In this paper, we comparatively analyze the Bures-Wasserstein (BW) geometry with the popular Affine-Invariant (AI) geometry for Riemannian optimization on the symmetric positive definite (SPD) matrix manifold. Our study begins with an observation that the BW metric has a linear dependence on SPD matrices in contrast to the quadratic dependence of the AI metric. We build on this to show that the BW metric is a more suitable and robust choice for several Riemannian optimization problems over ill-conditioned SPD matrices. We show that the BW geometry has a non-negative curvature, which further improves convergence rates of algorithms over the non-positively curved AI geometry. Finally, we verify that several popular cost functions, which are known to be geodesic convex under the AI geometry, are also geodesic convex under the BW geometry. Extensive experiments on various applications support our findings.

Fréchet regression, or conditional Barycenters, is a flexible framework for modeling relationships between covariates (usually Euclidean) and response variables on general metric spaces, e.g., probability distributions or positive definite matrices. However, in contrast to classical barycenter problems, computing conditional counterparts in many non-Euclidean spaces remains an open challenge, as they yield non-convex optimization problems with an affine structure. In this work, we study the existence and computation of conditional barycenters, specifically in the space of positive-definite matrices with the Bures-Wasserstein metric. We provide a sufficient condition for the existence of a minimizer of the conditional barycenter problem that characterizes the regression range of extrapolation. Moreover, we further characterize the optimization landscape, proving that under this condition, the objective is free of local maxima. Additionally, we develop a projection-free and provably correct algorithm for the approximate computation of first-order stationary points. Finally, we provide a stochastic reformulation that enables the use of off-the-shelf stochastic Riemannian optimization methods for large-scale setups. Numerical experiments validate the performance of the proposed methods on regression problems of real-world biological networks and on large-scale synthetic Diffusion Tensor Imaging problems.

Our study begins with an observation that the BW metric has alinear dependence on SPD matrices in contrast tothe quadratic dependence ofthe AI metric.

In this paper, we comparatively analyze the Bures-Wasserstein (BW) geometry with the popular Affine-Invariant (AI) geometry for Riemannian optimization on the symmetric positive definite (SPD) matrix manifold. Our study begins with an observation that the BW metric has a linear dependence on SPD matrices in contrast to the quadratic dependence of the AI metric. We build on this to show that the BW metric is a more suitable and robust choice for several Riemannian optimization problems over ill-conditioned SPD matrices. We show that the BW geometry has a non-negative curvature, which further improves convergence rates of algorithms over the non-positively curved AI geometry. Finally, we verify that several popular cost functions, which are known to be geodesic convex under the AI geometry, are also geodesic convex under the BW geometry. Extensive experiments on various applications support our findings.

We introduce a novel, efficient framework for clustering data on high-dimensional, non-Euclidean manifolds that overcomes the computational challenges associated with standard intrinsic methods. The key innovation is the use of the $p$-Fréchet map $F^p : \mathcal{M} \to \mathbb{R}^\ell$ -- defined on a generic metric space $\mathcal{M}$ -- which embeds the manifold data into a lower-dimensional Euclidean space $\mathbb{R}^\ell$ using a set of reference points $\{r_i\}_{i=1}^\ell$, $r_i \in \mathcal{M}$. Once embedded, we can efficiently and accurately apply standard Euclidean clustering techniques such as k-means. We rigorously analyze the mathematical properties of $F^p$ in the Euclidean space and the challenging manifold of $n \times n$ symmetric positive definite matrices $\mathit{SPD}(n)$. Extensive numerical experiments using synthetic and real $\mathit{SPD}(n)$ data demonstrate significant performance gains: our method reduces runtime by up to two orders of magnitude compared to intrinsic manifold-based approaches, all while maintaining high clustering accuracy, including scenarios where existing alternative methods struggle or fail.

