arXiv research

Uniform bounds for eigenvalues of Hodge Laplacian on manifolds with lower Ricci curvature.

A new method achieves optimal uniformity in designs with minimal flexibility.

Uniform bounds for Green's function on Kähler manifolds derived from complex Monge-Ampère equations.

Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.

We derive a uniform bound for the total betti number of a closed manifold in terms of a Ricci curvature lower bound, a conjugate radius lower bound and a diameter upper bound. The result is based on an angle version of Toponogov comparison estimate for small triangles in a complete manifold with a Ricci curvature lower…

Improved uniform convergence bound with fat-shattering dimension reduces sample complexity gap.

We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of compact manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then $Y$ has a universal cover. We then show that, for $i$ sufficiently large, the fundamental group of $M_i$ has a surjective homeomorphis…

We design and mathematically analyze sampling-based algorithms for regularized loss minimization problems that are implementable in popular computational models for large data, in which the access to the data is restricted in some way. Our main result is that if the regularizer's effect does not become negligible as th…

The paper achieves nearly optimal regret bounds for contextual multinomial logit bandits.

In this paper we describe the topology of 4-dimensional closed orientable Riemannian manifolds with a uniform lower bound of sectional curvature and with a uniform upper bound of diameter which collapse to metric spaces of lower dimensions. This enables us to understand the set of homeomorphism classes of closed orient…

In this paper we determine the topology of three-dimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.

We investigate regularization of riemannian metrics by mollification. Assuming both-sided bounds on the Ricci tensor and a lower injectivity radius bound we obtain a uniform estimate on the change of the sectional curvature. Actually, our result holds for any metric with a uniform bound on the $W^{2,p}$-harmonic radius…

Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the stability operator. When the latter assumption is replaced by a uniform lower bound on …

We prove some criteria for uniform K-stability of log Fano pairs. In particular, we show that uniform K-stability is equivalent to $β$-invariant having a positive lower bound. Then we study the relation between optimal destabilization conjecture and the conjectural equivalence between uniform K-stability and K-stabilit…

We study the equation $Δ_g u -\frac{n-2}{4(n-1)}R(g)u+Ku^p=0 (1+ζ\leq p \leq \frac{n+2}{n-2})$ on locally conformally flat compact manifolds $(M^n,g)$. We prove the following: (i) When the scalar curvature $R(g)>0$ and the dimension $n \geq 4$, under suitable conditions on $K$, all positive solutions $u$ have uniform u…

The paper provides a uniform lower bound for intersection numbers of psi-classes on moduli spaces.

We prove the existence of positive lower bounds on the Cheeger constants of manifolds of the form $X/Γ$ where $X$ is a contractible Riemannian manifold and $Γ<\Isom(X)$ is a discrete subgroup, typically with infinite co-volume. The existence depends on the $L^2$-Betti numbers of $Γ$, its subgroups and of a uniform latt…

No algorithm outperforms uniform sampling in A/B testing.

For sequences of warped product metrics on a $3$-torus satisfying the scalar curvature bound $R_j \geq -\frac{1}{j}$, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger Intrin…

Uniform proof for Ricci flows on complete manifolds.

Uniformly positive scalar curvature implies a lower bound on injectivity radius.

Surveying Ricci flow for weak lower scalar curvature bounds.

In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for the associated quasiconformal homogeneity constants across all closed hyperbolic…

We provide a lower bound for the uniform exponential growth rate of closed nonflat nonpositively curved 3-manifold groups. A detailed study of the uniform exponential growth rate of closed 3-manifold groups is also presented.

We prove a uniform version of the Tits alternative. As a consequence, we obtain uniform lower bounds for the Cheeger constant of Cayley grahs of finitely generated non virtually solvable linear groups in arbitrary characteristic. Also we show that the algebraic entropy of discrete subgroups of a given Lie group is unif…

Sharp lower bound found for geodesic ball eigenvalues.

New algorithm for learning mixtures with mostly uniform weights, improving on previous bounds.

Study on signatures of positive braids with bounds derived.

Extends Ricci flow theory to Kato-type curvature bounds, proving manifold properties.

We present Rotated Adaptive Tetra-iterated Quantizer (RATQ), a fixed-length quantizer for gradients in first order stochastic optimization. RATQ is easy to implement and involves only a Hadamard transform computation and adaptive uniform quantization with appropriately chosen dynamic ranges. For noisy gradients with al…

Quantitative Sobolev extensions lead to Neumann heat kernel bounds.

Example surfaces with curvature bounds but no Laplacian eigenvalue lower bound.

In this note, we prove a uniform distance distortion estimate for Ricci flows with uniformly bounded scalar curvature, independent of the lower bound of the initial $μ$-entropy. Our basic principle tells that once correctly renormalized, the metric-measure quantities obey similar estimates as in the non-collapsing case…

Classifies 2-uniform maps on torus with formulas and asymptotic bounds.

New stability bounds for SGD on nonsmooth convex losses.

The paper proves unique geodesics on hyperbolic surfaces and finds lower bounds.

A closed hyperbolic Riemann surface M is said to be K-quasiconformally homogeneous if there exists a transitive family F of K-quasiconformal homeomorphisms. Further, if all [f] in F act trivially on H1(M;Z), we say M is Torelli-K-quasiconformally homogeneous. We prove the existence of a uniform lower bound on K for Tor…

Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.

Study uniform rates for estimating Gaussian mixtures without separation assumption.

New algorithm learns policies without uniform overlap assumption.

Using polar convex bodies and the $C_0$-bounds from Guan and Ni \cite{PL}, we obtain a uniform lower bound on the Gauss curvature of the normalized solution of the Gauss curvature flow without using Chow's Harnack inequality \cite{Ch2}.

Paper establishes tight lower bounds for minimizing certain smooth and convex functions.

Uniform systole bounds for arithmetic orbifolds and number fields.

In this paper, we first prove a folklore conjecture on a greatest lower bound of the Calabi energy in all Kähler manifold. Similar result in algebriac setting was obtained by S. K. Donaldson. Secondly, we give an upper/lower bound estimate of the K energy in terms of the geodesic distance and the Calabi energy. This is…

Stability of timelike Ricci bounds in low-regularity spacetimes.

The paper proves uniqueness of evolving graphs by mean curvature flow under specific conditions.

Motivated by a recent work of X. Chen and M. Zhu (Commun. Math. Stat., 1 (2013) 369-385), we establish a Trudinger-Moser inequality on compact Riemannian surface without boundary. The proof is based on blow-up analysis together with Carleson-Chang's result (Bull. Sci. Math. 110 (1986) 113-127). This inequality is diffe…

We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of complete manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature then $Y$ has a universal cover.