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Intersection property, interaction decomposition, regionalized optimization and applications.
Preprints and early-stage research may not have been peer reviewed yet.
Abstract
PhD thesis
We define the notion of a measure family: a pre-cosheaf of finite measures over a finite set; every joint measure on a product of finite sets has an associated measure family. To each measure family there is an associated index, or “Euler characteristic”, related to the Tsallis deformation of mutual information. This index is further categorified by a (weighted) simplicial complex whose topology retains information about the correlations between various subsystems.
To perform an accurate protein synthesis, ribosomes accomplish complex tasks involving the long-range communication between its functional centres such as the peptidyl transfer centre, the tRNA bindings sites and the peptide exit tunnel. How information is transmitted between these sites remains one of the major challenges in current ribosome research. Many experimental studies have revealed that some r-proteins play essential roles in remote communication and the possible involvement of r-protein networks in these processes have been recently proposed. Our phylogenetic, structural and mathematical study reveals that of the three kingdom’s r-protein networks converged towards non-random graphs where r-proteins collectively coevolved to optimize interconnection between functional centres. The massive acquisition of conserved aromatic residues at the interfaces and along the extensions of the newly connected eukaryotic r-proteins also highlights that a strong selective pressure acts on their sequences probably for the formation of new allosteric pathways in the network.
For 15 years, the eukaryote Tree of Life (eToL) has been divided into five to eight major groupings, known as 'supergroups'. However, the tree has been profoundly rearranged during this time. The new eToL results from the widespread application of phylogenomics and numerous discoveries of major lineages of eukaryotes, mostly free-living heterotrophic protists. The evidence that supports the tree has transitioned from a synthesis of molecular phylogenetics and biological characters to purely molecular phylogenetics. Most current supergroups lack defining morphological or cell-biological characteristics, making the supergroup label even more arbitrary than before. Going forward, the combination of traditional culturing with maturing culture-free approaches and phylogenomics should accelerate the process of completing and resolving the eToL at its deepest levels.
This paper outlines a program in what one might call spectral sheaf theory—an extension of spectral graph theory to cellular sheaves. By lifting the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular cell complex, one can relate spectral data to the sheaf cohomology and cell structure in a manner reminiscent of spectral graph theory. This work gives an exploratory introduction, and includes discussion of eigenvalue interlacing, sparsification, effective resistance, synchronization, and sheaf approximation. These results and subsequent applications are prefaced by an introduction to cellular sheaves and Laplacians.
In the past few decades, studies on translation have converged towards the metaphor of a “ribosome nanomachine”; they also revealed intriguing ribosome properties challenging this view. Many studies have shown that to perform an accurate protein synthesis in a fluctuating cellular environment, ribosomes sense, transfer information and even make decisions. This complex “behaviour” that goes far beyond the skills of a simple mechanical machine has suggested that the ribosomal protein networks could play a role equivalent to nervous circuits at a molecular scale to enable information transfer and processing during translation. We analyse here the significance of this analogy and establish a preliminary link between two fields: ribosome structure-function studies and the analysis of information processing systems. This cross-disciplinary analysis opens new perspectives about the mechanisms of information transfer and processing in ribosomes and may provide new conceptual frameworks for the understanding of the behaviours of unicellular organisms.
We define and study a probability monad on the category of complete metric spaces and short maps. It assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich-Wasserstein distance. This monad is analogous to the Giry monad on the category of Polish spaces, and it extends a construction due to van Breugel for compact and for 1-bounded complete metric spaces.
We prove that this Kantorovich monad arises from a colimit construction on finite power-like constructions, which formalizes the intuition that probability measures are limits of finite samples. The proof relies on a criterion for when an ordinary left Kan extension of lax monoidal functors is a monoidal Kan extension.
The colimit characterization allows the development of integration theory and the treatment of measures on spaces of measures, without measure theory.
We also show that the category of algebras of the Kantorovich monad is equivalent to the category of closed convex subsets of Banach spaces with short affine maps as morphisms.
In this paper we develop a novel mathematical formalism for the modeling of neural information networks endowed with additional structure in the form of assignments of resources, either computational or metabolic or informational. The starting point for this construction is the notion of summing functors and of Segal's Gamma-spaces in homotopy theory. The main results in this paper include functorial assignments of concurrent/distributed computing architectures and associated binary codes to networks and their subsystems, a categorical form of the Hopfield network dynamics, which recovers the usual Hopfield equations when applied to a suitable category of weighted codes, a functorial assignment to networks of corresponding information structures and information cohomology, and a cohomological version of integrated information.
This thesis extends in several directions the cohomological study of information theory pioneered by Baudot and Bennequin. We introduce a topos-theoretical notion of statistical space and then study several cohomological invariants. Information functions and related objects appear as distinguished cohomology classes; the corresponding cocycle equations encode recursive properties of these functions. Information has thus topological meaning and topology serves as a unifying framework.Part I discusses the geometrical foundations of the theory. Information structures are introduced as categories that encode the relations of refinement between different statistical observables. We study products and coproducts of information structures, as well as their representation by measurable functions or hermitian operators. Every information structure gives rise to a ringed site; we discuss in detail the definition of information cohomology using the homological tools developed by Artin, Grothendieck, Verdier and their collaborators.Part II studies the cohomology of discrete random variables. Information functions—Shannon entropy, Tsallis alpha-entropy, Kullback-Leibler divergence—appear as 1-cocycles for appropriate modules of probabilistic coefficients (functions of probability laws). In the combinatorial case (functions of histograms), the only 0-cocycle is the exponential function, and the 1-cocycles are generalized multinomial coefficients (Fontené-Ward). There is an asymptotic relation between the combinatorial and probabilistic cocycles.Part III studies in detail the q-multinomial coefficients, showing that their growth rate is connected to Tsallis 2-entropy (quadratic entropy). When q is a prime power, these q-multinomial coefficients count flags of finite vector spaces with prescribed length and dimensions. We obtain a combinatorial explanation for the nonadditivity of the quadratic entropy and a frequentist justification for the maximum entropy principle with Tsallis statistics. We introduce a discrete-time stochastic process associated to the q-binomial probability distribution that generates finite vector spaces (flags of length 2). The concentration of measure on certain typical subspaces allows us to extend Shannon's theory to this setting.Part IV discusses the generalization of information cohomology to continuous random variables. We study the functoriality properties of conditioning (seen as disintegration) and its compatibility with marginalization. The cohomological computations are restricted to the real valued, gaussian case. When coordinates are fixed, the 1-cocycles are the differential entropy as well as generalized moments. When computations are done in a coordinate-free manner, with the so-called grassmannian categories, we recover as the only degree-one cohomology classes the entropy and the dimension. This constitutes a novel algebraic characterization of differential entropy.
We develop Markov categories as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs. This means that we treat the following concepts in purely abstract categorical terms: conditioning and disintegration; various versions of conditional independence and its standard properties; conditional products; almost surely; sufficient statistics; versions of theorems on sufficient statistics due to Fisher–Neyman, Basu, and Bahadur.
Besides the conceptual clarity offered by our categorical setup, its main advantage is that it provides a uniform treatment of various types of probability theory, including discrete probability theory, measure-theoretic probability with general measurable spaces, Gaussian probability, stochastic processes of either of these kinds, and many others.
