arXiv subjects

Nicolas Courty

Publications and source records attributed to Nicolas Courty.

At least 19 recordsLinked to original sources

arXiv2026

A${}^2$BM: Alignment-Aware Bridge Matching for Image-to-Image Translation

Paired image-to-image translation underpins a wide range of computer vision tasks, including image editing, sensor translation, and domain adaptation. Bridge matching and flow matching have recently emerged as powerful frameworks, extending diffusion models to arbitrary source and target distributions. However, their standard formulations assume perfectly aligned training pairs, treating all source-target correspondences as equally reliable. In practice, real-world applications often involve weakly aligned pairs due to changes of acquisition conditions, including e.g. asynchronous captures, different illuminations, or misregistration. In this work, we introduce Alignment-Aware Bridge Matching (A${}^2$BM), a bridge matching method that leverages image pairs alignment during training. By incorporating alignment scores, the model learns to disentangle true semantic correspondences from misalignment artifacts. At inference time, we use the alignment score as a control variable over translation fidelity, with strongly aligned outputs obtained when prompting the model with the highest alignment score. We validate A${}^2$BM on both controlled synthetic experiments and on challenging real-world tasks, including cross-sensor super-resolution and pixel-space unsupervised domain adaptation. In all settings, A${}^2$BM consistently improves translation fidelity over strong GAN-, diffusion-, and Schr{\"o}dinger bridge-based baselines, establishing alignment conditioning as a principled solution for image translation models with weakly aligned data.

cs.CV↗

arXiv2026

Take It or Leave It: Intent-Controlled Partial Optimal Transport

While optimal transport (OT) enforces a rigid constraint by requiring two measures to be matched exactly, partial optimal transport relaxes this requirement by allowing mass to remain unmatched through a global budget, scalar rebate, or uniform rejection rule. However, many applications call for more structured, pointwise rejection mechanisms, where the decision to leave mass unmatched depends on side-specific reliability, support geometry, or external information about which components should participate in the comparison. We introduce \emph{intent-controlled partial optimal transport} (IC-POT), a targeted generalization of partial transport that replaces the global rejection paradigm with pointwise rejection costs over both measures. We show that the resulting optimization problem admits a dual interpretation in terms of local acceptance thresholds and can be solved by recasting it as a balanced Kantorovich OT problem on an augmented support. Beyond theoretical analysis, we demonstrate the practical relevance of IC-POT in settings where rejection is driven by side information. In positive-unlabeled learning and open-partial domain adaptation, incorporating pointwise rejection rules that encode statistical structure improves fixed baseline pipelines. Finally, we motivate the use of IC-POT with a geophysical practical case: multi-modal satellite ocean measurements, for which physical and sensors priors naturally inform the rejection mechanism and define the retrieved comparable signal information.

cs.LG↗

arXiv2026

Spherical Harmonic Optimal Transport: Application to Climate Models Comparisons

Optimal transport provides a powerful framework for comparing measures while respecting the geometry of their support, but comes with an expensive computational cost, hindering its potential application to real world use cases. On manifolds, convolutional algorithms based on the heat kernel have been proposed to alleviate this cost, but their theoretical properties remain largely unexplored. We establish that the heat kernel cost converges to the optimal transport cost as time vanishes in the balanced and unbalanced cases. In the specific case of the 2-sphere $\mathbb{S}^2$, we ensure that the associated Sinkhorn divergences retains the desirable geometric and analytic properties of classical optimal transport discrepancies. Moreover, we leverage the harmonic structure of the sphere to derive a fast Sinkhorn algorithm, requiring only $\mathcal{O}(n)$ memory and $\mathcal{O}(n^{3/2})$ time per iteration, with fully dense GPU-friendly operations. We validate its computational efficiency on synthetic data, and discuss its potential use in the evaluation of global climate models, providing both spatial and seasonal insights into models performances.

cs.LG↗

arXiv2025

Improving Multimodal Distillation for 3D Semantic Segmentation under Domain Shift

