The Poincar\'e-Boltzmann Machine: from Statistical Physics to Machine Learning and back

Assessing where and how information is stored in biological networks (such as neuronal and genetic networks) is a central task both in neuroscience and in molecular genetics, but most available tools focus on the network's structure as opposed to its function. Here we introduce a new information-theoretic tool: "information fragmentation analysis" that, given full phenotypic data, allows us to localize information in complex networks, determine how fragmented (across multiple...

Identifying the relevant coarse-grained degrees of freedom in a complex physical system is a key stage in developing powerful effective theories in and out of equilibrium. The celebrated renormalization group provides a framework for this task, but its practical execution in unfamiliar systems is fraught with ad hoc choices, whereas machine learning approaches, though promising, often lack formal interpretability. Recently, the optimal coarse-graining in a statistical system ...

Why complex structures emerge in a real world environment is a fascinating question which has not found any fully satisfactory answer at the point of today. At the border between statistical physics, machine learning and network theory, we provide some computational empirical evidence indicating that the emergence of complex structures in a real world environment is the macroscopic thermodynamic result of the optimal trade-off between the ability to encode information about t...

This paper introduces several fundamental concepts in information theory from the perspective of their origins in engineering. Understanding such concepts is important in neuroscience for two reasons. Simply applying formulae from information theory without understanding the assumptions behind their definitions can lead to erroneous results and conclusions. Furthermore, this century will see a convergence of information theory and neuroscience; information theory will expand ...

The combinatorial basis of entropy, given by Boltzmann, can be written $H = N^{-1} \ln \mathbb{W}$, where $H$ is the dimensionless entropy, $N$ is the number of entities and $\mathbb{W}$ is number of ways in which a given realization of a system can occur (its statistical weight). This can be broadened to give generalized combinatorial (or probabilistic) definitions of entropy and cross-entropy: $H=\kappa (\phi(\mathbb{W}) +C)$ and $D=-\kappa (\phi(\mathbb{P}) +C)$, where $\m...

We formulate an info-clustering paradigm based on a multivariate information measure, called multivariate mutual information, that naturally extends Shannon's mutual information between two random variables to the multivariate case involving more than two random variables. With proper model reductions, we show that the paradigm can be applied to study the human genome and connectome in a more meaningful way than the conventional algorithmic approach. Not only can info-cluster...

It is known that statistical model selection as well as identification of dynamical equations from available data are both very challenging tasks. Physical systems behave according to their underlying dynamical equations which, in turn, can be identified from experimental data. Explaining data requires selecting mathematical models that best capture the data regularities. The existence of fundamental links among physical systems, dynamical equations, experimental data and sta...

Information is a key concept in evolutionary biology. Information is stored in biological organism's genomes, and used to generate the organism as well as to maintain and control it. Information is also "that which evolves". When a population adapts to a local environment, information about this environment is fixed in a representative genome. However, when an environment changes, information can be lost. At the same time, information is processed by animal brains to survive ...

An antithetical concept, adaptive symmetry, to conservative symmetry in physics is proposed to understand the deep neural networks (DNNs). It characterizes the invariance of variance, where a biotic system explores different pathways of evolution with equal probability in absence of feedback signals, and complex functional structure emerges from quantitative accumulation of adaptive-symmetries breaking in response to feedback signals. Theoretically and experimentally, we char...

Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we...