Human activity recognition is an important research topic in pattern recognition field. It has been the subject of many studies in the past two decades because of its importance in numerous areas such as security, health, daily activity, energy consumption and robotics. Recently, some works on the recognition of hand gestures or human actions from skeletal data are based on the modeling of the skeleton's movement as manifold-based representation and proposed deep neural networks on this structure [1, 2, 3]. These approaches demonstrated their potential in the processing of skeletal data. Most of them are applied on offline human action recognition which is useful in time-limited tasks. However, in many applications, simply recognizing a single gesture in a given segmented sequence is not enough, especially in monitoring systems and virtual-reality devices which need to detect human movements moment by moment in continuous videos. In these online recognition systems, it is important to detect the existence of an action as early as possible after its beginning. It is also essential to determine the nature of the movement within a sequence of frames, without having information about the number of gestures present within the video, their starting times or their durations, unlike the segmented action recognition. In this paper, we propose to use a manifold-based model in order to build an online motion recognition system that detects and identifies different human activities in unsegmented skeletal sequences.

In our article, we introduce a Riemannian batch normalization (batch-norm) algorithm, which generalizes the one used in Euclidean nets.

Many modern applications involve predicting structured, non-Euclidean outputs such as probability distributions, networks, and symmetric positive-definite matrices. These outputs are naturally modeled as elements of general metric spaces, where classical regression techniques that rely on vector space structure no longer apply. We introduce E2M (End-to-End Metric regression), a deep learning framework for predicting metric space-valued outputs. E2M performs prediction via a weighted Fréchet means over training outputs, where the weights are learned by a neural network conditioned on the input. This construction provides a principled mechanism for geometry-aware prediction that avoids surrogate embeddings and restrictive parametric assumptions, while fully preserving the intrinsic geometry of the output space. We establish theoretical guarantees, including a universal approximation theorem that characterizes the expressive capacity of the model and a convergence analysis of the entropy-regularized training objective. Through extensive simulations involving probability distributions, networks, and symmetric positive-definite matrices, we show that E2M consistently achieves state-of-the-art performance, with its advantages becoming more pronounced at larger sample sizes. Applications to human mortality distributions and New York City taxi networks further demonstrate the flexibility and practical utility of the framework.

Foundation models for electroencephalography (EEG) signals have recently demonstrated success in learning generalized representations of EEGs, outperforming specialized models in various downstream tasks. However, many of these models lack transparency in their pretraining dynamics and offer limited insight into how well EEG information is preserved within their embeddings. For successful clinical integration, EEG foundation models must ensure transparency in pretraining, downstream fine-tuning, and the interpretability of learned representations. Current approaches primarily operate in the temporal domain, overlooking advancements in digital signal processing that enable the extraction of deterministic and traceable features, such as wavelet-based representations. We propose MENDR (Manifold Explainable Neural Data Representations), a filter bank-based EEG foundation model built on a novel Riemannian Manifold Transformer architecture to resolve these issues. MENDR learns symmetric positive definite matrix embeddings of EEG signals and is pretrained on a large corpus comprising over 4,000 hours of EEG data, decomposed via discrete wavelet packet transforms into multi-resolution coefficients. MENDR significantly enhances interpretability by visualizing symmetric positive definite embeddings as geometric ellipsoids and supports accurate reconstruction of EEG signals from learned embeddings. Evaluations across multiple clinical EEG tasks demonstrate that MENDR achieves near state-of-the-art performance with substantially fewer parameters, underscoring its potential for efficient, interpretable, and clinically applicable EEG analysis.

The goal of this paper is to show how different machine learning tools on the Riemannian manifold $\mathcal{P}_d$ of Symmetric Positive Definite (SPD) matrices can be united under a probabilistic framework. For this, we will need several Gaussian distributions defined on $\mathcal{P}_d$. We will show how popular classifiers on $\mathcal{P}_d$ can be reinterpreted as Bayes Classifiers using these Gaussian distributions. These distributions will also be used for outlier detection and dimension reduction. By showing that those distributions are pervasive in the tools used on $\mathcal{P}_d$, we allow for other machine learning tools to be extended to $\mathcal{P}_d$.