arXiv research

A new PGA algorithm ensures stable, robust, and noise-immune solutions for non-negative inverse problems.

New method finds optimal learning rates for neural nets.

Proposes a new method for posterior sampling using MMD with negative distance kernel.

A new method using negative-shifted gradient descent improves overparameterized linear regression by avoiding structural limitations of negative ridge endpoints.

New gradient flows for non-negative and probability measures combining optimal transport and interaction forces.

Morse theory connects low energy submanifolds in 3-sphere.

Paper proposes an adversarial sampling method for efficient extreme classification.

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…

Study on gradient steady Kähler Ricci solitons with specific curvature properties.

The study shows ends of shrinking gradient $ρ$-Einstein solitons are non-parabolic.

New kernel improves MMDs with theoretical guarantees for gradient flows.

GOLS finds activation functions affect training robustness, especially ReLU.

Paper extends Aronson-Bénilan estimates for porous medium equations on manifolds with negative curvature.

In this paper, we study the gradient estimates of Li-Yau-Hamilton type for positive solutions to both drifting heat equation and the simple nonlinear heat equation problem $u_t-Δu=au\log u, \ \ u>0$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative (Bakry-Emery)-Ricci curvature. Here…

We improve the well known local gradient estimate of Cheng and Yau in the case when Ricci curvature has a negative lower bound.

Gradient flow studies Spin(7)-structures on compact 8-manifolds.

DCGD improves training of PINNs by adjusting gradients to avoid negative inner products.

A1GM method improves efficiency in reconstructing missing data using KL divergence.

Although the word-popularity based negative sampler has shown superb performance in the skip-gram model, the theoretical motivation behind oversampling popular (non-observed) words as negative samples is still not well understood. In this paper, we start from an investigation of the gradient vanishing issue in the skip…

The study proves triviality and rigidity results for Ricci solitons and estimates their conjugate radius.

Paper proves equivalences in portfolio optimization with new risk measures.

New degree theory proves existence of solitons on 4D manifolds.

In this paper, we study the gradient estimate for positive solutions to the following nonlinear heat equation problem $u_t-Δu=au\log u+Vu, \ \ u>0$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative Ricci curvature. Here $a\leq 0$ is a constant, $V$ is a smooth function on $M$ with $-…

A new approach improves numerical tabular data imputation by addressing diffusion models' limitations.

Advances geometric structure flows, proving short-time existence and uniqueness for various flows.

Games generalize the single-objective optimization paradigm by introducing different objective functions for different players. Differentiable games often proceed by simultaneous or alternating gradient updates. In machine learning, games are gaining new importance through formulations like generative adversarial netwo…

DG improves policy gradients by weighting actions with a sigmoid of advantage and surprisal.

Study growth rates of harmonic functions on curved surfaces.

In this note, we complete the classification of the geometry of non-compact two-dimensional gradient Ricci solitons. As a consequence, we obtain two corollaries: First, a complete two-dimensional gradient Ricci soliton has bounded curvature. Second, we give examples of complete two-dimensional expanding Ricci solitons …

Sharp Liouville theorem for minimal graphs on manifolds with nonnegative Ricci curvature.

Constructs new steady gradient Ricci solitons for higher dimensions.

In this paper, we prove that the $L^p$ essential spectra of the Laplacian on functions are $[0,+\infty)$ on a non-compact complete Riemannian manifold with non-negative Ricci curvature at infinity. The similar method applies to gradient shrinking Ricci soliton, which is similar to non-compact manifold with non-negative…

Accelerated gradient (AG) methods are breakthroughs in convex optimization, improving the convergence rate of the gradient descent method for optimization with smooth functions. However, the analysis of AG methods for non-convex optimization is still limited. It remains an open question whether AG methods from convex o…

Non-negative matrix factorization is a basic tool for decomposing data into the feature and weight matrices under non-negativity constraints, and in practice is often solved in the alternating minimization framework. However, it is unclear whether such algorithms can recover the ground-truth feature matrix when the wei…

By extending Koiso's examples to the non-compact case, we construct complete gradient Kahler-Ricci solitons of various types on certain holomorphic line bundles over compact Kahler-Einstein manifolds. Moreover, a uniformization result on steady gradient Kahler-Ricci solitons with non-negative Ricci curvature is obtaine…

Eigenfunction gradients on curved spaces imply rigid structure.

In this paper, we prove that complete gradient steady Kähler-Ricci solitons with harmonic Bochner tensor are necessarily Kähler-Ricci flat, i.e., Calabi-Yau, and that complete gradient shrinking (or expanding) Kähler-Ricci solitons with harmonic Bochner tensor must be isometric to a quotient of $N^k\times \mathbb{C}^{n…

This article shows that if the negative part of Ricci curvature lies in the Kato class, the heat kernel satisfies a Li-Yau type estimate. Additionally, using the resulting heat kernel bound, we show that the obtained heat kernel estimate leads to bounds on the first Betti number only depending on the Kato constant.

Characterizes gradient Yamabe solitons with specific conditions.

In this paper we consider the Martin compactification, associated with the operator $\mathcal{L} = Δ-1$, of a complete non-compact surface $(Σ^2, ds^2)$ with negative curvature. In particular, we investigate positive eigenfunctions with eigenvalue one of the Laplace operator $Δ$ of $(Σ^2, ds^2)$ and prove a uniqueness …

Minimal graphs grow slowly on curved spaces, proving constant solutions.

Every connected, weighted graph with non-negative curvature has exactly two ends.

Corrected CBOW performs similarly to Skip-gram.

We prove gradient estimates for hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1},$ expanding by negative powers of a certain class of homogeneous curvature functions. We obtain optimal gradient estimates for hypersurfaces evolving by certain powers $p>1$ of $F^{-1}$ and smooth convergence of the properly rescale…

In this paper, we prove some classification results for four-dimensional gradient Ricci solitons. For a four-dimensional gradient shrinking Ricci soliton with $div^4Rm^\pm=0$, we show that it is either Einstein or a finite quotient of $\mathbb{R}^4$, $\mathbb{S}^2\times\mathbb{R}^2$ or $\mathbb{S}^3\times\mathbb{R}$. T…

Stochastic gradient Langevin dynamics (SGLD) is a computationally efficient sampler for Bayesian posterior inference given a large scale dataset. Although SGLD is designed for unbounded random variables, many practical models incorporate variables with boundaries such as non-negative ones or those in a finite interval.…

GBHT uses gradient boosting for density estimation with theoretical guarantees.

Sharp gradient estimates for positive Ricci curvature manifolds.