arXiv research

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

Paper analyzes solutions to quasilinear elliptic equations on manifolds using Nash-Moser iteration.

In this paper, we will establish an elliptic local Li-Yau gradient estimate for weak solutions of the heat equation on metric measure spaces with generalized Ricci curvature bounded from below. One of its main applications is a sharp gradient estimate for the logarithm of heat kernels. These results seem new even for s…

A new interpolation method speeds up neural ODE training.

We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with low…

The paper provides new gradient estimates for solutions to a nonlinear elliptic equation on smooth metric measure spaces.

AdaptOn achieves logarithmic regret in adaptive control of unknown partially observable linear systems.

Uniform-in-time analysis for Stein Variational Gradient Descent across various metrics.

The purpose of this work is to study some monotone functionals of the heat kernel on a complete Riemannian manifold with nonnegative Ricci curvature. In particular, we show that on these manifolds, the gradient estimate of Li and Yau, the gradient estimate of Ni, the monotonicity of the Perelman's entropy and the volum…

Paper proposes an $ε$-policy gradient for online pricing, reducing regret to $\mathcal{O}(\sqrt{T})$.

New loss function helps learn unstable dynamical systems.

New method for unbiased regression reduces excess risk.

We consider the problem of optimizing a high-dimensional convex function using stochastic zeroth-order queries. Under sparsity assumptions on the gradients or function values, we present two algorithms: a successive component/feature selection algorithm and a noisy mirror descent algorithm using Lasso gradient estimate…

An equivalent definition of entropic Ricci curvature on discrete spaces was given in terms of the global gradient estimate. With a particular choice of the density function $ρ$, we obtain a localized gradient estimate, which in turns allow us to derive a Bonnet-Myers type diameter bound for graphs with positive entropi…

We consider estimating a piecewise-constant image, or a gradient-sparse signal on a general graph, from noisy linear measurements. We propose and study an iterative algorithm to minimize a penalized least-squares objective, with a penalty given by the "l_0-norm" of the signal's discrete graph gradient. The method proce…

This paper focuses on projection-free methods for solving smooth Online Convex Optimization (OCO) problems. Existing projection-free methods either achieve suboptimal regret bounds or have high per-iteration computational costs. To fill this gap, two efficient projection-free online methods called ORGFW and MORGFW are …

In the first part of this paper, we prove local interior and boundary gradient estimates for p-harmonic functions on general Riemannian manifolds. With these estimates, following the strategy in recent work of R. Moser, we prove an existence theorem for weak solutions to the level set formulation of the 1/H (inverse me…

Paper proposes nested MLMC for SNPE with intractable likelihoods.

We propose a novel, efficient approach for distributed sparse learning in high-dimensions, where observations are randomly partitioned across machines. Computationally, at each round our method only requires the master machine to solve a shifted ell_1 regularized M-estimation problem, and other workers to compute the g…

Gradient flows for knot energies ensure long-term existence of knotted loops.

Policy gradient algorithm with variable learning rates achieves near-optimal performance in multi-arm bandit problems.

The paper improves sparse Gaussian processes by optimizing predictive loss.

REGS samples from unnormalized distributions using gradient flow and neural networks.

Adaptive gradient methods have become recently very popular, in particular as they have been shown to be useful in the training of deep neural networks. In this paper we have analyzed RMSProp, originally proposed for the training of deep neural networks, in the context of online convex optimization and show $\sqrt{T}$-…

We propose a nonconvex estimator for joint multivariate regression and precision matrix estimation in the high dimensional regime, under sparsity constraints. A gradient descent algorithm with hard thresholding is developed to solve the nonconvex estimator, and it attains a linear rate of convergence to the true regres…

Paper develops probabilistic bounds for a stochastic gradient algorithm in non-convex problems.

We consider the flows generated by generic gradients of Morse maps of a closed connected manifold $M$ to a circle. To each such flow we associate an invariant counting the closed orbits of the flow. Each closed orbit is counted with the weight derived from its index and homotopy class. The resulting invariant is called…

New method aligns diffusion models for inference-time properties without retraining.

We consider derivative-free algorithms for stochastic and non-stochastic convex optimization problems that use only function values rather than gradients. Focusing on non-asymptotic bounds on convergence rates, we show that if pairs of function values are available, algorithms for $d$-dimensional optimization that use …

This work shows linear convergence for two-layer neural networks in mean-field regime.

We introduce a new version of a curvature-dimension inequality for non-negative curvature. We use this inequality to prove a logarithmic Li-Yau inequality on finite graphs. To formulate this inequality, we introduce a non-linear variant of the calculus of Bakry and Émery. In the case of manifolds, the new calculus and …

Study inequalities on hyperbolic spaces and Riemannian manifolds using symmetrization and heat semigroup.

In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves…

We study inequalities related to the heat kernel for the hypoelliptic sublaplacian on an H-type Lie group. Specifically, we obtain precise pointwise upper and lower bounds on the heat kernel function itself. We then apply these bounds to derive an estimate on the gradient of solutions of the heat equation, which is kno…

LMC algorithm converges to target in Chi-squared and Renyi divergence.

Novel algorithm reduces privacy noise in machine learning.

We establish the longtime existence and convergence results of the mean curvature flow of entire Lagrangian graphs in Pseudo-Euclidean space which is related to Logarithmic gradient flow.

Stochastic gradient descent based algorithms are typically used as the general optimization tools for most deep learning models. A Restricted Boltzmann Machine (RBM) is a probabilistic generative model that can be stacked to construct deep architectures. For RBM with Bernoulli inputs, non-Euclidean algorithm such as st…

Researchers propose a non-monotone quantum natural gradient for quantum systems.

AIHT improves online high-dimensional quantile regression by separating support discovery and refinement.

The computational cost of training with softmax cross entropy loss grows linearly with the number of classes. For the settings where a large number of classes are involved, a common method to speed up training is to sample a subset of classes and utilize an estimate of the loss gradient based on these classes, known as…

Sharp upper diameter limit found for Ricci solitons.

E-LDA offers faster, interpretable LDA topic models.

In this paper, we propose and analyze zeroth-order stochastic approximation algorithms for nonconvex and convex optimization, with a focus on addressing constrained optimization, high-dimensional setting and saddle-point avoiding. To handle constrained optimization, we first propose generalizations of the conditional g…

This work improves policy evaluation and selection using logarithmic smoothing for pessimistic off-policy estimation.

The study analyzes the performance of statistical estimators under stability and computational efficiency.

Logarithmic regret achieved in continuous-time linear-quadratic reinforcement learning.

The paper proves various inequalities on gradient shrinking Ricci solitons.