Salem Said

Isabella Costa Maia, Pedro L. C. Rodrigues, Salem Said et al.

Cross-domain EEG decoding remains challenging despite advances in Riemannian deep learning: covariance matrices from different subjects occupy systematically distinct regions of the SPD manifold, yet existing domain adaptation methods either require target-domain calibration data or learn subject-specific components that cannot generalise across domains. We propose dynamic Stiefel routing: a pool of $K$ expert projection filters on the Stiefel manifold, each specialised for a different region of the SPD manifold, with each input covariance routed to the most appropriate filter via cross-attention, adapting the subspace projection per sample. A central finding is that this approach, implemented naively, provably collapses to ensemble averaging: when routing weights are uniform, the adaptive filter reduces exactly to an equal-contribution combination of experts, indistinguishable from a single fixed filter. Three structural properties break this degeneracy: a symmetric anchor $W_{\mathrm{base}} \in \mathrm{St}(n,k)$ that removes proximity bias among experts; a frozen domain-discriminative query encoder that decouples routing from task optimisation; and a decoupled key alignment loss that trains expert keys toward stable domain attractors. Together they produce the first genuinely committed and domain-structured routing on SPD manifolds, with consistent gains across three datasets: balanced accuracy improves from $0.773\to 0.823$, $0.757\to 0.809$, and $0.801\to 0.839$, with the alignment strategy determined automatically by a single data-driven rule and no dataset-specific hyperparameter search.

Dmitry Eremeev, Salem Said, Viacheslav Borovitskiy

Applying kernel methods to matchings is challenging due to their discrete, non-Euclidean nature. In this paper, we develop a principled framework for constructing geometric kernels that respect the natural geometry of the space of matchings. To this end, we first provide a complete characterization of stationary kernels, i.e. kernels that respect the inherent symmetries of this space. Because the class of stationary kernels is too broad, we specifically focus on the heat and Matérn kernel families, adding an appropriate inductive bias of smoothness to stationarity. While these families successfully extend widely popular Euclidean kernels to matchings, evaluating them naively incurs a prohibitive super-exponential computational cost. To overcome this difficulty, we introduce and analyze a novel, sub-exponential algorithm leveraging zonal polynomials for efficient kernel evaluation. Finally, motivated by the known bijective correspondence between matchings and phylogenetic trees-a crucial data modality in biology-we explore whether our framework can be seamlessly transferred to the space of trees, establishing novel negative results and identifying a significant open problem.

Florent Bouchard, Nils Laurent, Salem Said et al.

In this paper, the issue of averaging data on a manifold is addressed. While the Fréchet mean resulting from Riemannian geometry appears ideal, it is unfortunately not always available and often computationally very expensive. To overcome this, R-barycenters have been proposed and successfully applied to Stiefel and Grassmann manifolds. However, R-barycenters still suffer severe limitations as they rely on iterative algorithms and complicated operators. We propose simpler, yet efficient, barycenters that we call RL-barycenters. We show that, in the setting relevant to most applications, our framework yields astonishingly simple barycenters: arithmetic means projected onto the manifold. We apply this approach to the Stiefel and Grassmann manifolds. On simulated data, our approach is competitive with respect to existing averaging methods, while computationally cheaper.

Elias N. Zois, Moises Diaz, Salem Said et al.

Offline handwritten signature verification remains a challenging task, particularly in writer-independent settings where models must generalize across unseen individuals. Recent developments have highlighted the advantage of geometrically inspired representations, such as covariance descriptors on Riemannian manifolds. However, past or present, handcrafted or data-driven methods usually depend on real-world signature datasets for classifier training. We introduce a quasi-synthetic data generation framework leveraging the Riemannian geometry of Symmetric Positive Definite matrices (SPD). A small set of genuine samples in the SPD space is the seed to a Riemannian Gaussian Mixture which identifies Riemannian centers as synthetic writers and variances as their properties. Riemannian Gaussian sampling on each center generates positive as well as negative synthetic SPD populations. A metric learning framework utilizes pairs of similar and dissimilar SPD points, subsequently testing it over on real-world datasets. Experiments conducted on two popular signature datasets, encompassing Western and Asian writing styles, demonstrate the efficacy of the proposed approach under both intra- and cross- dataset evaluation protocols. The results indicate that our quasi-synthetic approach achieves low error rates, highlighting the potential of generating synthetic data in Riemannian spaces for writer-independent signature verification systems.

Franziskus Steinert, Salem Said, Cyrus Mostajeran

The universality properties of kernels characterize the class of functions that can be approximated in the associated reproducing kernel Hilbert space and are of fundamental importance in the theoretical underpinning of kernel methods in machine learning. In this work, we establish fundamental tools for investigating universality properties of kernels in Riemannian symmetric spaces, thereby extending the study of this important topic to kernels in non-Euclidean domains. Moreover, we use the developed tools to prove the universality of several recent examples from the literature on positive definite kernels defined on Riemannian symmetric spaces, thus providing theoretical justification for their use in applications involving manifold-valued data.

Alain Durmus, Pablo Jiménez, Éric Moulines et al.

This paper studies fixed step-size stochastic approximation (SA) schemes, including stochastic gradient schemes, in a Riemannian framework. It is motivated by several applications, where geodesics can be computed explicitly, and their use accelerates crude Euclidean methods. A fixed step-size scheme defines a family of time-homogeneous Markov chains, parametrized by the step-size. Here, using this formulation, non-asymptotic performance bounds are derived, under Lyapunov conditions. Then, for any step-size, the corresponding Markov chain is proved to admit a unique stationary distribution, and to be geometrically ergodic. This result gives rise to a family of stationary distributions indexed by the step-size, which is further shown to converge to a Dirac measure, concentrated at the solution of the problem at hand, as the step-size goes to 0. Finally, the asymptotic rate of this convergence is established, through an asymptotic expansion of the bias, and a central limit theorem.

Alain Durmus, Pablo Jiménez, Éric Moulines et al.

This paper analyzes the convergence for a large class of Riemannian stochastic approximation (SA) schemes, which aim at tackling stochastic optimization problems. In particular, the recursions we study use either the exponential map of the considered manifold (geodesic schemes) or more general retraction functions (retraction schemes) used as a proxy for the exponential map. Such approximations are of great interest since they are low complexity alternatives to geodesic schemes. Under the assumption that the mean field of the SA is correlated with the gradient of a smooth Lyapunov function (possibly non-convex), we show that the above Riemannian SA schemes find an ${\mathcal{O}}(b_\infty + \log n / \sqrt{n})$-stationary point (in expectation) within ${\mathcal{O}}(n)$ iterations, where $b_\infty \geq 0$ is the asymptotic bias. Compared to previous works, the conditions we derive are considerably milder. First, all our analysis are global as we do not assume iterates to be a-priori bounded. Second, we study biased SA schemes. To be more specific, we consider the case where the mean-field function can only be estimated up to a small bias, and/or the case in which the samples are drawn from a controlled Markov chain. Third, the conditions on retractions required to ensure convergence of the related SA schemes are weak and hold for well-known examples. We illustrate our results on three machine learning problems.