arXiv research

New volume comparison theorem for gradient Ricci almost solitons.

The paper proves gradient and comparison inequalities for RCD spaces.

Gradient estimates for special harmonic functions on manifolds.

Paper develops methods for estimating gradients of Finslerian Schrödinger equations.

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.

Sharp gradient estimates for positive Ricci curvature manifolds.

In this paper, we first obtain the sub-Laplacian comparison theorem in a complete noncompact pseudohermitian manifold of vanishing torsion (i.e. Sasakian manifold). Secondly, we derive the sub-gradient estimate for positive pseudoharmonic functions in a complete noncompact pseudohermitian manifold which satisfies the C…

Proves volume comparison and monotonicity for Bakry-Émery Ricci curvature.

Derives gradient estimate for a specific nonlinear parabolic equation on Finsler manifolds.

Paper extends knowledge gradient for preferential BO, overcoming computational challenges.

Selecting an optimizer is a central step in the contemporary deep learning pipeline. In this paper, we demonstrate the sensitivity of optimizer comparisons to the hyperparameter tuning protocol. Our findings suggest that the hyperparameter search space may be the single most important factor explaining the rankings obt…

Global gradient estimates for Fisher-KPP equation on Finsler metric measure spaces.

The paper studies volume and area comparisons in non-compact 3-manifolds with non-negative scalar curvature.

We consider a condition on the Ricci curvature involving vector fields, which is broader than the Bakry-Émery Ricci condition. Under this condition volume comparison, Laplacian comparison, isoperimetric inequality and gradient bounds are proven on the manifold. Specializing to the Bakry-Émery Ricci curvature condition,…

The paper explores inequalities between eigenvalues on Riemannian manifolds.

The paper proves geometric comparisons on metric measure spaces with integral Bakry-Émery Ricci tensor bounds.

The paper develops a method to estimate consumer preferences from observed rankings.

New Harnack inequality for heat equation on compact manifolds.

XGBoost is a scalable ensemble technique based on gradient boosting that has demonstrated to be a reliable and efficient machine learning challenge solver. This work proposes a practical analysis of how this novel technique works in terms of training speed, generalization performance and parameter setup. In addition, a…

We study the problem of supervised linear dimensionality reduction, taking an information-theoretic viewpoint. The linear projection matrix is designed by maximizing the mutual information between the projected signal and the class label (based on a Shannon entropy measure). By harnessing a recent theoretical result on…

Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.

In the lorentzian product $\Bbb G^n\times\Bbb R_1,$ we give a comparison between the $f$-volume of an entire $f$-maximal graph and the $f$-volume of the hyperbolic $H_r^+$ under the assumption that the gradient of the function defining the graph is bounded away from 1. As a consequence, we obtain a Bernstein type theor…

Understanding the flow of information in Deep Neural Networks (DNNs) is a challenging problem that has gain increasing attention over the last few years. While several methods have been proposed to explain network predictions, there have been only a few attempts to compare them from a theoretical perspective. What is m…

The paper proves Laplacian comparison theorems for modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.

The report studies ranking from pairwise comparisons in graphs, achieving optimal error bounds and proposing efficient algorithms.

The paper proves conditions for Einstein solitons to split into line and manifold.

Paper compares different models for time-to-event analysis.

Studied SGD convergence under weak conditions.

SGLRW improves robustness of stochastic gradient MCMC methods.

Develops comparison methods for semilinear elliptic problems on Riemannian manifolds with Ricci lower bound.

Introduces Alexandrov spaces with curvature below, covering various theorems.

GD outperforms ridge regression and SGD in linear regression problems.

This paper provides a block coordinate descent algorithm to solve unconstrained optimization problems. In our algorithm, computation of function values or gradients is not required. Instead, pairwise comparison of function values is used. Our algorithm consists of two steps; one is the direction estimate step and the o…

We study the geodesic equation for the Dirichlet (gradient) metric in the space of Kaehler potentials. We first solve the initial value problem for the geodesic equation of the combination metric, including the gradient metric. We then discuss a comparison theorem between it and the Calabi metric. As geometric motivati…

In this work and the supporting Part II, we examine the performance of stochastic sub-gradient learning strategies under weaker conditions than usually considered in the literature. The new conditions are shown to be automatically satisfied by several important cases of interest including SVM, LASSO, and Total-Variatio…

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

Identifies a gradient flow to solve kernel learning problems with noise reduction.

A new amortized variance-reduced gradient (AVRG) algorithm was developed in \cite{ying2017convergence}, which has constant storage requirement in comparison to SAGA and balanced gradient computations in comparison to SVRG. One key advantage of the AVRG strategy is its amenability to decentralized implementations. In th…

BSVGD improves sampling for multimodal distributions using branching.

Recent advances in stochastic gradient techniques have made it possible to estimate posterior distributions from large datasets via Markov Chain Monte Carlo (MCMC). However, when the target posterior is multimodal, mixing performance is often poor. This results in inadequate exploration of the posterior distribution. A…

We proved a matrix Li-Yau-Hamilton type gradient estimates for the positive solutin of the heat equation on complete Kaehler manifolds with nonnegative bisectional curvature. As a consequence we obtain a comparison theorem for the distance function under this curvature assumption.

In this short note, using Günther's volume comparison theorem and Yokota's gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton $(M^{n},g,f)$ with sectional curvature $K(g)<A$ and ${\rm Vol}_{f}(M)\geq v$ for some uniform constant $A,v$, there exists…

DBQPG improves policy gradient estimation with fewer samples.

In this paper, we analyze a real-valued reflected backward stochastic differential equation (RBSDE) with an unbounded obstacle and an unbounded terminal condition when its generator $f$ has quadratic growth in the $z$-variable. In particular, we obtain existence, comparison, and stability results, and consider the opti…

This paper provides lower bounds on the convergence rate of Derivative Free Optimization (DFO) with noisy function evaluations, exposing a fundamental and unavoidable gap between the performance of algorithms with access to gradients and those with access to only function evaluations. However, there are situations in w…

We propose a variance reduction framework for variational inference using the Multilevel Monte Carlo (MLMC) method. Our framework is built on reparameterized gradient estimators and "recycles" parameters obtained from past update history in optimization. In addition, our framework provides a new optimization algorithm …

MAGE optimizes policies using action gradients from model-based learning.

Rank aggregation based on pairwise comparisons over a set of items has a wide range of applications. Although considerable research has been devoted to the development of rank aggregation algorithms, one basic question is how to efficiently collect a large amount of high-quality pairwise comparisons for the ranking pur…