A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Unique metric found for discrete curvature on spherical cone-metrics.
A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying …
We define Discrete Quasi-Einstein metrics (DQE-metrics) as the critical points of discrete total curvature functional on triangulated 3-manifolds. We study DQE-metrics by introducing some combinatorial curvature flows. We prove that these flows produce solutions which converge to discrete quasi-Einstein metrics when th…
The paper proves a theorem for discretizing Gaussian curvature on surfaces.
A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature me…
We introduce the discrete Einstein metrics as critical points of discrete energy on triangulated 3-manifolds, and study them by discrete curvature flow of second (fourth) order. We also study the convergence of the discrete curvature flow. Discrete curvature flow of second order is an analogue of smooth Ricci flow.
The paper introduces a new discretization of Gaussian curvature on surfaces.
In this short note, we prove that the space of all admissible piecewise linear metrics parameterized by length square on a triangulated manifolds is a convex cone. We further study Regge's Einstein-Hilbert action and give a much more reasonable definition of discrete Einstein metric than our former version in \cite{G}.…
The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form of a Laplacian on …
The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.
The paper studies singularities in discrete indefinite affine minimal surfaces.
In this paper, we introduce a parameterized discrete curvature (-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is established for the parameterized discrete curvature, which generalizes the discrete uni…
The paper proposes a new way to approximate Riemannian metrics using discrete wall systems.
Existence and uniqueness of discrete Einstein metrics on trees proven.
Study discrete analog of zeta-determinant maximization on triangulated surfaces.
New interpretation of discrete conformality using polyhedral convex hulls.
The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.
Rigidity theorem for discrete metric spaces embedded in Riemannian surfaces.
Inspired by the Weierstrass representation of smooth affine minimal surfaces with indefinite metric, we propose a constructive process producing a large class of discrete surfaces that we call discrete affine minimal surfaces. We show that they are critical points of an affine area functional defined on the space of qu…
New metrics on curve spaces improve shape analysis.
This paper completes the classification of discrete conformal structures on surfaces.
Study on Brownian motion on discrete curve spaces, proving stochastic completeness.
The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
We prove an inequality that generalizes the Fan-Taussky-Todd discrete analog of the Wirtinger inequality. It is equivalent to an estimate on the spectral gap of a weighted discrete Laplacian on the circle. The proof uses a geometric construction related to the discrete isoperimetric problem on the surface of a cone. In…
This study introduces balanced DRPS and OrderedLogitNN for better QDE of discrete-level questions.
Study on deforming discrete conformal structures on surfaces with boundaries.
New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.
StochasticRank optimizes ranking metrics efficiently and guarantees global convergence.
This work revisits, from a geometric perspective, the notion of discrete connection on a principal bundle, introduced by M. Leok, J. Marsden and A. Weinstein. It provides precise definitions of discrete connection, discrete connection form and discrete horizontal lift and studies some of their basic properties and rela…
New algorithm estimates intrinsic dimension of discrete datasets.
We prove that the isoperimetric inequalities in the euclidean and hyperbolic plane hold for all euclidean, respectively hyperbolic, cone-metrics on a disk with singularities of negative curvature. This is a discrete analog of the theorems of Weil and Bol that deal with Riemannian metrics of curvature bounded from above…
Derives Mirror Descent from gradient flow on a Riemannian manifold.
Study shows convergence of anticanonically balanced metrics to Kähler-Einstein metrics on Fano manifolds.
Neural networks learn discrete tasks on continuous data via emergent geometry.
New method learns discrete graph diffusion via free-energy gradient flows.
Positive-curvature metrics on trees identified for specific configurations.
We analyze convergence of Fermat distances and their application in clustering.
In \cite{Luo0}, Feng Luo conjectured that the discrete Yamabe flow will converge to the constant curvature PL-metric after finite number of surgeries on the triangulation. In this paper, we prove that the flow can always be extended (without surgeries) to a solution that converges exponentially fast to the constant cur…
Contractible Vietoris-Rips complexes for integer n proved using discrete Morse theory.
In this paper, we generalize our results in \cite{GX3} to triangulated surfaces in hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gaussian curvature by dividing the classical discrete Gauss curvature by an area element, which …
New metrics produce discrete zero sets for nondegenerate harmonic forms.
The main result of this paper is a discrete Lawson correspondence between discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. This is a correspondence between two discrete isothermic surfaces. We show that this correspondence is an isometry in the following sense: it preserves the metric coefficients int…
Score based learning (SBL) is a promising approach for learning Bayesian networks in the discrete domain. However, when employing SBL in the continuous domain, one is either forced to move the problem to the discrete domain or use metrics such as BIC/AIC, and these approaches are often lacking. Discretization can have …
We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric…
Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean curvature, normal and co-normal vector fields and cubic form, and the…
The paper proves the existence of a complete holomorphic vector field on a complex manifold with a Kähler-Einstein metric.
Let K be an irreducible and reversible Markov kernel on a finite set X. We construct a metric W on the set of probability measures on X and show that with respect to this metric, the law of the continuous time Markov chain evolves as the gradient flow of the entropy. This result is a discrete counterpart of the Wassers…
This paper develops efficient bounds on the Wasserstein metric for discrete measures.