apply differential topology

This paper establishes a categorical correspondence between the geometric notions of *local connectedness* and *properness* and the logical constructs of Σ-types (existential quantification) and Π-types (universal quantification) in dependent type theory. Using higher-categorical machinery and the framework of Grothendieck fibrations, it systematically reveals, for the first time, an isomorphism between *smooth/proper morphisms* and *Σ/Π-constructions*, and naturally extends this correspondence to arbitrary Grothendieck fibrations, endowing them with intrinsic geometric semantics. The approach unifies classical examples—including topological fibrations and étale/proper morphisms in algebraic geometry—and generalizes to novel homotopical geometric semantics. Its core contribution is a transferable structural paradigm bridging geometry and logic, realized via a bidirectional formal correspondence. All principal results are rigorously proven against external literature.

This paper addresses whether the persistent homology of an unknown $mathbb{R}^n$-valued function $f$ on a metric space $X$ can be reliably approximated from a finite sample—containing pairwise distances and function values—extending prior theoretical guarantees, which exist only for $n=1$. Method: We generalize the function–geometry bifiltration to arbitrary $n$, assuming $f$ is Lipschitz continuous and $X$ is a regular geodesic metric space. Our approach establishes provable approximation error bounds for multiparameter persistent homology, explicitly characterizing how scale parameters affect estimation accuracy—going beyond mere stability analysis. Contribution/Results: We provide the first theoretically grounded, general framework for high-dimensional topological data analysis, overcoming the limitations of single-parameter persistence. The results enable robust modeling of multiscale and multidirectional topological features, with quantifiable approximation guarantees for multiparameter persistent homology under realistic sampling assumptions.

This work addresses stochastic optimization of nonsmooth *tame* functions on Riemannian manifolds, motivated by training challenges in deep learning arising from geometric constraints and nondifferentiable components. We propose a reparameterized stochastic gradient descent (SGD) algorithm incorporating a contraction mapping to guarantee that all iterates remain strictly on the manifold. For the first time, we unify tame geometry theory with Riemannian stochastic optimization, rigorously linking the subdifferential properties of tame functions on manifolds to the convergence behavior of SGD. Under mild regularity assumptions—specifically, requiring only weak continuity of the generalized gradient—we establish almost-sure global convergence of the algorithm for general nonsmooth objectives under diminishing step sizes. This provides the first theoretically rigorous and practically implementable convergence framework for modern machine learning models involving both geometric constraints and nonsmooth structures.

This work addresses the joint geometric and photometric alignment problem in image matching. We formulate a variational model grounded in nonlinear elasticity theory: images are modeled as bounded open subsets of ℝⁿ equipped with vector-valued intensity fields, and optimal deformations are obtained by minimizing an integral energy functional subject to orientation-preserving diffeomorphism constraints. Our key contributions include: (i) the introduction of a novel polyconvex energy functional dependent on second-order derivatives of the transformation—overcoming the classical limitation of relying solely on the Jacobian determinant; (ii) a rigorous existence proof for minimizers under bounded measurable intensity fields; and (iii) the first necessary and sufficient condition for affine-invariant recovery, leading to a new model that uniquely recovers the ground-truth deformation for any pair of affinely related images. The framework integrates measure-theoretic analysis, landmark embedding, and variational methods, ensuring both theoretical rigor and high matching accuracy.

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.

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.

This study addresses statistical ill-posedness in distributional models—such as non-identifiability, singular Fisher information, and moment indeterminacy—arising from geometric degeneracies. It introduces transversality theory from differential topology to systematically characterize well-behaved models through the geometric properties of kernel-induced feature maps. Leveraging tools including Sard’s theorem, smooth mappings, jet spaces, and parametric transversality, the work unifies the explanation of six classical pathological phenomena, including representation collapse and the Behrens–Fisher problem, demonstrating that these issues are non-generic under typical conditions. The analysis provides a geometric perspective and theoretical foundation for robust statistical modeling, while presenting the underlying topological machinery in an accessible, pedagogically oriented manner.

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.

This work proposes a pedagogical framework for introducing topological data analysis to students of mathematics and computer science, balancing mathematical rigor with accessibility. Departing from conventional metric-space-based approaches, the framework models data as information-carrying functions and foregrounds the role of the observer along with symmetry constraints. It naturally bridges persistent homology and symmetry-aware modeling in machine learning through group equivariant non-expansive operators (GENEOs). By integrating persistent homology, algebraic topology, and monodromy theory from two-parameter persistence, the approach forms a self-contained instructional system that significantly enhances conceptual clarity and cross-disciplinary applicability, making it well-suited for advanced undergraduate and graduate instruction.

This work addresses the theoretical gap in large-stepsize gradient descent for over-parameterized least squares problems with vector outputs, where conventional theory requires stepsizes smaller than twice the inverse sharpness—yet practitioners routinely use larger stepsizes near flat minima manifolds without rigorous justification. By integrating tools from dynamical systems, differential geometry, and singular partial differential equations, this study establishes the first convergence theory for large-stepsize gradient descent extended from isolated flat minima to general flat minima manifolds. The main contributions include a unified normal form and convergence analysis, the discovery that the set of minimizers in deep matrix factorization possesses a fiber bundle structure over a product of spheres satisfying the Morse–Bott condition, and a novel method for solving the associated singular PDEs, thereby providing a rigorous theoretical foundation for large-stepsize optimization.