arXiv subjects

Grégoire Sergeant-Perthuis

Publications and source records attributed to Grégoire Sergeant-Perthuis.

14 recordsLinked to original sources

arXiv2026

Exact Algebraic Computation of Learning Coefficients for Two-Dimensional Singular Models

Classical information criteria such as the Bayesian Information Criterion (BIC) rely on regularity assumptions that break down for singular models, leading to incorrect model selection in settings such as deep learning. The Widely Applicable Bayesian Information Criterion (WBIC) relies on local learning coefficients $λ$, which in the analytic case coincides with local Real Log Canonical Thresholds (RLCT) of the Kullback-Leibler divergence of the model, to capture correct marginal likelihood asymptotics. Exact computation of the learning coefficients has been limited to special cases, and only sampling-based estimation methods are generally applicable. We present the first deterministic algorithm that computes local RLCTs exactly for any two-dimensional model whose Kullback-Leibler distance is contact equivalent to a polynomial, derive a bound on its complexity, and demonstrate its effectiveness for a broad class of models, with applications including polynomial neural networks. Beyond providing ground truth to calibrate sampling-based estimators, exact computation reveals algebraic structure in learning coefficients that sampling cannot and out-speeds it in the shallow regime.

cs.LG↗

arXiv2026

Implementation of reinforcement learning in chemical reaction networks: application to phototaxis as curiosity-driven exploration

Living systems navigate environments using noisy and incomplete sensory signals. In unicellular algae, phototaxis is often modeled as a mechanistic run--tumble process driven by stimulus--response rules. However, such descriptions overlook how organisms actively sample their environment to reduce sensory ambiguity. From a minimal cognition perspective, we reframe this navigation as a subjective, information-driven sensorimotor process. To this end, we propose a framework linking a Partially Observable Markov Decision Process (POMDP) with biochemical reaction dynamics. Environmental variables are hidden, while the cell updates a minimal internal state from each observation through a memoryless Bayesian step. These internal dynamics balance orienting toward light with exploratory reorientation and can be implemented through Chemical-Reaction-Network Ordinary Differential Equations (CRN--ODEs). Our model includes a biophysical observation process for photoreception and a chemically computable polynomial bound on information gain. Using Inverse Reinforcement Learning (IRL) on 30 experimentally recorded Chlamydomonas trajectories, we infer the behavioral objective consistent with observed phototactic motion and benchmark the resulting dynamics with standard Stochastic Simulation Algorithm (SSA) baselines. Our model reproduces the empirical alignment-to-light distribution, comparable to objective SSA baselines on this dataset. Within this framework, run--tumble alternation emerges as an information-acquisition strategy: tumbling reorients the cell to sample new sensory configurations and resolve sensor ambiguity, demonstrating how intracellular biochemical networks can support adaptive information-seeking behavior in cellular navigation.

cs.LG↗

arXiv2025

Minima and Critical Points of the Bethe Free Energy Are Invariant Under Deformation Retractions of Factor Graphs

In graphical models, factor graphs, and more generally energy-based models, the interactions between variables are encoded by a graph, a hypergraph, or, in the most general case, a partially ordered set (poset). Inference on such probabilistic models cannot be performed exactly due to cycles in the underlying structures of interaction. Instead, one resorts to approximate variational inference by optimizing the Bethe free energy. Critical points of the Bethe free energy correspond to fixed points of the associated Belief Propagation algorithm. A full characterization of these critical points for general graphs, hypergraphs, and posets with a finite number of variables is still an open problem. We show that, for hypergraphs and posets with chains of length at most 1, changing the poset of interactions of the probabilistic model to one with the same homotopy type induces a bijection between the critical points of the associated free energy. This result extends and unifies classical results that assume specific forms of collapsibility to prove uniqueness of the critical points of the Bethe free energy.

stat.ML↗

arXiv2025

On the Functoriality of Belief Propagation Algorithms on finite Partially Ordered Sets

Undirected graphical models are a widely used class of probabilistic models in machine learning that capture prior knowledge or putative pairwise interactions between variables. Those interactions are encoded in a graph for pairwise interactions; however, generalizations such as factor graphs account for higher-degree interactions using hypergraphs. Inference on such models, which is performed by conditioning on some observed variables, is typically done approximately by optimizing a free energy, which is an instance of variational inference. The Belief Propagation algorithm is a dynamic programming algorithm that finds critical points of that free energy. Recent efforts have been made to unify and extend inference on graphical models and factor graphs to more expressive probabilistic models. A synthesis of these works shows that inference on graphical models, factor graphs, and their generalizations relies on the introduction of presheaves and associated invariants (homology and cohomology groups).We propose to study the impact of the transformation of the presheaves onto the associated message passing algorithms. We show that natural transformations between presheaves associated with graphical models and their generalizations, which can be understood as coherent binning of the set of values of the variables, induce morphisms between associated message-passing algorithms. It is, to our knowledge, the first result on functoriality of the Loopy Belief Propagation.

