Related papers: Kriging Riemannian Data via Random Domain Decompos…

To improve the predictability of complex computational models in the experimentally-unknown domains, we propose a Bayesian statistical machine learning framework utilizing the Dirichlet distribution that combines results of several…

Barren plateaus present a major challenge in the training of variational quantum algorithms (VQAs), particularly for large-scale discretizations of nonlinear partial differential equations. In this work, we introduce a domain decomposition…

This paper focuses on the construction of accurate and predictive data-driven reduced models of large-scale numerical simulations with complex dynamics and sparse training datasets. In these settings, standard, single-domain approaches may…

Modern generative modeling methods have demonstrated strong performance in learning complex data distributions from clean samples. In many scientific and imaging applications, however, clean samples are unavailable, and only noisy or…

Reformulating computer vision problems over Riemannian manifolds has demonstrated superior performance in various computer vision applications. This is because visual data often forms a special structure lying on a lower dimensional space…

Performing Bayesian inference on large spatio-temporal models requires extracting inverse elements of large sparse precision matrices for marginal variances, as well as estimating model hyperparameters. Although direct matrix factorizations…

We propose a domain decomposition method for the efficient simulation of nonlocal problems. Our approach is based on a multi-domain formulation of a nonlocal diffusion problem where the subdomains share "nonlocal" interfaces of the size of…

A local approach to the time integration of PDEs by exponential methods is proposed, motivated by theoretical estimates by A.Iserles on the decay of off-diagonal terms in the exponentials of sparse matrices. An overlapping domain…

Fr\'echet regression extends classical regression methods to non-Euclidean metric spaces, enabling the analysis of data relationships on complex structures such as manifolds and graphs. This work establishes a rigorous theoretical analysis…

This paper is a short summary of our recent work on the medians and means of probability measures in Riemannian manifolds. Firstly, the existence and uniqueness results of local medians are given. In order to compute medians in practical…

We propose a natural intrinsic extension of ridge regression from Euclidean spaces to general Riemannian manifolds for time-series prediction. Our approach combines Riemannian least-squares fitting via B\'ezier curves, empirical covariance…

This paper proposes a general framework of Riemannian adaptive optimization methods. The framework encapsulates several stochastic optimization algorithms on Riemannian manifolds and incorporates the mini-batch strategy that is often used…

A central issue of the science of complex systems is the quantitative characterization of complexity. In the present work we address this issue by resorting to information geometry. Actually we propose a constructive way to associate to a -…

We propose a new Riemannian gradient descent method for computing spherical area-preserving mappings of topological spheres using a Riemannian retraction-based framework with theoretically guaranteed convergence. The objective function is…

This paper studies the problem of distributed Riemannian optimization over a network of agents whose cost functions are geodesically smooth but possibly geodesically non-convex. Extending a well-known distributed optimization strategy…

Modeling non-Euclidean data is drawing extensive attention along with the unprecedented successes of deep neural networks in diverse fields. Particularly, a symmetric positive definite matrix is being actively studied in computer vision,…

Nonlinear dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. However, many popular methods can fail dramatically, even on simple two-dimensional manifolds, due to problems such as…

Bayesian model-based spatial clustering methods are widely used for their flexibility in estimating latent clusters with an unknown number of clusters while accounting for spatial proximity. Many existing methods are designed for clustering…

This paper develops the first decentralized online Riemannian optimization algorithm on Hadamard manifolds. Our algorithm, the decentralized projected Riemannian gradient descent, iteratively performs local updates using projected…

Optimization with orthogonality constraints frequently arises in various fields such as machine learning. Riemannian optimization offers a powerful framework for solving these problems by equipping the constraint set with a Riemannian…