arXiv research

Improved Local SGD convergence for general convex objectives with bounded second-order heterogeneity.

Local asymptotic minimax risk bounds in a locally asymptotically mixture of normal family of distributions have been investigated under asymmetric loss functions and the asymptotic distribution of the optimal estimator that attains the bound has been obtained.

The paper proves ACC for local volumes under boundedness conditions.

Proves convergence of gradient Ricci shrinkers with uniform bounds.

Proves boundedness of log Fano cone singularities with bounded local volumes.

Study groups acting on trees with specific local actions, proving cohomology vanishing or infinite.

We prove a pseudolocality type theorem for compact Ricci Flow under local integral bounds of curvature. The main tool is Local Ricci Flow introduced by Deane Yang in [4] and Pseudolocality Theorem of Perelman in [3]. We also study L^p bounds for the derivatives of curvature and smooth extension of Local Ricci Flow.

Optimal lower bounds for eigenvalues of Dirac-Witten operator on certain submanifolds.

Local Hardy spaces defined for Riemannian manifolds with bounded geometry.

Optimizes bounds for threefold singularity volumes.

Study shows boundedness of klt singularities in 3D or with bounded Kollár components.

This paper provides a general result on controlling local Rademacher complexities, which captures in an elegant form to relate the complexities with constraint on the expected norm to the corresponding ones with constraint on the empirical norm. This result is convenient to apply in real applications and could yield re…

Enhances Ricci flow theorem with scalar curvature bound.

Homological algebra used to study local equivalence of complex rings.

We give lower bounds for the first Dirichilet eigenvalues for domains in submanifolds with locally bounded mean curvatures. These bounds depend on the injectivity radius, sectional curvature (upperbound) of the ambient space and on the mean curvature of the submanifold. For submanifolds fo Hadamard manifolds these lowe…

The thesis clarifies when local updates outperform centralized methods in heterogeneous data environments.

This paper fills in local bounds for Spearman's footrule and Gini's gamma measures of association.

A local cut point is by definition a point that disconnectes its sufficiently small neighborhood. We show that there exists an upper bound for the degree of a local cut point in a metric measure space satisfying the generalized Bishop--Gromov inequality. As a corollary, we obtain an upper bound for the number of ends o…

We prove a general connection between the communication complexity of two-player games and the sample complexity of their multi-player locally private analogues. We use this connection to prove sample complexity lower bounds for locally differentially private protocols as straightforward corollaries of results from com…

This article shows that under locally uniformly integral bounds of the negative part of Ricci curvature the heat kernel admits a Gaussian upper bound for small times. This provides general assumptions on the geometry of a manifold such that certain function spaces are in the Kato class. Additionally, the results imply …

We obtain a local volume growth for complete, noncompact Riemannian manifolds with small integral bounds and with Bach tensor having finite $L^2$ norm in dimension 4.

This paper achieves optimal regret bounds for locally private linear contextual bandit.

The paper calculates bounds on the local Lipschitz constants of neural network layers.

The study extends convergence theorems for Ricci-limit spaces with bounded curvature.

Study of mapping class groups on infinite graphs, focusing on their large-scale geometry.

In this work, we (partially) generalize two classical tools in study of collapsed manifolds with bounded sectional curvature: a (singular) fibration theorem by Fukaya (1987) and Cheeger-Fukaya-Gromov (1992), and the stability for isometric compact Lie group actions on manifolds by Palais (1961) and Grove-Karcher (1973)…

We propose a new localized inference algorithm for answering marginalization queries in large graphical models with the correlation decay property. Given a query variable and a large graphical model, we define a much smaller model in a local region around the query variable in the target model so that the marginal dist…

Locally private mechanisms' output divergence bounds derived.

We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the loca…

We study a basic private estimation problem: each of $n$ users draws a single i.i.d. sample from an unknown Gaussian distribution, and the goal is to estimate the mean of this Gaussian distribution while satisfying local differential privacy for each user. Informally, local differential privacy requires that each data …

Study shows local topologies of certain geometric spaces.

Ricci flow smooths locally collapsing manifolds with controlled curvature.

Motivated by Perelman's Pseudo Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost-euclidean isoperimetric inequality. The result is actu…

We give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in $N \times \mathbb{R}$, where $N$ is an $n$-dimensional complete Riemannian manifold with radial sectional curvature $K_{N} \leq κ$. When the immersion is minimal our estimates are sharp. We also show that …

We prove pointwise bounds for $L^2$ eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with $\mathbb{Q}$-rank one if the corresponding eigenvalues lie below the continuous part of the $L^2$ spectrum. Furthermore, we use these bounds in order to obtain some results concerning the $L^p$ spectrum.

Local knots can't bound smaller surfaces in rational homology 3-spheres.

The study tightens risk bounds for mixtures of experts using local differential privacy.

We derive an upper bound on the local Rademacher complexity of $\ell_p$-norm multiple kernel learning, which yields a tighter excess risk bound than global approaches. Previous local approaches aimed at analyzed the case $p=1$ only while our analysis covers all cases $1\leq p\leq\infty$, assuming the different feature …

New SGD covering technique yields dimension-independent generalization bounds.

Study finds lower bounds on flat cycles in congruence covers of symmetric spaces.

New method certifies global robustness of neural networks efficiently.

The study bounds Hausdorff measure of flat singular points in area-minimizing currents.

Paper tightens statistical aggregation results using local complexity.

New study shows deep networks generalize well due to loss surface geometry.

A new algorithm estimates mean under varying user data sizes with local differential privacy.

Characterizes orbifolds with upper curvature bounds as reflectofolds.

Relative cup-length defined for non-Morse functions on manifolds.

This note is a continuation of the author's paper \cite{Li}. We prove that if the metric $g$ of a 4-manifold has bounded Ricci curvature and the curvature has no local concentration everywhere, then it can be smoothed to a metric with bounded sectional curvature. Here we don't assume the bound for local Sobolev constan…