Semantic segmentation networks trained under full supervision for one type of lidar fail to generalize to unseen lidars without intervention. To reduce the performance gap under domain shifts, a recent trend is to leverage vision foundation models (VFMs) providing robust features across domains. In this work, we conduct an exhaustive study to identify recipes for exploiting VFMs in unsupervised domain adaptation for semantic segmentation of lidar point clouds. Building upon unsupervised image-to-lidar knowledge distillation, our study reveals that: (1) the architecture of the lidar backbone is key to maximize the generalization performance on a target domain; (2) it is possible to pretrain a single backbone once and for all, and use it to address many domain shifts; (3) best results are obtained by keeping the pretrained backbone frozen and training an MLP head for semantic segmentation. The resulting pipeline achieves state-of-the-art results in four widely-recognized and challenging settings. The code will be available at: https://github.com/valeoai/muddos.

cs.CV↗

arXiv2025

Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing

The Busemann function has recently found much interest in a variety of geometric machine learning problems, as it naturally defines projections onto geodesic rays of Riemannian manifolds and generalizes the notion of hyperplanes. As several sources of data can be conveniently modeled as probability distributions, it is natural to study this function in the Wasserstein space, which carries a rich formal Riemannian structure induced by Optimal Transport metrics. In this work, we investigate the existence and computation of Busemann functions in Wasserstein space, which admits geodesic rays. We establish closed-form expressions in two important cases: one-dimensional distributions and Gaussian measures. These results enable explicit projection schemes for probability distributions on $\mathbb{R}$, which in turn allow us to define novel Sliced-Wasserstein distances over Gaussian mixtures and labeled datasets. We demonstrate the efficiency of those original schemes on synthetic datasets as well as transfer learning problems.

cs.LG↗

arXiv2025

Bridging Arbitrary and Tree Metrics via Differentiable Gromov Hyperbolicity

Trees and the associated shortest-path tree metrics provide a powerful framework for representing hierarchical and combinatorial structures in data. Given an arbitrary metric space, its deviation from a tree metric can be quantified by Gromov's $\delta$-hyperbolicity. Nonetheless, designing algorithms that bridge an arbitrary metric to its closest tree metric is still a vivid subject of interest, as most common approaches are either heuristical and lack guarantees, or perform moderately well. In this work, we introduce a novel differentiable optimization framework, coined DeltaZero, that solves this problem. Our method leverages a smooth surrogate for Gromov's $\delta$-hyperbolicity which enables a gradient-based optimization, with a tractable complexity. The corresponding optimization procedure is derived from a problem with better worst case guarantees than existing bounds, and is justified statistically. Experiments on synthetic and real-world datasets demonstrate that our method consistently achieves state-of-the-art distortion.

cs.LG↗

arXiv2024

Train Till You Drop: Towards Stable and Robust Source-free Unsupervised 3D Domain Adaptation

We tackle the challenging problem of source-free unsupervised domain adaptation (SFUDA) for 3D semantic segmentation. It amounts to performing domain adaptation on an unlabeled target domain without any access to source data; the available information is a model trained to achieve good performance on the source domain. A common issue with existing SFUDA approaches is that performance degrades after some training time, which is a by product of an under-constrained and ill-posed problem. We discuss two strategies to alleviate this issue. First, we propose a sensible way to regularize the learning problem. Second, we introduce a novel criterion based on agreement with a reference model. It is used (1) to stop the training when appropriate and (2) as validator to select hyperparameters without any knowledge on the target domain. Our contributions are easy to implement and readily amenable for all SFUDA methods, ensuring stable improvements over all baselines. We validate our findings on various 3D lidar settings, achieving state-of-the-art performance. The project repository (with code) is: github.com/valeoai/TTYD.

cs.CV↗

arXiv2024

Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds

While many Machine Learning methods were 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. Additionally, we derive non-parametric schemes to minimize these new distances by approximating their Wasserstein gradient flows.

cs.LG↗

arXiv2024

Non-Euclidean Sliced Optimal Transport Sampling

In machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well-dispersed collection of samples. Providing a formal metric for measuring the distance between probability measures on general spaces, Optimal Transport (OT) emerges as a pivotal theoretical framework within this context. However, the associated computational burden is prohibitive in most real-world scenarios. Leveraging the simple structure of OT in 1D, Sliced Optimal Transport (SOT) has appeared as an efficient alternative to generate samples in Euclidean spaces. This paper pushes the boundaries of SOT utilization in computational geometry problems by extending its application to sample densities residing on more diverse mathematical domains, including the spherical space Sd , the hyperbolic plane Hd , and the real projective plane Pd . Moreover, it ensures the quality of these samples by achieving a blue noise characteristic, regardless of the dimensionality involved. The robustness of our approach is highlighted through its application to various geometry processing tasks, such as the intrinsic blue noise sampling of meshes, as well as the sampling of directions and rotations. These applications collectively underscore the efficacy of our methodology.

