arXiv subjects

Rémi Flamary

Publications and source records attributed to Rémi Flamary.

At least 37 records · Page 2Linked to original sources

arXiv2023

Convolutional Monge Mapping Normalization for learning on sleep data

In many machine learning applications on signals and biomedical data, especially electroencephalogram (EEG), one major challenge is the variability of the data across subjects, sessions, and hardware devices. In this work, we propose a new method called Convolutional Monge Mapping Normalization (CMMN), which consists in filtering the signals in order to adapt their power spectrum density (PSD) to a Wasserstein barycenter estimated on training data. CMMN relies on novel closed-form solutions for optimal transport mappings and barycenters and provides individual test time adaptation to new data without needing to retrain a prediction model. Numerical experiments on sleep EEG data show that CMMN leads to significant and consistent performance gains independent from the neural network architecture when adapting between subjects, sessions, and even datasets collected with different hardware. Notably our performance gain is on par with much more numerically intensive Domain Adaptation (DA) methods and can be used in conjunction with those for even better performances.

eess.SP↗

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

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

Aligning individual brains with Fused Unbalanced Gromov-Wasserstein

Individual brains vary in both anatomy and functional organization, even within a given species. Inter-individual variability is a major impediment when trying to draw generalizable conclusions from neuroimaging data collected on groups of subjects. Current co-registration procedures rely on limited data, and thus lead to very coarse inter-subject alignments. In this work, we present a novel method for inter-subject alignment based on Optimal Transport, denoted as Fused Unbalanced Gromov Wasserstein (FUGW). The method aligns cortical surfaces based on the similarity of their functional signatures in response to a variety of stimulation settings, while penalizing large deformations of individual topographic organization. We demonstrate that FUGW is well-suited for whole-brain landmark-free alignment. The unbalanced feature allows to deal with the fact that functional areas vary in size across subjects. Our results show that FUGW alignment significantly increases between-subject correlation of activity for independent functional data, and leads to more precise mapping at the group level.

q-bio.NC↗

arXiv2023

Entropic Wasserstein Component Analysis

Dimension reduction (DR) methods provide systematic approaches for analyzing high-dimensional data. A key requirement for DR is to incorporate global dependencies among original and embedded samples while preserving clusters in the embedding space. To achieve this, we combine the principles of optimal transport (OT) and principal component analysis (PCA). Our method seeks the best linear subspace that minimizes reconstruction error using entropic OT, which naturally encodes the neighborhood information of the samples. From an algorithmic standpoint, we propose an efficient block-majorization-minimization solver over the Stiefel manifold. Our experimental results demonstrate that our approach can effectively preserve high-dimensional clusters, leading to more interpretable and effective embeddings. Python code of the algorithms and experiments is available online.

stat.ML↗

arXiv2023

Unbalanced CO-Optimal Transport

Optimal transport (OT) compares probability distributions by computing a meaningful alignment between their samples. CO-optimal transport (COOT) takes this comparison further by inferring an alignment between features as well. While this approach leads to better alignments and generalizes both OT and Gromov-Wasserstein distances, we provide a theoretical result showing that it is sensitive to outliers that are omnipresent in real-world data. This prompts us to propose unbalanced COOT for which we provably show its robustness to noise in the compared datasets. To the best of our knowledge, this is the first such result for OT methods in incomparable spaces. With this result in hand, we provide empirical evidence of this robustness for the challenging tasks of heterogeneous domain adaptation with and without varying proportions of classes and simultaneous alignment of samples and features across single-cell measurements.

stat.ML↗

arXiv2022

Wind power predictions from nowcasts to 4-hour forecasts: a learning approach with variable selection

We study short-term prediction of wind speed and wind power (every 10 minutes up to 4 hours ahead). Accurate forecasts for these quantities are crucial to mitigate the negative effects of wind farms' intermittent production on energy systems and markets. We use machine learning to combine outputs from numerical weather prediction models with local observations. The former provide valuable information on higher scales dynamics while the latter gives the model fresher and location-specific data. So as to make the results usable for practitioners, we focus on well-known methods which can handle a high volume of data. We study first variable selection using both a linear technique and a nonlinear one. Then we exploit these results to forecast wind speed and wind power still with an emphasis on linear models versus nonlinear ones. For the wind power prediction, we also compare the indirect approach (wind speed predictions passed through a power curve) and the indirect one (directly predict wind power).

cs.LG↗

arXiv2022

Time Series Alignment with Global Invariances

Multivariate time series are ubiquitous objects in signal processing. Measuring a distance or similarity between two such objects is of prime interest in a variety of applications, including machine learning, but can be very difficult as soon as the temporal dynamics and the representation of the time series, {\em i.e.} the nature of the observed quantities, differ from one another. In this work, we propose a novel distance accounting both feature space and temporal variabilities by learning a latent global transformation of the feature space together with a temporal alignment, cast as a joint optimization problem. The versatility of our framework allows for several variants depending on the invariance class at stake. Among other contributions, we define a differentiable loss for time series and present two algorithms for the computation of time series barycenters under this new geometry. We illustrate the interest of our approach on both simulated and real world data and show the robustness of our approach compared to state-of-the-art methods.

