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.

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 paper addresses classification of curve-shaped data by proposing a Riemannian geometric framework for linear and quadratic discriminant analysis (LDA/QDA) on shape manifolds. Methodologically, shapes are represented via the square-root velocity function (SRVF), and the infinite-dimensional shape manifold is locally linearized through tangent-space projection. Riemannian means and covariances are estimated in the tangent space, and dimensionality reduction is achieved using Fourier basis coefficients. The resulting classifier is invariant to translation, rotation, and scaling. The key contribution is the first systematic adaptation of classical LDA/QDA to the tangent plane of shape space, overcoming fundamental limitations of Euclidean approaches in preserving shape invariances. Experiments demonstrate superior classification performance on both synthetic datasets and real biomedical shapes—including cortical sulci, the corpus callosum, and midline facial features in fetal alcohol syndrome.

This work addresses the problem of energy estimation for signal propagation on submanifolds of Riemannian manifolds. Methodologically, it introduces a geometric signal modeling paradigm wherein signals are defined intrinsically as submanifolds—not scalar-valued functions—thereby unifying signal theory with differential geometry. The approach integrates submanifold theory, energy functional analysis, configuration space topology, and graph embedding techniques to derive, for the first time, explicit upper and lower bounds on signal propagation energy within canonical geometric parameter spaces—including Gaussian distribution manifolds and point configuration spaces. It further systematically characterizes how temporal evolution and graph embeddings affect energy constraints. The primary contribution is the establishment of the first Riemannian-geometric framework for energy analysis of submanifold-valued signals, providing both a rigorous theoretical foundation and computationally tractable quantification tools for geometric signal processing.

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 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 the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.

This work addresses the limitations of traditional geometric processing methods, which rely on manifold assumptions and struggle with non-manifold geometries featuring singular structures such as sharp features, self-intersections, or branches. The authors propose a novel “tangent blow-up” representation, introducing for the first time the algebraic geometry concept of blow-up into geometric processing. By jointly embedding each spatial point together with its tangent plane into the product space of Euclidean space and a Grassmannian manifold, the method iteratively disambiguates coincident points that differ in tangential or higher-order contact. Within this lifted domain, discrete gradient, divergence, and Laplace operators are rigorously defined. This structured representation enables a natural extension of classical differential operators to singular points, demonstrating effectiveness in tasks including geodesic computation, segmentation, parameterization, and curvature estimation.

This work addresses the variational minimization of complex functionals in differential geometry by systematically introducing physics-informed neural networks (PINNs) into a general differential geometric framework for the first time. By directly embedding geometric functionals into the loss function of neural networks, the approach enables end-to-end numerical solutions without requiring mesh discretization. A rigorous correspondence is established between geometric variational problems and deep learning optimization objectives, allowing the method to effectively handle high-dimensional and nonlinear geometric structures. The study reproduces and integrates three representative works, demonstrating the feasibility and advantages of PINNs in solving intricate geometric variational problems, thereby offering a novel paradigm at the intersection of differential geometry and artificial intelligence.