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

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 using 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 distance 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.

Disambiguation of One-Shot Visual Classification Tasks: A Simplex-Based Approach

The field of visual few-shot classification aims at transferring the state-of-the-art performance of deep learning visual systems onto tasks where only a very limited number of training samples are available. The main solution consists in training a feature extractor using a large and diverse dataset to be applied to the considered few-shot task. Thanks to the encoded priors in the feature extractors, classification tasks with as little as one example (or "shot'') for each class can be solved with high accuracy, even when the shots display individual features not representative of their classes. Yet, the problem becomes more complicated when some of the given shots display multiple objects. In this paper, we present a strategy which aims at detecting the presence of multiple and previously unseen objects in a given shot. This methodology is based on identifying the corners of a simplex in a high dimensional space. We introduce an optimization routine and showcase its ability to successfully detect multiple (previously unseen) objects in raw images. Then, we introduce a downstream classifier meant to exploit the presence of multiple objects to improve the performance of few-shot classification, in the case of extreme settings where only one shot is given for its class. Using standard benchmarks of the field, we show the ability of the proposed method to slightly, yet statistically significantly, improve accuracy in these settings.

Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projections

It has been shown beneficial for many types of data which present an underlying hierarchical structure to be embedded in hyperbolic spaces. Consequently, many tools of machine learning were extended to such spaces, but only few discrepancies to compare probability distributions defined over those spaces exist. Among the possible candidates, optimal transport distances are well defined on such Riemannian manifolds and enjoy strong theoretical properties, but suffer from high computational cost. On Euclidean spaces, sliced-Wasserstein distances, which leverage a closed-form of the Wasserstein distance in one dimension, are more computationally efficient, but are not readily available on hyperbolic spaces. In this work, we propose to derive novel hyperbolic sliced-Wasserstein discrepancies. These constructions use projections on the underlying geodesics either along horospheres or geodesics. We study and compare them on different tasks where hyperbolic representations are relevant, such as sampling or image classification.

Spatial Graph Signal Interpolation with an Application for Merging BCI Datasets with Various Dimensionalities

BCI Motor Imagery datasets usually are small and have different electrodes setups. When training a Deep Neural Network, one may want to capitalize on all these datasets to increase the amount of data available and hence obtain good generalization results. To this end, we introduce a spatial graph signal interpolation technique, that allows to interpolate efficiently multiple electrodes. We conduct a set of experiments with five BCI Motor Imagery datasets comparing the proposed interpolation with spherical splines interpolation. We believe that this work provides novel ideas on how to leverage graphs to interpolate electrodes and on how to homogenize multiple datasets.

Geometry-preserving lie group integrators for differential equations on the manifold of symmetric positive definite matrices

In many applications, one encounters signals that lie on manifolds rather than a Euclidean space. In particular, covariance matrices are examples of ubiquitous mathematical objects that have a non Euclidean structure. The application of Euclidean methods to integrate differential equations lying on such objects does not respect the geometry of the manifold, which can cause many numerical issues. In this paper, we propose to use Lie group methods to define geometry-preserving numerical integration schemes on the manifold of symmetric positive definite matrices. These can be applied to a number of differential equations on covariance matrices of practical interest. We show that they are more stable and robust than other classical or naive integration schemes on an example.

Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flows

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. We further constrain the path leading to the estimated Monge map 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 cost for several existing models without affecting the model performance.

Active Few-Shot Classification: a New Paradigm for Data-Scarce Learning Settings

We consider a novel formulation of the problem of Active Few-Shot Classification (AFSC) where the objective is to classify a small, initially unlabeled, dataset given a very restrained labeling budget. This problem can be seen as a rival paradigm to classical Transductive Few-Shot Classification (TFSC), as both these approaches are applicable in similar conditions. We first propose a methodology that combines statistical inference, and an original two-tier active learning strategy that fits well into this framework. We then adapt several standard vision benchmarks from the field of TFSC. Our experiments show the potential benefits of AFSC can be substantial, with gains in average weighted accuracy of up to 10% compared to state-of-the-art TFSC methods for the same labeling budget. We believe this new paradigm could lead to new developments and standards in data-scarce learning settings.

Preserving Fine-Grain Feature Information in Classification via Entropic Regularization

Labeling a classification dataset implies to define classes and associated coarse labels, that may approximate a smoother and more complicated ground truth. For example, natural images may contain multiple objects, only one of which is labeled in many vision datasets, or classes may result from the discretization of a regression problem. Using cross-entropy to train classification models on such coarse labels is likely to roughly cut through the feature space, potentially disregarding the most meaningful such features, in particular losing information on the underlying fine-grain task. In this paper we are interested in the problem of solving fine-grain classification or regression, using a model trained on coarse-grain labels only. We show that standard cross-entropy can lead to overfitting to coarse-related features. We introduce an entropy-based regularization to promote more diversity in the feature space of trained models, and empirically demonstrate the efficacy of this methodology to reach better performance on the fine-grain problems. Our results are supported through theoretical developments and empirical validation.

Spherical Sliced-Wasserstein

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: density estimation on the sphere, variational inference or hyperspherical auto-encoders.

Preventing Manifold Intrusion with Locality: Local Mixup

Mixup is a data-dependent regularization technique that consists in linearly interpolating input samples and associated outputs. It has been shown to improve accuracy when used to train on standard machine learning datasets. However, authors have pointed out that Mixup can produce out-of-distribution virtual samples and even contradictions in the augmented training set, potentially resulting in adversarial effects. In this paper, we introduce Local Mixup in which distant input samples are weighted down when computing the loss. In constrained settings we demonstrate that Local Mixup can create a trade-off between bias and variance, with the extreme cases reducing to vanilla training and classical Mixup. Using standardized computer vision benchmarks , we also show that Local Mixup can improve test accuracy.