math.ST↗

arXiv2024

Compositional statistical mechanics, entropy and variational inference

In this document, we aim to gather various results related to a compositional/categorical approach to rigorous Statistical Mechanics. Rigorous Statistical Mechanics is centered on the mathematical study of statistical systems. Central concepts in this field have a natural expression in terms of diagrams in a category that couples measurable maps and Markov kernels. We showed that statistical systems are particular representations of partially ordered sets (posets), that we call A-specifications, and expressed their phases, i.e., Gibbs measures, as invariants of these representations. It opens the way to the use of homological algebra to compute phases of statistical systems. Two central results of rigorous Statistical Mechanics are, firstly, the characterization of extreme Gibbs measures as it relates to the zero-one law for extreme Gibbs measures, and, secondly, their variational principle which states that for translation invariant Hamiltonians, Gibbs measures are the minima of the Gibbs free energy. We showed how the characterization of extreme Gibbs measures extends to A-specifications; we proposed an Entropy functional for A-specifications and gave a message-passing algorithm, that generalized the belief propagation algorithm of graphical models, to find critical points of the associated variational free energy.

math-ph↗

arXiv2023

Influence of the Geometry of the world model on Curiosity Based Exploration

In human spatial awareness, 3-D projective geometry structures information integration and action planning through perspective taking within an internal representation space. The way different perspectives are related and transform a world model defines a specific perception and imagination scheme. In mathematics, such collection of transformations corresponds to a 'group', whose 'actions' characterize the geometry of a space. Imbuing world models with a group structure may capture different agents' spatial awareness and affordance schemes. We used group action as a special class of policies for perspective-dependent control. We explored how such geometric structure impacts agents' behavior, comparing how the Euclidean versus projective groups act on epistemic value in active inference, drive curiosity, and exploration behaviors. We formally demonstrate and simulate how the groups induce distinct behaviors in a simple search task. The projective group's nonlinear magnification of information transformed epistemic value according to the choice of frame, generating behaviors of approach toward an object of interest. The projective group structure within the agent's world model contains the Projective Consciousness Model, which is know to capture key features of consciousness. On the other hand, the Euclidean group had no effect on epistemic value : no action was better than the initial idle state. In structuring a priori an agent's internal representation, we show how geometry can play a key role in information integration and action planning.

cs.AI↗

arXiv2023

On Non-Linear operators for Geometric Deep Learning

This work studies operators mapping vector and scalar fields defined over a manifold $\mathcal{M}$, and which commute with its group of diffeomorphisms $\text{Diff}(\mathcal{M})$. We prove that in the case of scalar fields $L^p_ω(\mathcal{M,\mathbb{R}})$, those operators correspond to point-wise non-linearities, recovering and extending known results on $\mathbb{R}^d$. In the context of Neural Networks defined over $\mathcal{M}$, it indicates that point-wise non-linear operators are the only universal family that commutes with any group of symmetries, and justifies their systematic use in combination with dedicated linear operators commuting with specific symmetries. In the case of vector fields $L^p_ω(\mathcal{M},T\mathcal{M})$, we show that those operators are solely the scalar multiplication. It indicates that $\text{Diff}(\mathcal{M})$ is too rich and that there is no universal class of non-linear operators to motivate the design of Neural Networks over the symmetries of $\mathcal{M}$.

cs.LG↗

arXiv2022

Regionalized Optimization

We propose a theoretical framework for non redundant reconstruction of a global loss from a collection of local ones under constraints given by a functor; we call this loss the regionalized loss in honor to Yedidia, Freeman, Weiss' celebrated article `Constructing free-energy approximations and generalized belief propagation algorithms' where a first example of regionalized loss, for entropy and the marginal functor, is built. We show how one can associate to these regionalized losses message passing algorithms for finding their critical points. It is a natural mathematical framework for optimization problems where there are multiple points of views on a dataset and replaces message passing algorithms as canonical ways of finding the optima of these problems. We explain how Generalized Belief propagation algorithms fall into the framework we propose and propose novel message passing algorithms for noisy channel networks.

cs.AI↗

arXiv2021

Combining the Projective Consciousness Model and Virtual Humans to assess ToM capacity in Virtual Reality: a proof-of-concept

