Author name cluster

Lucas Drumetz

Possible papers associated with this exact author name in Arrow. This page groups case-insensitive exact name matches and is not a full identity disambiguation profile.

7 papers

2 author rows

TMLR Journal 2025 Journal Article

Augmented Invertible Koopman Autoencoder for long-term time series forecasting

Anthony Frion
Lucas Drumetz
Mauro Dalla Mura
Guillaume Tochon
Abdeldjalil AISSA EL BEY

Following the introduction of Dynamic Mode Decomposition and its numerous extensions, many neural autoencoder-based implementations of the Koopman operator have recently been proposed. This class of methods appears to be of interest for modeling dynamical systems, either through direct long-term prediction of the evolution of the state or as a powerful embedding for downstream methods. In particular, a recent line of work has developed invertible Koopman autoencoders (IKAEs), which provide an exact reconstruction of the input state thanks to their analytically invertible encoder, based on coupling layer normalizing flow models. We identify that the conservation of the dimension imposed by the normalizing flows is a limitation for the IKAE models, and thus we propose to augment the latent state with a second, non-invertible encoder network. This results in our new model: the Augmented Invertible Koopman AutoEncoder (AIKAE). We demonstrate the relevance of the AIKAE through a series of long-term time series forecasting experiments, on satellite image time series as well as on a benchmark involving predictions based on a large lookback window of observations.

PDF Details

TMLR Journal 2025 Journal Article

FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing flows and the Feynman-Kac Formula

Naoufal EL BEKRI
Lucas Drumetz
Franck VERMET

Solving the Fokker-Planck equation for high-dimensional complex dynamical systems remains a pivotal yet challenging task due to the intractability of analytical solutions and the limitations of traditional numerical methods. In this work, we present FlowKac, a novel approach that reformulates the Fokker-Planck equation using the Feynman-Kac formula, allowing to query the solution at a given point via the expected values of stochastic paths. A key innovation of FlowKac lies in its adaptive stochastic sampling scheme which significantly reduces the computational complexity while maintaining high accuracy. This sampling technique, coupled with a time-indexed normalizing flow, designed for capturing time-evolving probability densities, enables robust sampling of collocation points, resulting in a flexible and mesh-free solver. This formulation mitigates the curse of dimensionality and enhances computational efficiency and accuracy, which is particularly crucial for applications that inherently require dimensions beyond the conventional three. We validate the robustness and scalability of our method through various experiments on a range of stochastic differential equations, demonstrating significant improvements over existing techniques.

PDF Details

JMLR Journal 2025 Journal Article

Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds

Clément Bonet
Lucas Drumetz
Nicolas Courty

While many Machine Learning methods have been developed or transposed on Riemannian manifolds to tackle data with known non-Euclidean geometry, Optimal Transport (OT) methods on such spaces have not received much attention. The main OT tool on these spaces is the Wasserstein distance, which suffers from a heavy computational burden. On Euclidean spaces, a popular alternative is the Sliced-Wasserstein distance, which leverages a closed-form solution of the Wasserstein distance in one dimension, but which is not readily available on manifolds. In this work, we derive general constructions of Sliced-Wasserstein distances on Cartan-Hadamard manifolds, Riemannian manifolds with non-positive curvature, which include among others Hyperbolic spaces or the space of Symmetric Positive Definite matrices. Then, we propose different applications such as classification of documents with a suitably learned ground cost on a manifold, and data set comparison on a product manifold. Additionally, we derive non-parametric schemes to minimize these new distances by approximating their Wasserstein gradient flows. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2025. ( edit, beta )

PDF Details

ICML Conference 2023 Conference Paper

Sliced-Wasserstein on Symmetric Positive Definite Matrices for M/EEG Signals

Clément Bonet
Benoît Malézieux
Alain Rakotomamonjy
Lucas Drumetz
Thomas Moreau 0001
Matthieu Kowalski
Nicolas Courty

