arXiv research

The Thurston spine's properties are studied in relation to Morse-Smale complexes.

New method improves submodular maximization for machine learning applications.

Machine learning predicts liquid water properties from cluster data.

The paper studies sprays on Hamel-Funk functions and their properties.

Paper defines Farey Recursive Functions and explores their properties.

In this article, existence results concerning temporal functions with additional properties on a globally hyperbolic manifold are obtained. These properties are certain bounds on geometric quantities as lapse and shift. The results are linked to completeness properties and the existence of closed isometric embeddings i…

Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit …

In the paper, the author studies properties of three functions relating to the exponential function and the existence of partitions of unity, including accurate and explicit computation of their derivatives, analyticity, complete monotonicity, logarithmically complete monotonicity, absolute monotonicity, and the like.

The article studies critical points of a new energy functional in higher dimensions.

Study uses property elicitation to understand how fairness regularizers affect optimal decisions.

Establishes relationships between prudence and stability properties of risk functionals.

Paper explores RKHS properties for derivative and integral operators.

The staircase property aids deep learning by guiding hierarchical feature learning.

The stochastic gradient descent has been widely used for solving composite optimization problems in big data analyses. Many algorithms and convergence properties have been developed. The composite functions were convex primarily and gradually nonconvex composite functions have been adopted to obtain more desirable prop…

We study here the action of subgroups of PSL(2,R) on the space of harmonic functions on the unit disc bounded by a common constant, as well as the relationship this action has with the foliated Liouville problem: Given a foliation of a compact manifold by Riemannian leaves and a leafwise harmonic continuous function on…

We introduce a geometrically transparent strict saddle property for nonsmooth functions. This property guarantees that simple proximal algorithms on weakly convex problems converge only to local minimizers, when randomly initialized. We argue that the strict saddle property may be a realistic assumption in applications…

Study shows infoGAN's generalization error bound for two-layer networks.

We give a proof of the Donnelly-Fefferman growth bound of Laplace-Beltrami eigenfunctions which is probably the easiest and the most elementary one. Our proof also gives new quantitative geometric estimates in terms of curvature bounds which improve and simplify previous work by Garofalo and Lin. The proof is based on …

In this paper, we study volume growth, Liouville theorem and the local gradient estimate for $f$-harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral properties of f-harmonic functions and harmonic functions on such manifolds.

The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous…

Study expands classical harmonic function results to Riemannian manifolds.

The study finds monotonic properties of harmonic functions on 3-manifolds with a flat end.

In this short note, we study the asymptotic property of Huisken's functional for mean curvature flow on the minimal submanifolds of Euclidean space. We prove that the limit of Huisken's functional equals to the extrinsic asymptotic volume ratio on the minimal submanifold of Euclidean space.

Introduces Hurewicz fibrations for embedding maps of orbifold charts.

The paper studies continuous submodular functions and their optimization.

The authors define a class of functions on Riemannian manifolds, which is called geodesic semilocal E-preinvex functions, as a generalization of geodesic semilocal E-convex and geodesic semi E-preinvex functions and some of its properties are established. Furthermore, a nonlinear fractional multiobjective programming i…

The paper proves rigidity of certain Dirac operators using theta functions.

Regularizers change the geometric properties of loss functions in neural networks.

The paper extends sequences while preserving statistical properties using a mixture model.

This research extends quasiplurisubharmonic functions on compact Kähler manifolds.

The paper examines functional properties on manifolds with very negative curvature.

This note establishes several integral identities relating certain metric properties of level hypersurfaces of Morse functions.

We show that a subspace $S$ of the space of real analytical functions on a manifold that satisfies certain regularity properties is contained in the set of solutions of a linear elliptic differential equation. The regularity properties are that $S$ is closed in $L^2(M)$ and that if a sequence of functions $f_n$ in $S$ …

UDENet and ResNet can approximate any function, with ODENet showing UAP for continuous functions.

SONATA algorithm converges to solutions of nonconvex smooth functions with KL property.

New probabilistic method constructs Kähler-Einstein metrics and suggests zero-free properties of zeta functions.

Researchers examine global properties of a scalar curvature functional to solve the prescribed Ricci curvature problem.

A property, or statistical functional, is said to be elicitable if it minimizes expected loss for some loss function. The study of which properties are elicitable sheds light on the capabilities and limitations of point estimation and empirical risk minimization. While recent work asks which properties are elicitable, …

Study of harmonic functions on infinite penny graphs.

Introduces Floer functions and Floerfolds for intrinsic properties.

The paper examines how isoparametric foliations affect the Pompeiu property in compact Riemannian manifolds.

We explore the efficacy of using a novel activation function in Artificial Neural Networks (ANN) in characterizing exoplanets into different classes. We call this Saha-Bora Activation Function (SBAF) as the motivation is derived from long standing understanding of using advanced calculus in modeling habitability score …

We prove a Liouville property for any $f$-harmonic function with polynomial growth on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ when the Bakry-Émery Ricci curvature is nonnegative and its diameter of geodesic sphere has sublinear growth.

The self-concordant-like property of a smooth convex function is a new analytical structure that generalizes the self-concordant notion. While a wide variety of important applications feature the self-concordant-like property, this concept has heretofore remained unexploited in convex optimization. To this end, we deve…

The paper investigates how activation functions impact the training of Neural ODEs, leading to global convergence.

Cubic-regularized Newton's method (CR) is a popular algorithm that guarantees to produce a second-order stationary solution for solving nonconvex optimization problems. However, existing understandings of the convergence rate of CR are conditioned on special types of geometrical properties of the objective function. In…

Proves regularity of extremal function on compact Kähler manifolds.

Harder's reduction theory provides filtrations of euclidean buildings that allow one to deduce cohomological and homological properties of S-arithmetic groups over global function fields. In this survey I will sketch the main points of Harder's reduction theory starting from Weil's geometry of numbers and the Riemann-R…