stat.ML↗

arXiv2022

Sliced Gromov-Wasserstein

Recently used in various machine learning contexts, the Gromov-Wasserstein distance (GW) allows for comparing distributions whose supports do not necessarily lie in the same metric space. However, this Optimal Transport (OT) distance requires solving a complex non convex quadratic program which is most of the time very costly both in time and memory. Contrary to GW, the Wasserstein distance (W) enjoys several properties (e.g. duality) that permit large scale optimization. Among those, the solution of W on the real line, that only requires sorting discrete samples in 1D, allows defining the Sliced Wasserstein (SW) distance. This paper proposes a new divergence based on GW akin to SW. We first derive a closed form for GW when dealing with 1D distributions, based on a new result for the related quadratic assignment problem. We then define a novel OT discrepancy that can deal with large scale distributions via a slicing approach and we show how it relates to the GW distance while being $O(n\log(n))$ to compute. We illustrate the behavior of this so called Sliced Gromov-Wasserstein (SGW) discrepancy in experiments where we demonstrate its ability to tackle similar problems as GW while being several order of magnitudes faster to compute.

stat.ML↗

arXiv2022

Learning to Predict Graphs with Fused Gromov-Wasserstein Barycenters

This paper introduces a novel and generic framework to solve the flagship task of supervised labeled graph prediction by leveraging Optimal Transport tools. We formulate the problem as regression with the Fused Gromov-Wasserstein (FGW) loss and propose a predictive model relying on a FGW barycenter whose weights depend on inputs. First we introduce a non-parametric estimator based on kernel ridge regression for which theoretical results such as consistency and excess risk bound are proved. Next we propose an interpretable parametric model where the barycenter weights are modeled with a neural network and the graphs on which the FGW barycenter is calculated are additionally learned. Numerical experiments show the strength of the method and its ability to interpolate in the labeled graph space on simulated data and on a difficult metabolic identification problem where it can reach very good performance with very little engineering.

stat.ML↗

arXiv2022

Multi-source Domain Adaptation via Weighted Joint Distributions Optimal Transport

The problem of domain adaptation on an unlabeled target dataset using knowledge from multiple labelled source datasets is becoming increasingly important. A key challenge is to design an approach that overcomes the covariate and target shift both among the sources, and between the source and target domains. In this paper, we address this problem from a new perspective: instead of looking for a latent representation invariant between source and target domains, we exploit the diversity of source distributions by tuning their weights to the target task at hand. Our method, named Weighted Joint Distribution Optimal Transport (WJDOT), aims at finding simultaneously an Optimal Transport-based alignment between the source and target distributions and a re-weighting of the sources distributions. We discuss the theoretical aspects of the method and propose a conceptually simple algorithm. Numerical experiments indicate that the proposed method achieves state-of-the-art performance on simulated and real-life datasets.

cs.LG↗

arXiv2022

Template based Graph Neural Network with Optimal Transport Distances

Current Graph Neural Networks (GNN) architectures generally rely on two important components: node features embedding through message passing, and aggregation with a specialized form of pooling. The structural (or topological) information is implicitly taken into account in these two steps. We propose in this work a novel point of view, which places distances to some learnable graph templates at the core of the graph representation. This distance embedding is constructed thanks to an optimal transport distance: the Fused Gromov-Wasserstein (FGW) distance, which encodes simultaneously feature and structure dissimilarities by solving a soft graph-matching problem. We postulate that the vector of FGW distances to a set of template graphs has a strong discriminative power, which is then fed to a non-linear classifier for final predictions. Distance embedding can be seen as a new layer, and can leverage on existing message passing techniques to promote sensible feature representations. Interestingly enough, in our work the optimal set of template graphs is also learnt in an end-to-end fashion by differentiating through this layer. After describing the corresponding learning procedure, we empirically validate our claim on several synthetic and real life graph classification datasets, where our method is competitive or surpasses kernel and GNN state-of-the-art approaches. We complete our experiments by an ablation study and a sensitivity analysis to parameters.

cs.LG↗

arXiv2022

Sliding window strategy for convolutional spike sorting with Lasso : Algorithm, theoretical guarantees and complexity

Spike sorting is a class of algorithms used in neuroscience to attribute the time occurences of particular electric signals, called action potential or spike, to neurons. We rephrase this problem as a particular optimization problem : Lasso for convolutional models in high dimension. Lasso (i.e. least absolute shrinkage and selection operator) is a very generic tool in machine learning that help us to look for sparse solutions (here the time occurrences). However, for the size of the problem at hand in this neuroscience context, the classical Lasso solvers are failing. We present here a new and much faster algorithm. Making use of biological properties related to neurons, we explain how the particular structure of the problem allows several optimizations, leading to an algorithm with a temporal complexity which grows linearly with respect to the size of the recorded signal and can be performed online. Moreover the spatial separability of the initial problem allows to break it into subproblems, further reducing the complexity and making possible its application on the latest recording devices which comprise a large number of sensors. We provide several mathematical results: the size and numerical complexity of the subproblems can be estimated mathematically by using percolation theory. We also show under reasonable assumptions that the Lasso estimator retrieves the true time occurrences of the spikes {with large probability}. Finally the theoretical time complexity of the algorithm is given. Numerical simulations are also provided in order to illustrate the efficiency of our approach.

