apply differential geometry

This work addresses the gap between abstract Riemannian geometry and practical algorithmic implementation by systematically developing a computationally tractable geometric framework for Riemannian optimization. Focusing on canonical matrix manifolds—Stiefel, Grassmann, and symmetric positive-definite (SPD) manifolds—it explicitly derives core geometric structures, including tangent spaces, metric tensors, Levi-Civita connections, curvature operators, and geodesics, all expressed in coordinate- and matrix-based forms amenable to numerical computation. Furthermore, it provides closed-form expressions for the Riemannian gradient, Hessian, exponential map, and retraction operators. To the best of our knowledge, this is the first unified formulation that translates classical differential-geometric constructions into a consistent, implementation-ready framework, thereby bridging theory and practice and offering a rigorous foundation for efficient and accurate algorithm design in Riemannian optimization and geometric machine learning.

Computing geodesics on Riemannian and Finsler manifolds typically relies on numerical approximations, which suffer from poor stability, slow convergence, and limited scalability to high dimensions. This paper introduces the first formulation of geodesic computation as a discrete optimal control problem, unified within a nonlinear optimization framework that incorporates manifold-aware gradient methods—applicable to both Riemannian and Finsler geometries. Theoretically, we establish global convergence and local quadratic convergence rates. Empirically, on high-dimensional manifolds arising in information geometry and generative modeling, our method achieves an average 2.3× speedup over state-of-the-art solvers, with superior accuracy, enhanced numerical stability, and milder growth in computational complexity with respect to dimensionality.

This work addresses the lack of effective support for Kendall’s 3D shape space in existing Python libraries—such as Geomstats—which has hindered the practical application of Riemannian geometry in three-dimensional shape analysis. We present the first systematic implementation of an efficient and user-friendly Python toolkit tailored specifically to Kendall’s 3D shape space, enabling scale-, position-, and orientation-invariant shape modeling and statistical analysis. By providing a ready-to-use, open-source solution for shape statistics on manifolds, this contribution fills a critical software gap in advanced 3D shape analysis, substantially lowering the barrier to entry for researchers, improving computational efficiency, and enhancing the reproducibility of results.

This work addresses manifold learning for the space of absolutely continuous probability measures $mathcal{P}_{mathrm{a.c.}}(Omega)$—endowed with the Wasserstein-2 metric—where $Omega subset mathbb{R}^d$ is compact and convex. We propose the first extrinsic, distance-based implicit manifold modeling framework grounded solely in Wasserstein distances. Methodologically, we construct locally linearizable non-flat Wasserstein submanifolds and estimate tangent spaces via spectral analysis of the covariance operator associated with optimal transport maps, using only pairwise Wasserstein distances between samples. Theoretically, we prove that, as sample density tends to infinity, the distance graph asymptotically recovers the intrinsic metric structure of the manifold. Empirically, tangent spaces are reconstructed with high accuracy. Our key contribution lies in transcending Euclidean assumptions: we establish the first rigorous theoretical and algorithmic foundation for distance-driven manifold learning directly in the Wasserstein space.

Traditional calculus and Riemannian geometry struggle to handle non-manifold, noisy real-world data. This work proposes a data-driven framework grounded in diffusion processes that, for the first time, systematically realizes computable formulations of vector calculus and key geometric objects—such as geodesic distances, curvatures, vector field flows, and solutions to partial differential equations—within the paradigm of diffusion geometry. The framework further integrates topological tools from de Rham cohomology and Morse theory. Leveraging efficient numerical linear algebra techniques, it achieves substantial improvements in computational accuracy, noise robustness, and scalability, demonstrating exceptional numerical stability, low computational complexity, and strong robustness across a range of geometric and topological tasks.

This work addresses the limitations of traditional Euclidean dimensionality reduction methods in effectively handling data intrinsically residing on nonlinear Riemannian manifolds—such as hyperspheres or the manifold of symmetric positive-definite matrices. By extending classical techniques like principal component analysis and discriminant analysis into a Riemannian geometric framework, the study proposes geometry-aware nonlinear dimensionality reduction approaches grounded in geodesic distances, tangent space mappings, and intrinsic statistical measures. These include Principal Geodesic Analysis (PGA) and manifold-based discriminant analysis. Experimental results demonstrate that the proposed methods significantly outperform their Euclidean counterparts on benchmark datasets embedded in curved spaces, achieving superior preservation of intrinsic manifold structure, enhanced quality of low-dimensional embeddings, and improved downstream classification performance.

This work addresses optimization problems defined over products of simplices, such as low-rank learning of discrete multivariate probability distributions and function data registration based on the Square-Root Velocity Function (SRVF) representation. To tackle the inherent constraints, the authors propose a smooth reparameterization that is strictly convex element-wise, transforming the constrained problem into an unconstrained optimization over a Riemannian manifold. The resulting problem is solved via Riemannian gradient descent (RGD). Theoretical analysis shows that this reparameterization maps second-order KKT points on the manifold to weak second-order KKT points of the original problem, ensuring theoretical soundness while enhancing computational efficiency. Experiments demonstrate that RGD significantly outperforms projected gradient descent (PGD), achieving more accurate shape-preserving registration in functional data and efficiently solving probability tensor decomposition tasks.

Existing manifold-based deep learning approaches are often limited to specific manifolds, rely on Euclidean approximations, or involve computationally expensive and numerically unstable geometric operations. This work proposes a unified Riemannian deep learning framework whose core innovations include generalizing batch normalization to Lie and rotation groups, extending multinomial logistic regression to arbitrary Riemannian manifolds, and designing an adaptive, efficient metric for symmetric positive definite (SPD) matrices. The framework integrates generalized batch normalization, Riemannian multinomial regression, unconstrained modeling in hyperbolic space, Busemann function learning, and Cholesky/Log-Euclidean geometries. Theoretical analysis and experiments demonstrate that the proposed method achieves superior performance and computational efficiency across diverse tasks in computer vision, signal processing, graph learning, and genomics.