arXiv research

Our goal in this paper is to develop an effective estimator of fractal dimension. We survey existing ideas in dimension estimation, with a focus on the currently popular method of Grassberger and Procaccia for the estimation of correlation dimension. There are two major difficulties in estimation based on this method. …

Study bounds derivatives of solutions to a specific equation on domains.

Estimates mean curvature flow with geometric bounds.

In this note, a gradient estimate for the complex Monge-Ampere equation is established. It differs from previous estimates of Yau, Hanani, Blocki, P. Guan, B. Guan - Q. Li in that it is pointwise, and depends only on the infimum of the solution instead of its $C^0$ norm.

We establish fundamental results for a parabolic flow of Riemannian metrics introduced by Bahuaud-Helliwell in arXiv:1010:4287v1 which is based on the Fefferman-Graham ambient obstruction tensor. First, we obtain local $L^2$ smoothing estimates for the curvature tensor and use them to prove pointwise smoothing estimate…

Paper describes profiles of multivariate normal distributions and novel estimators for mutual information.

Estimates interactions between modalities for multimodal data.

We propose a new model for supervised learning to rank. In our model, the relevance labels are assumed to follow a categorical distribution whose probabilities are constructed based on a scoring function. We optimize the training objective with respect to the multivariate categorical variables with an unbiased and low-…

Study dispersive estimates for Schrödinger and wave equations on a cone with specific metric.

We provide a pointwise confidence bound for non-linear least-squares with fixed design.

Paper studies optimal federated learning for nonparametric regression with privacy constraints.

This paper introduces a new clustering technique, called {\em dimensional clustering}, which clusters each data point by its latent {\em pointwise dimension}, which is a measure of the dimensionality of the data set local to that point. Pointwise dimension is invariant under a broad class of transformations. As a resul…

The paper analyzes uncertainty quantification in sparse Gaussian process regression with a Brownian motion prior.

In recent years, kernel density estimation has been exploited by computer scientists to model machine learning problems. The kernel density estimation based approaches are of interest due to the low time complexity of either O(n) or O(n*log(n)) for constructing a classifier, where n is the number of sampling instances.…

We establish dimension-independent estimates related to heat operators e^{tL} on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply…

New Liouville-type results for CR Yamabe equation in Heisenberg group.

Estimating machine learning performance 'in the wild' is both an important and unsolved problem. In this paper, we seek to examine, understand, and predict the pointwise competence of classification models. Our contributions are twofold: First, we establish a statistically rigorous definition of competence that general…

We refine a metric bunching estimate for pinched manifolds.

Recently, B.-Y. Chen and O. J. Garay studied pointwise slant submanifolds of almost Hermitian manifolds. By using the notion of pointwise slant submanifolds, we investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian and cosymplectic manifolds. We prove that there exist no pr…

Study shows how discrete graph curvature relates to manifold curvature.

Improved spectral projection estimates on manifolds of non-positive curvature.

The purpose of this paper is to study pointwise pseudo-slant warped product submanifolds of a Kähler manifold $\widetilde{M}$. We derive the conditions of integrability and totally geodesic foliation for the distributions allied to the characterization of a pointwise pseudo-slant submanifold of $\widetilde{M}$. The nec…

We introduce the notions of pointwise almost h-slant submanifolds and pointwise almost h-semi-slant submanifolds as a generalization of slant submanifolds, pointwise slant submanifolds, semi-slant submanifolds, and pointwise semi-slant submanifolds. We have characterizations and investigate the integrability of distrib…

A debiasing method improves nonparametric regression's statistical properties.

As a generalization of slant submanifolds and semi-slant submanifolds, we introduce the notions of pointwise slant submanifolds and pointwise semi-slant sunmanifolds of an almost contact metric manifold. We obtain a characterization at each notion, investigate the topological properties of pointwise slant submanifolds,…

Study properties of pointwise k-slant submanifolds in Kähler manifolds.

Paper analyzes faster convergence rates for reinforcement learning from offline data.

Estimates heat kernel gradients on fractal-like cable systems.

Paper proposes Pcomp classification for binary classification with pairwise confidence comparisons.

The paper challenges the use of decision trees for pointwise inference due to slow convergence rates.

Develops asymptotic theory for deep Cox models to enable valid inference.

Let (M, g) be a compact smooth Riemannian manifold. We obtain new off-diagonal estimates as λ tend to infinity for the remainder in the pointwise Weyl Law for the kernel of the spectral projector of the Laplacian onto functions with frequency at most λ. A corollary is that, when rescaled around a non self-focal point, …

We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions. To the best of our knowledge, this is the first such estimates without assuming smallness of first derivatives of the defining map. An immediate application is a convergence theorem of the mean curvature flow of the graph…

We study pointwise and $L^p$ gradient estimates of the heat kernel, on manifolds that may have some amount of negative Ricci curvature, provided it is not too negative (in an integral sense) at infinity. We also prove uniform boundedness results on $L^p$ spaces for the heat operator of the Hodge Laplacian on differenti…

The paper studies a new type of submanifolds in product spaces.

Sharp gradient estimates extended to surfaces with lower Ricci curvature.

Estimate arrival times in random recursive trees using iterated Jordan centralities.

We show that harmonic spinors obey a strengthened version of the well-known pointwise Kato inequality for sections of a vector bundle with a connection. We then prove a decay estimate for eigenspinors using this Kato-Yau estimate and resulting differential inequality. We briefly describe some applications to gauge theo…

Proves Hessian estimates for special Lagrangian equation with new proofs.

We consider the problem of nonparametric regression under shape constraints. The main examples include isotonic regression (with respect to any partial order), unimodal/convex regression, additive shape-restricted regression, and constrained single index model. We review some of the theoretical properties of the least …

Introduces $k$-slant distributions and submanifolds in various geometric settings.

Study on slant submanifolds with new conditions and transitivity.

New Hessian estimates for heat equations on manifolds.

In this paper, we study rotational surfaces of elliptic, hyperbolic and parabolic type with pointwise 1-type Gauss map which have spacelike profile curve in four dimensional pseudo Euclidean space E4-2 and obtain some characterizations for these rotational surfaces to have pointwise 1-type Gauss map.

We study a class of Riemannian manifolds with respect to the covariant derivative of their curvature tensors. We introduce geometrically the class of directed Riemannian manifolds of pointwise constant relative sectional curvature and give a tensor characterization for such manifolds. We prove that all rotational hyper…

In this paper, we proved a compactness result about Riemannian manifolds with an arbitrary pointwisely pinched Ricci curvature tensor.

The paper studies geometric properties of a specific type of submanifolds in Kaehler manifolds.

The paper establishes lower bounds on injectivity radius and constructs metrics with bounded geometry.