cs.GR↗

arXiv2024

Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein

Unsupervised learning aims to capture the underlying structure of potentially large and high-dimensional datasets. Traditionally, this involves using dimensionality reduction (DR) methods to project data onto lower-dimensional spaces or organizing points into meaningful clusters (clustering). In this work, we revisit these approaches under the lens of optimal transport and exhibit relationships with the Gromov-Wasserstein problem. This unveils a new general framework, called distributional reduction, that recovers DR and clustering as special cases and allows addressing them jointly within a single optimization problem. We empirically demonstrate its relevance to the identification of low-dimensional prototypes representing data at different scales, across multiple image and genomic datasets.

cs.LG↗

arXiv2023

Interpolating between Clustering and Dimensionality Reduction with Gromov-Wasserstein

We present a versatile adaptation of existing dimensionality reduction (DR) objectives, enabling the simultaneous reduction of both sample and feature sizes. Correspondances between input and embedding samples are computed through a semi-relaxed Gromov-Wasserstein optimal transport (OT) problem. When the embedding sample size matches that of the input, our model recovers classical popular DR models. When the embedding's dimensionality is unconstrained, we show that the OT plan delivers a competitive hard clustering. We emphasize the importance of intermediate stages that blend DR and clustering for summarizing real data and apply our method to visualize datasets of images.

cs.LG↗

arXiv2023

Optimal Transport with Adaptive Regularisation

Regularising the primal formulation of optimal transport (OT) with a strictly convex term leads to enhanced numerical complexity and a denser transport plan. Many formulations impose a global constraint on the transport plan, for instance by relying on entropic regularisation. As it is more expensive to diffuse mass for outlier points compared to central ones, this typically results in a significant imbalance in the way mass is spread across the points. This can be detrimental for some applications where a minimum of smoothing is required per point. To remedy this, we introduce OT with Adaptive RegularIsation (OTARI), a new formulation of OT that imposes constraints on the mass going in or/and out of each point. We then showcase the benefits of this approach for domain adaptation.

cs.LG↗

arXiv2023

Match-And-Deform: Time Series Domain Adaptation through Optimal Transport and Temporal Alignment

While large volumes of unlabeled data are usually available, associated labels are often scarce. The unsupervised domain adaptation problem aims at exploiting labels from a source domain to classify data from a related, yet different, target domain. When time series are at stake, new difficulties arise as temporal shifts may appear in addition to the standard feature distribution shift. In this paper, we introduce the Match-And-Deform (MAD) approach that aims at finding correspondences between the source and target time series while allowing temporal distortions. The associated optimization problem simultaneously aligns the series thanks to an optimal transport loss and the time stamps through dynamic time warping. When embedded into a deep neural network, MAD helps learning new representations of time series that both align the domains and maximize the discriminative power of the network. Empirical studies on benchmark datasets and remote sensing data demonstrate that MAD makes meaningful sample-to-sample pairing and time shift estimation, reaching similar or better classification performance than state-of-the-art deep time series domain adaptation strategies.

cs.LG↗

arXiv2023

Fast Optimal Transport through Sliced Wasserstein Generalized Geodesics

Wasserstein distance (WD) and the associated optimal transport plan have been proven useful in many applications where probability measures are at stake. In this paper, we propose a new proxy of the squared WD, coined min-SWGG, that is based on the transport map induced by an optimal one-dimensional projection of the two input distributions. We draw connections between min-SWGG and Wasserstein generalized geodesics in which the pivot measure is supported on a line. We notably provide a new closed form for the exact Wasserstein distance in the particular case of one of the distributions supported on a line allowing us to derive a fast computational scheme that is amenable to gradient descent optimization. We show that min-SWGG is an upper bound of WD and that it has a complexity similar to as Sliced-Wasserstein, with the additional feature of providing an associated transport plan. We also investigate some theoretical properties such as metricity, weak convergence, computational and topological properties. Empirical evidences support the benefits of min-SWGG in various contexts, from gradient flows, shape matching and image colorization, among others.