Relating explicit psychological mechanisms and observable behaviours is a central aim of psychological and behavioural science. We implemented the principles of the Projective Consciousness Model into artificial agents embodied as virtual humans, as a proof-of-concept for a methodological framework aimed at simulating behaviours and assessing underlying psychological parameters, in the context of experiments in virtual reality. We focus on simulating the role of Theory of Mind (ToM) in the choice of strategic behaviours of approach and avoidance to optimise the satisfaction of agents' preferences. We designed an experiment in a virtual environment that could be used with real humans, allowing us to classify behaviours as a function of order of ToM, up to the second order. We show that our agents demonstrate expected behaviours with consistent parameters of ToM in this experiment. We also show that the agents can be used to estimate correctly each other order of ToM. A similar approach could be used with real humans in virtual reality experiments not only to enable human participants to interact with parametric, virtual humans as stimuli, but also as a mean of inference to derive model-based psychological assessments of the participants.

q-bio.NC↗

arXiv2021

Interaction decomposition for Hilbert spaces

The decomposition into interaction subspaces is an important result for graphical models and plays a central role for results on the linearized marginal problem; similarly the Chaos decomposition plays an important role in statistical physics at the thermodynamic limit and in probability theory in general. We unify and extend both constructions by defining and characterizing decomposable functors from a well-founded poset to the category of Hilbert spaces, completing previous work on decomposable collections of vector subspaces and presheaves.

math.CT↗

arXiv2021

Interaction decomposition for presheaves

Consider a collection of vector subspaces of a given vector space and a collection of projectors on these vector spaces, can we decompose the vector space into a product of vector subspaces such that the projectors are isomorphic to projections? We provide an answer to this question by extending the relation between the intersection property and the interaction decomposition (done in the injective case) to the projective case. This enables us to classify the decompositions of interactions for factor spaces. We then extend these results for presheaves from a poset to the category of modules by adding the data of a section functor when it exists.

math.CT↗

arXiv2021

Intersection property and interaction decomposition

The decomposition into interaction subspaces is a hierarchical decomposition of the spaces of cylindrical functions of a finite product space, also called factor spaces. It is an important construction in graphical models and a standard way to prove the Hammersley-Clifford theorem that relates Markov fields to Gibbs fields and plays a central role in Kellerer's result for the linearized marginal problem. We define an intersection of sum property, or simply intersection property, and show that it characterizes collections of vector subspaces over a poset that can be hierarchically decomposed into direct sums, giving therefore a general setting for such construction to hold. We will call this generalization the interaction decomposition. The intersection property is the Bayesian intersection property when specified to factor spaces which, under this new perspective on the interaction decomposition, appears to be a structure property. An application is the extension of the decomposition into interaction subspaces for any product of any set.

math.RA↗

arXiv2021

Bayesian/Graphoid intersection property for factorisation spaces

We remark that Pearl's Graphoid intersection property, also called intersection property in Bayesian networks, is a particular case of a general intersection property, in the sense of intersection of coverings, for factorisation spaces, also coined as factorisation models, factor graphs or by Lauritzen in his reference book 'Graphical Models' as hierarchical model subspaces. A particular case of this intersection property appears in Lauritzen's book as a consequence of the decomposition into interaction subspaces; the novel proof that we give of this result allows us to extend it in the most general setting. It also allows us to give a direct and new proof of the Hammersley-Clifford theorem transposing and reducing it to a corresponding statement for graphs, justifying formally the geometric intuition of independency, and extending it to non finite graphs. This intersection property is the starting point for a generalization of the decomposition into interaction subspaces to collections of vector spaces.

math.ST↗

arXiv2020

Extra-fine sheaves and interaction decompositions

We introduce an original notion of extra-fine sheaf on a topological space, and a variant (hyper-extra-fine) for which Čech cohomology in strictly positive degree vanishes. We provide a characterization of such sheaves when the topological space is a partially ordered set (poset) equipped with the Alexandrov topology. Then we further specialize our results to some sheaves of vector spaces and injective maps, where extra-fineness is (essentially) equivalent to the decomposition of the sheaf into a direct sum of subfunctors, known as interaction decomposition, and can be expressed by a sum-intersection condition. We use these results to compute the dimension of the space of global sections when the presheaves are freely generated over a functor of sets, generalizing classical counting formulae for the number of solutions of the linearized marginal problem (Kellerer and Matúš). We finish with a comparison theorem between the Čech cohomology associated to a covering and the topos cohomology of the poset with coefficients in the presheaf, which is also the cohomology of a cosimplicial local system over the nerve of the poset. For that, we give a detailed treatment of cosimplicial local systems on simplicial sets. The appendixes present presheaves, sheaves and Čech cohomology, and their application to the marginal problem.

math.AT↗