measure-preserving mapping

This work addresses the challenge of learning probability distributions on the simplex for categorical data modeling. The proposed method introduces a Euclidean density modeling framework based on a smooth bijective mapping: it constructs, for the first time, a smooth bijection from the open simplex to ℝ^(K−1) that preserves Aitchison geometry; integrates Dirichlet interpolation to enable continuous representation of discrete categories; and employs flow matching for efficient density estimation. Crucially, the inverse mapping exactly recovers the distribution on the original simplex, circumventing the computational complexity inherent in Riemannian manifold modeling. Experiments on synthetic and real-world datasets demonstrate that the approach achieves or surpasses state-of-the-art baselines in both distributional fidelity and sample quality, while maintaining theoretical rigor and practical generalizability.

This paper establishes a rigorous mathematical analogy between conditional probability and geometric path lifting in the Wasserstein space. Method: It introduces the notion of submetry into the Wasserstein metric structure for the first time, proving that conditional probability corresponds bijectively to cost-preserving lifts of optimal transport plans; it further develops a “lens” framework within weighted categories to model conditional probability as structured morphisms. The approach integrates optimal transport theory, measure theory on standard Borel spaces, pseudometric geometry, and weighted category theory. Contributions: (1) A categorical characterization of the Wasserstein distance, whose value arises as the solution to an optimization problem over weighted-category morphisms; (2) A one-to-one correspondence between conditional probabilities and cost-preserving lifts; (3) A unification of optimal transport, metric geometry, and category theory, yielding an intrinsic geometric interpretation of probabilistic structures.

This paper addresses the lack of an information-theoretic characterization and a point-to-set principle for finite-state dimension (FS dimension). It introduces a precise quantification of the information content of real numbers under finite precision, integrating effective dimension theory, finite-state automaton complexity, and relativized algorithmic information theory. The work establishes, for the first time, a point-to-set principle for FS dimension and thereby defines a robust notion of relative normality. It then rigorously proves the equivalence between relative normality and FS dimension. The main contributions are: (1) an information-theoretic, exact definition of FS dimension; (2) a necessary and sufficient characterization linking relative normality to FS dimension; and (3) a foundational framework enabling future investigation of its equidistribution properties.

Modeling the joint distribution of high-dimensional discrete variables (e.g., binary or categorical data) suffers from exponential computational complexity as the number of categories grows, severely limiting scalability. Method: This paper proposes a generative model based on the Random Assignment Flow (RAF), which models discrete distributions via measure transport on a statistical submanifold. It uniquely integrates *e*-connection geodesics from information geometry with conditional Riemannian flow matching, enabling simulation-free, end-to-end training. Contribution/Results: The model achieves linear time complexity in the number of affinity function parameters—bypassing the exponential barrier of conventional methods—while supporting efficient sampling and exact log-likelihood evaluation. Empirical validation on structured image annotation demonstrates superior scalability: performance degradation under increasing category count is markedly slower than state-of-the-art baselines, confirming its exceptional scalability and practical utility for large-scale discrete modeling.

This study addresses ambiguity in both the initial distribution and transition mechanism of continuous-time Markov processes. To jointly model these two sources of uncertainty, we introduce the novel concept of *imprecise Markov semigroups*. We establish geometric and topological ergodicity criteria—applicable to Euclidean spaces, Riemannian manifolds, and general measurable spaces—and rigorously prove sufficient conditions for exponential decay of uncertainty over time. Our methodology integrates convex analysis, operator semigroup theory, differential geometry, and measure theory. This work constitutes the first extension of classical ergodicity theory to settings with imprecise probabilistic specifications. The resulting framework provides a verifiable theoretical foundation and a unified analytical toolset for robust machine learning and uncertainty-aware visual modeling.

This work investigates whether “straight” generative flows—those realizable via a single forward pass—can transform a source distribution into a target distribution under endpoint-independent conditions. By constructing a stochastic process that interpolates between the two distributions, the authors derive an ordinary differential equation (ODE) flow through its conditional expectation and characterize straightness via zero acceleration. The study reveals a sharp dichotomy: explicit straight flows exist for any pair of Gaussian endpoints, yet no such flow can exist for sufficiently separated multimodal targets. The analysis establishes a profound connection between sample-path behavior and the spatiotemporal geometry of the flow map, provides multiple equivalent partial differential equation (PDE) characterizations, and delineates the precise existence boundary of straight generative flows through an impossibility theorem.

This work addresses the challenge of verifying mathematical proofs generated by large language models by formally encoding, for the first time, an entire advanced undergraduate probability textbook—including its measure-theoretic foundations—into Lean. To bridge the semantic gap between the textbook’s exposition and the abstract formalism of the Mathlib library, the authors introduce an “interface lemma” strategy. Combined with structured proof engineering and formalization techniques specific to measure theory, this approach yields a reusable, machine-verifiable infrastructure spanning fourteen textbook chapters. The resulting formalization not only provides rigorous verification of all stated theorems and explicit articulation of their assumptions but also establishes a robust foundation for reliable AI-assisted mathematics, educational applications, and future formalization efforts in probability theory.

This work addresses the challenge of representing persistence diagrams and related measures under partial optimal transport (POT) by introducing "Persistence Spheres"—an explicit embedding that maps integrable measures on the upper half-plane into a space of continuous functions on the sphere, leveraging convex geometry and ReLU integrals. This representation is the first in topological machine learning to be stable with respect to the POT₁ distance, linear, and endowed with a continuous inverse that is compactly supported everywhere. It inherently encodes a deletion mechanism, naturally capturing the optimal transport behavior to the diagonal under persistence-aware costs without requiring hyperparameter tuning. Empirical evaluations demonstrate that Persistence Spheres match or outperform established baselines—including persistence images, landscapes, splines, and sliced Wasserstein kernels—across clustering, regression, and classification tasks on diverse data types such as functional data, time series, graphs, meshes, and point clouds.

Homogeneous spaces—realized as quotients of Lie groups—pose a challenge for existing flow-matching methods due to the absence of explicit metrics and geodesic structures. This work proposes an intrinsic framework that lifts the target distribution to the underlying Lie group and performs Euclidean flow matching on the corresponding Lie algebra, thereby entirely circumventing the need for a predefined metric or geodesics. The approach requires only a Lie group action and a local section, eliminating any reliance on Riemannian geometric computations and substantially simplifying the generative modeling pipeline. Experiments demonstrate that this method yields more efficient, concise, and scalable generative models on homogeneous spaces while preserving intrinsic geometric fidelity.