When dealing with electro or magnetoencephalography records, many supervised prediction tasks are solved by working with covariance matrices to summarize the signals. Learning with these matrices requires the usage of Riemanian geometry to account for their structure. In this paper, we propose a new method to deal with distributions of covariance matrices, and demonstrate its computational efficiency on M/EEG multivariate time series. More specifically, we define a Sliced-Wasserstein distance between measures of symmetric positive definite matrices that comes with strong theoretical guarantees. Then, we take advantage of its properties and kernel methods to apply this discrepancy to brain-age prediction from MEG data, and compare it to state-of-the-art algorithms based on Riemannian geometry. Finally, we show that it is an efficient surrogate to the Wasserstein distance in domain adaptation for Brain Computer Interface applications.

Details

ICLR Conference 2023 Conference Paper

Spherical Sliced-Wasserstein

Clément Bonet
Paul Berg
Nicolas Courty
François Septier
Lucas Drumetz
Minh-Tan Pham

Many variants of the Wasserstein distance have been introduced to reduce its original computational burden. In particular the Sliced-Wasserstein distance (SW), which leverages one-dimensional projections for which a closed-form solution of the Wasserstein distance is available, has received a lot of interest. Yet, it is restricted to data living in Euclidean spaces, while the Wasserstein distance has been studied and used recently on manifolds. We focus more specifically on the sphere, for which we define a novel SW discrepancy, which we call spherical Sliced-Wasserstein, making a first step towards defining SW discrepancies on manifolds. Our construction is notably based on closed-form solutions of the Wasserstein distance on the circle, together with a new spherical Radon transform. Along with efficient algorithms and the corresponding implementations, we illustrate its properties in several machine learning use cases where spherical representations of data are at stake: sampling on the sphere, density estimation on real eath data or hyperspherical auto-encoders.

Details

TMLR Journal 2023 Journal Article

Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flows

Guillaume Morel
Lucas Drumetz
Simon Benaïchouche
Nicolas Courty
François Rousseau

Normalizing Flows (NF) are powerful likelihood-based generative models that are able to trade off between expressivity and tractability to model complex densities. A now well established research avenue leverages optimal transport (OT) and looks for Monge maps, i.e. models with minimal effort between the source and target distributions. This paper introduces a method based on Brenier's polar factorization theorem to transform any trained NF into a more OT-efficient version without changing the final density. We do so by learning a rearrangement of the source (Gaussian) distribution that minimizes the OT cost between the source and the final density. The Gaussian preserving transformation is implemented with the construction of high dimensional divergence free functions and the path leading to the estimated Monge map is further constrained to lie on a geodesic in the space of volume-preserving diffeomorphisms thanks to Euler's equations. The proposed method leads to smooth flows with reduced OT costs for several existing models without affecting the model performance.

PDF Details

TMLR Journal 2022 Journal Article

Efficient Gradient Flows in Sliced-Wasserstein Space

Clément Bonet
Nicolas Courty
François Septier
Lucas Drumetz

Minimizing functionals in the space of probability distributions can be done with Wasser- stein gradient flows. To solve them numerically, a possible approach is to rely on the Jordan–Kinderlehrer–Otto (JKO) scheme which is analogous to the proximal scheme in Euclidean spaces. However, it requires solving a nested optimization problem at each it- eration, and is known for its computational challenges, especially in high dimension. To alleviate it, very recent works propose to approximate the JKO scheme leveraging Brenier’s theorem, and using gradients of Input Convex Neural Networks to parameterize the density (JKO-ICNN). However, this method comes with a high computational cost and stability is- sues. Instead, this work proposes to use gradient flows in the space of probability measures endowed with the sliced-Wasserstein (SW) distance. We argue that this method is more flex- ible than JKO-ICNN, since SW enjoys a closed-form differentiable approximation. Thus, the density at each step can be parameterized by any generative model which alleviates the computational burden and makes it tractable in higher dimensions.

PDF Details