math.ST↗

arXiv2022

Semi-relaxed Gromov-Wasserstein divergence with applications on graphs

Comparing structured objects such as graphs is a fundamental operation involved in many learning tasks. To this end, the Gromov-Wasserstein (GW) distance, based on Optimal Transport (OT), has proven to be successful in handling the specific nature of the associated objects. More specifically, through the nodes connectivity relations, GW operates on graphs, seen as probability measures over specific spaces. At the core of OT is the idea of conservation of mass, which imposes a coupling between all the nodes from the two considered graphs. We argue in this paper that this property can be detrimental for tasks such as graph dictionary or partition learning, and we relax it by proposing a new semi-relaxed Gromov-Wasserstein divergence. Aside from immediate computational benefits, we discuss its properties, and show that it can lead to an efficient graph dictionary learning algorithm. We empirically demonstrate its relevance for complex tasks on graphs such as partitioning, clustering and completion.

cs.LG↗

arXiv2021

Factored couplings in multi-marginal optimal transport via difference of convex programming

Optimal transport (OT) theory underlies many emerging machine learning (ML) methods nowadays solving a wide range of tasks such as generative modeling, transfer learning and information retrieval. These latter works, however, usually build upon a traditional OT setup with two distributions, while leaving a more general multi-marginal OT formulation somewhat unexplored. In this paper, we study the multi-marginal OT (MMOT) problem and unify several popular OT methods under its umbrella by promoting structural information on the coupling. We show that incorporating such structural information into MMOT results in an instance of a different of convex (DC) programming problem allowing us to solve it numerically. Despite high computational cost of the latter procedure, the solutions provided by DC optimization are usually as qualitative as those obtained using currently employed optimization schemes.

stat.ML↗

arXiv2021

Provably Convergent Working Set Algorithm for Non-Convex Regularized Regression

Owing to their statistical properties, non-convex sparse regularizers have attracted much interest for estimating a sparse linear model from high dimensional data. Given that the solution is sparse, for accelerating convergence, a working set strategy addresses the optimization problem through an iterative algorithm by incre-menting the number of variables to optimize until the identification of the solution support. While those methods have been well-studied and theoretically supported for convex regularizers, this paper proposes a working set algorithm for non-convex sparse regularizers with convergence guarantees. The algorithm, named FireWorks, is based on a non-convex reformulation of a recent primal-dual approach and leverages on the geometry of the residuals. Our theoretical guarantees derive from a lower bound of the objective function decrease between two inner solver iterations and shows the convergence to a stationary point of the full problem. More importantly, we also show that convergence is preserved even when the inner solver is inexact, under sufficient decay of the error across iterations. Our experimental results demonstrate high computational gain when using our working set strategy compared to the full problem solver for both block-coordinate descent or a proximal gradient solver.

cs.LG↗

arXiv2021

Optimal Transport for Conditional Domain Matching and Label Shift

We address the problem of unsupervised domain adaptation under the setting of generalized target shift (joint class-conditional and label shifts). For this framework, we theoretically show that, for good generalization, it is necessary to learn a latent representation in which both marginals and class-conditional distributions are aligned across domains. For this sake, we propose a learning problem that minimizes importance weighted loss in the source domain and a Wasserstein distance between weighted marginals. For a proper weighting, we provide an estimator of target label proportion by blending mixture estimation and optimal matching by optimal transport. This estimation comes with theoretical guarantees of correctness under mild assumptions. Our experimental results show that our method performs better on average than competitors across a range domain adaptation problems including \emph{digits},\emph{VisDA} and \emph{Office}. Code for this paper is available at \url{https://github.com/arakotom/mars_domain_adaptation}.

cs.LG↗

arXiv2021

Learning with minibatch Wasserstein : asymptotic and gradient properties

Optimal transport distances are powerful tools to compare probability distributions and have found many applications in machine learning. Yet their algorithmic complexity prevents their direct use on large scale datasets. To overcome this challenge, practitioners compute these distances on minibatches {\em i.e.} they average the outcome of several smaller optimal transport problems. We propose in this paper an analysis of this practice, which effects are not well understood so far. We notably argue that it is equivalent to an implicit regularization of the original problem, with appealing properties such as unbiased estimators, gradients and a concentration bound around the expectation, but also with defects such as loss of distance property. Along with this theoretical analysis, we also conduct empirical experiments on gradient flows, GANs or color transfer that highlight the practical interest of this strategy.

stat.ML↗