Related papers: Metric Limits in Categories with a Flow

We study two problems related to flow equivalence of shift spaces. The first problem, the classification of $S$-gap shifts up to flow equivalence, is partially solved with the establishment of a new invariant for the sofic $S$-gap shifts…

This paper considers metric spaces where distances between a pair of nodes are represented by distance intervals. The goal is to study methods for the determination of hierarchical clusters, i.e., a family of nested partitions indexed by a…

Analysis and processing of data is a vital part of our modern society and requires vast amounts of computational resources. To reduce the computational burden, compressing and approximating data has become a central topic. We consider the…

We use weights on objects in an abelian category to define what we call a path metric. We introduce three special classes of weight: those compatible with short exact sequences; those induced by their path metric; and those which bound…

Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces,…

Flow matching has emerged as a powerful generative framework, with recent few-step methods achieving remarkable inference acceleration. However, we identify a critical yet overlooked limitation: these models suffer from severe diversity…

After an introductory chapter on the quantum supersymmetric string, in which particular attention will be devoted to the techniques via which phenomenologically viable models can be obtained from the ultraviolet microscopic degrees of…

Depth is a concept that measures the `centrality' of a point in a given data cloud or in a given probability distribution. Every depth defines a family of so-called trimmed regions. For statistical applications it is desirable that with…

Given a martingale sequence of random fields that satisfies a natural assumption of boundedness, it is shown that the pointwise limit of this sequence can be modified in such a way that a certain class of moduli of continuity is preserved.…

Information flow analysis is a powerful technique for reasoning about the sensitive information exposed by a program during its execution. While past work has proposed information theoretic metrics (e.g., Shannon entropy, min-entropy,…

Collections of measures on compact metric spaces form a model category ("data complexes"), whose morphisms are marginalization integrals. The fibrant objects in this category represent collections of measures in which there is a measure on…

Riehl and Verity have established that for a quasi-category $A$ that admits limits, and a homotopy coherent monad on $A$ which does not preserve limits, the Eilenberg-Moore object still admits limits; this can be interpreted as a…

Flow matching has recently emerged as a promising alternative to diffusion-based generative models, offering faster sampling and simpler training by learning continuous flows governed by ordinary differential equations. Despite growing…

In many physical situations involving diverse length scales, waves or rays representing them travel through media characterized by spatially smooth, random, modest refactive index variations. "Primary" diffraction (by individual…

The sequential nature of autoregressive next-token prediction imposes a fundamental speed limit on large language models. While continuous flow models offer a path to parallel generation, they traditionally demand expensive iterative…

In this paper, we introduce a class of new logarithmic curvature flow. The flows are designed to embrace the monotonicity of the related functional, and the convergence of this flow would tackle the solvability of the weighted…

The percolation threshold for flow or conduction through voids surrounding randomly placed spheres is rigorously calculated. With large scale Monte Carlo simulations, we give a rigorous continuum treatment to the geometry of the…

In this paper, we propose a new numerical scheme for a spatially discrete model of constrained total variation flows, which are total variation flows whose values are constrained in a Riemannian manifold. The difficulty of this problem is…

In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial…

Building on the notion of normed category as suggested by Lawvere, we introduce notions of Cauchy convergence and cocompleteness which differ from proposals in previous works. Key to our approach is to treat them consequentially as…