their individual strategies. Maximizing the entropy for each of the community allows us to identify the analogy of the spatial distribution of the temperature. Temperature distribution determines the fluxes in the system and allows us to understand the sources and sinks for the market.

Having the recent census data from the Australian Bureau of Statistics we perform analysis for each community (city) within Australia. By doing this we build a map of Australia, which shows how “heated” different regions are with respect to each other. This information can be used by policy making agencies to properly react on market challenges.

Robots can Understand Physics from Fisher Information

A Defense Against Adversarial Examples based on Image Reconstruction by Variational Autoencoders

Self-Organised Transitions in Swarms with Turing Patterns

Mean fields in networks of interacting particles

Quantum Information Geometry and Stochastic Reconfiguration

Complex systems and Geometric structures

Loosely speaking a complex system is a measurable set $(\Xi,\Omega)$; $\Gamma(\Xi,\Omega)$ is the group of measurable isomorphisms of $\Xi$ (viz the group of efficient statistics.) An information Geometry of $(\Xi,\Omega)$ is a $\Gamma$-Geometry in a statistical model of $(\Xi,\Omega)$. Relevant informations are invariants of such a $\Gamma$-geometry. The relevancy of informations is linked with the existence of nice geometric structures in both $(\Xi,\Omega)$ and its chosen model. Among rich geometries in statistical models are the symplectic geometry, the geometry of Koszul, the bi-invariant Riemannian geometry in Lie groups, the left invariant symplectic geometry in Lie groups. The aim of the talk is to address those concerns, some related open geometric problems and a few recent contributions.

• Amari and Nagaoka: Methods of information geometry, AMS-Oxford monograph 91.
• Barbaresco F. GeometriC Theory of Heat from Souriau Lie group thermodynamics and Koszul geomtry; Entropy 2016.
• Nguiffo Boyom M. Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and cohomology; Entropy 12 vol 18 2016.
• Nguiffo Boyom M. Numerical properties of Koszul connections; arxiv.1708.01106. 3 august 2017.

The Failure of the MaxEnt Principle for the generalized entropies

On the Geometry of the Latent Space of Variational AutoEncoders: An Explanatory Analysis

On the Use of Natural Gradient for Variational AutoEncoders

Decomposing multivariate information

We propose a decomposition of multivariate information which is based on a generalisation of Amari's hierarchy over a lattice imposed on combinations of primary random variables, so-called "structures". While related, our construction differs from the well-known lattice construction of Williams/Beer's in that the quantities to be interpreted as information terms sit on the edges and that no variable set is distinguished as a predictor of others; all variables are on the same level, similar to the approach by Rosas et al. (2016). We show that this construction can address some of the questions posed by James & Crutchfield (2017)

Geometry of latent representations for word embeddings