stat.ML↗

arXiv2023

Joint multi-modal Self-Supervised pre-training in Remote Sensing: Application to Methane Source Classification

With the current ubiquity of deep learning methods to solve computer vision and remote sensing specific tasks, the need for labelled data is growing constantly. However, in many cases, the annotation process can be long and tedious depending on the expertise needed to perform reliable annotations. In order to alleviate this need for annotations, several self-supervised methods have recently been proposed in the literature. The core principle behind these methods is to learn an image encoder using solely unlabelled data samples. In earth observation, there are opportunities to exploit domain-specific remote sensing image data in order to improve these methods. Specifically, by leveraging the geographical position associated with each image, it is possible to cross reference a location captured from multiple sensors, leading to multiple views of the same locations. In this paper, we briefly review the core principles behind so-called joint-embeddings methods and investigate the usage of multiple remote sensing modalities in self-supervised pre-training. We evaluate the final performance of the resulting encoders on the task of methane source classification.

cs.CV↗

arXiv2023

Slicing Unbalanced Optimal Transport

Optimal transport (OT) is a powerful framework to compare probability measures, a fundamental task in many statistical and machine learning problems. Substantial advances have been made in designing OT variants which are either computationally and statistically more efficient or robust. Among them, sliced OT distances have been extensively used to mitigate optimal transport's cubic algorithmic complexity and curse of dimensionality. In parallel, unbalanced OT was designed to allow comparisons of more general positive measures, while being more robust to outliers. In this paper, we bridge the gap between those two concepts and develop a general framework for efficiently comparing positive measures. We notably formulate two different versions of sliced unbalanced OT, and study the associated topology and statistical properties. We then develop a GPU-friendly Frank-Wolfe like algorithm to compute the corresponding loss functions, and show that the resulting methodology is modular as it encompasses and extends prior related work. We finally conduct an empirical analysis of our loss functions and methodology on both synthetic and real datasets, to illustrate their computational efficiency, relevance and applicability to real-world scenarios including geophysical data.

cs.LG↗

arXiv2023

SNEkhorn: Dimension Reduction with Symmetric Entropic Affinities

Many approaches in machine learning rely on a weighted graph to encode the similarities between samples in a dataset. Entropic affinities (EAs), which are notably used in the popular Dimensionality Reduction (DR) algorithm t-SNE, are particular instances of such graphs. To ensure robustness to heterogeneous sampling densities, EAs assign a kernel bandwidth parameter to every sample in such a way that the entropy of each row in the affinity matrix is kept constant at a specific value, whose exponential is known as perplexity. EAs are inherently asymmetric and row-wise stochastic, but they are used in DR approaches after undergoing heuristic symmetrization methods that violate both the row-wise constant entropy and stochasticity properties. In this work, we uncover a novel characterization of EA as an optimal transport problem, allowing a natural symmetrization that can be computed efficiently using dual ascent. The corresponding novel affinity matrix derives advantages from symmetric doubly stochastic normalization in terms of clustering performance, while also effectively controlling the entropy of each row thus making it particularly robust to varying noise levels. Following, we present a new DR algorithm, SNEkhorn, that leverages this new affinity matrix. We show its clear superiority to state-of-the-art approaches with several indicators on both synthetic and real-world datasets.

cs.LG↗

arXiv2023

SALUDA: Surface-based Automotive Lidar Unsupervised Domain Adaptation

Learning models on one labeled dataset that generalize well on another domain is a difficult task, as several shifts might happen between the data domains. This is notably the case for lidar data, for which models can exhibit large performance discrepancies due for instance to different lidar patterns or changes in acquisition conditions. This paper addresses the corresponding Unsupervised Domain Adaptation (UDA) task for semantic segmentation. To mitigate this problem, we introduce an unsupervised auxiliary task of learning an implicit underlying surface representation simultaneously on source and target data. As both domains share the same latent representation, the model is forced to accommodate discrepancies between the two sources of data. This novel strategy differs from classical minimization of statistical divergences or lidar-specific domain adaptation techniques. Our experiments demonstrate that our method achieves a better performance than the current state of the art, both in real-to-real and synthetic-to-real scenarios.

cs.CV↗