Seminars and short courses

Propagation and generation of chaos is an important ingredient for rigorous control of applicability of kinetic theory, in general. Chaos is here understood as sufficient statistical independence of random variables related to the "kinetic" observables of the system. Cumulant hierarchy of these random variables thus often gives a way of controlling the evolution and degree of such independence, i.e., the degree of chaos in the system. In this talk, I will discuss our analysis of the cumulant hierarchy of the stochastic Kac model in the preprint [arxiv.org:2407.17068], a joint work with Aleksis Vuoksenmaa. We control generation of chaos via the magnitude of finite order cumulants of kinetic energies for arbitrary symmetric initial data, with the usual restriction of a fixed energy density. This allows estimating the accuracy of kinetic theory, uniformly in time and for any system with sufficiently large number of particles, N. We prove that the evolution of the system can be divided into three regimes: an initial regime of the length of at most O(ln N) in which the state of the system becomes chaotic, the kinetic regime which is determined by solutions to the Boltzmann-Kac equation, and an equilibrium regime.

In this talk I will review the notion of unbalanced optimal transport, introduced $\sim10$ years ago to handle mass variarions betwenn nonnegative measures. I will discuss the induced pseudo-Riemannian structure and gradient-flow evolution, corresponding to some class of parabolic reaction-diffusion equations. If time permits I will present two applications: a fitness-driven model from population dynamics, and a Hele-Shaw free boundary problem for tumor growth.

We prove that the space of symplectic embeddings of $n\geq 1$ standard balls, each of capacity at most $\frac{1}{n}$, into the standard complex projective plane $\mathbb{CP}^2$ is homotopy equivalent to the configuration space of $n$ points in $\mathbb{CP}^2$. Our techniques also suggest that for every $n \geq 9$, there may exist infinitely many homotopy types of spaces of symplectic ball embeddings.

Analogical reasoning is a powerful inductive mechanism, widely used in human cognition and increasingly applied in artificial intelligence. Formal frameworks for analogical inference have been developed for Boolean domains, where inference is provably sound for affine functions and approximately correct when close to affine. These results enabled the design of analogy-based classifiers. However, they do not extend to regression tasks or continuous domains.

In these series of seminars we will revisit analogical inference from a foundational perspective. After a brief motivation, we will first present a recently proposed formalism to model numerical analogies that relies on p-generalized means, and that enables a unifying framework that subsume the classical notions of arithmetic, geometric and harmonic analogies. We will derive several interesting properties such as transitivity of conformity, as well as present algorithmic approaches to detect and compute the parameter p.

In the second part of this series, we will leverage this unified formalism and lift analogical reasoning to real-valued domains and various ML&AI downstream tasks. In particular, we will see that it supports analogical inference over continuous functions, and thus both classification and regression tasks. We characterize the class of analogy-preserving functions in this setting and derive both worst-case and average-case error bounds under smoothness assumptions. If time allows, we will also discuss further applications, e.g., on image reconstruction and NLP downstream tasks.

These two seminars are based on several published and recently submitted by Miguel Couceiro and his collaborators, including Francisco Malaca and Francisco Vincente Cunha, respectively, graduate and undergraduate students at the DM@IST.

Some very recent References:

1. Francisco Malaca, Yves Lepage, Miguel Couceiro. Numerical analogies through generalized means:notion, properties and algorithmic approaches. Submitted.
2. Francisco Cunha, Yves Lepage, Zied Bouraoui, Miguel Couceiro. Generalizing Analogical Inference Across Boolean and Continuous Domains. Submitted.
3. Jakub Pillion, Miguel Couceiro, Yves Lepage. Analogical pooling for image reconstruction. Submitted.
4. Fadi Badra, Esteban Marquer, Marie-Jeanne Lesot, Miguel Couceiro, David Leake. EnergyCompress: A General Case Base Learning Strategy. To appear in IJCAI2025.
5. Yves Lepage, Miguel Couceiro. Any four real numbers are on all fours with analogy. CoRR abs/2407.18770 (2024)
6. Miguel Couceiro, Erkko Lehtonen. Galois theory for analogical classifiers. Ann. Math. Artif. Intell. 92(1): 29-47 (2024)
7. Pierre Monnin, Cherif-Hassan Nousradine, Lucas Jarnac, Laurel Zuckerman, Miguel Couceiro. KGPRUNE: A Web Application to Extract Subgraphs of Interest from Wikidata with Analogical Pruning. ECAI 2024: 4495-4498
8. Yves Lepage, Miguel Couceiro. Analogie et moyenne généralisée. JIAF-JFPDA 2024: 114-124
9. Lucas Jarnac, Miguel Couceiro, Pierre Monnin. Relevant Entity Selection: Knowledge Graph Bootstrapping via Zero-Shot Analogical Pruning. CIKM 2023: 934-944
10. N. Kumar, and S. Schockaert. Solving hard analogy questions with relation embedding chains. EMNLP 2023, 6224–6236. ACL

Analogical reasoning is a powerful inductive mechanism, widely used in human cognition and increasingly applied in artificial intelligence. Formal frameworks for analogical inference have been developed for Boolean domains, where inference is provably sound for affine functions and approximately correct when close to affine. These results enabled the design of analogy-based classifiers. However, they do not extend to regression tasks or continuous domains.

In these series of seminars we will revisit analogical inference from a foundational perspective. After a brief motivation, we will first present a recently proposed formalism to model numerical analogies that relies on p-generalized means, and that enables a unifying framework that subsume the classical notions of arithmetic, geometric and harmonic analogies. We will derive several interesting properties such as transitivity of conformity, as well as present algorithmic approaches to detect and compute the parameter p.

In the second part of this series, we will leverage this unified formalism and lift analogical reasoning to real-valued domains and various ML&AI downstream tasks. In particular, we will see that it supports analogical inference over continuous functions, and thus both classification and regression tasks. We characterize the class of analogy-preserving functions in this setting and derive both worst-case and average-case error bounds under smoothness assumptions. If time allows, we will also discuss further applications, e.g., on image reconstruction and NLP downstream tasks.

These two seminars are based on several published and recently submitted by Miguel Couceiro and his collaborators, including Francisco Malaca and Francisco Vincente Cunha, respectively, graduate and undergraduate students at the DM@IST.

The purpose of this talk is to explore how heterogeneity in the connectivity of an epidemiological network impacts the interaction between strains in a co-infection SIS framework.

In the first part, we will present an epidemiological model of co-infection with multiple strains under the assumption of homogeneous connectivity among hosts. We will show that, under the assumption of neutrality, this model satisfies a neutral null property, an ecologically important concept. Mathematically, this neutrality means that the $\omega$-limit set of any trajectory is a central manifold that can be parameterized by the proportion $(z_i)$ of each strain $i$.

This neutrality can be relaxed using a slow-fast argument, leading to an equation for $(z_i)$ that describes the slow dynamics on the central manifold. This equation is the well-known replicator equation, whose parameters are explicitly linked to macro-level ecological parameters. This connection allows us to understand pathogen ecology through the epidemiological dynamics at the host scale.

In the second part, we will explain how this approach can be extended when considering a heterogeneous network of host connectivity. After precisely describing this new model, we will show that the final replicator equation retains traces of this heterogeneity, providing a direct way to model how the complexity of host interactions affects the dynamics of a pathogen and its multiple variants.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.