arXiv research

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

We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.

Study of manifolds with nonnegative Ricci curvature and slow relative volume growth.

We prove a Liouville property for any $f$-harmonic function with polynomial growth on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ when the Bakry-Émery Ricci curvature is nonnegative and its diameter of geodesic sphere has sublinear growth.

Proves Calabi-Yau theorem for certain nonnegative curvature manifolds.

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

Study on homology growth and $\ell^2$-Betti numbers of Out(W_n).

The main results of this paper consists of two parts. Firstly, we obtain an almost rigidity theorem which says that on a RCD(0, N) space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Seco…

Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no nonconstant harmonic functions of sublinear growth. In the same paper, Yau used this resul…

We study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kazue. We also get a Cheng and Yau estimates for the gradient of harmonic functions.

Suppose X is any finite complex with vanishing L^2 Betti number. We prove upper bounds on the Betti numbers for regular coverings of X, sublinear in the order of covering. The bounds are sensitive to the Novikov-Shubin invariants of X, and are improved in the presence of a spectral gap.

The paper explores curvature-free effects in manifolds with volume growth and ends-counting.

The study proves manifolds with specific curvature and volume properties always split off a line at infinity.

New model allocates features sublinearly, improving model fit and performance.

Study volume growth and asymptotic cones of nonnegative Ricci curvature manifolds.

This paper concerns the $L^2$ essential spectrum of the Laplacian $Δ$ and the drift Laplacian $Δ_f$ on complete Riemannian manifolds endowed with a weighted measure $e^{-f}d\;vol_g$. We prove that the essential spectrum of the drift Laplacian $Δ_f$ is $[0,+\infty)$ provided the Bakry-Émery curvature tensor $Ric_f$ is …

We study both function theoretic and spectral properties on complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-Émery Ricci curvature. Among other things, we derive a gradient estimate for positive $f$-harmonic functions and obtain as a consequence the strong Liouville property under…

The paper extends a Liouville theorem to biharmonic functions on manifolds with nonnegative Ricci curvature.

New algorithm reduces prediction error in online learning without knowing base measure.

What are appropriate geometric conditions ensuring that a complete Riemannian 2-cylinder without conjugate points is flat? Examples with nonpositive curvature show that one has to assume that the ends of the cylinder open sublinearly. We show that sublinear growth of the ends is indeed sufficient if it is measured by t…

GP-PSRL achieves sublinear regret for continuous control with unbounded state space.

Sketch-GNN reduces GNN training time and memory usage to sublinear scales.

The paper shows how sublinear biLipschitz equivalences affect Morse boundaries of metric spaces.

Non-unique option pricing in Heston model analyzed mathematically.

In this paper, we prove sharp gradient estimates for a positive solution to the heat equation $u_t=Δu+au\log u$ in complete noncompact Riemannian manifolds. As its application, we show that if $u$ is a positive solution of the equation $u_t=Δu$ and $\log u$ is of sublinear growth in both spatial and time directions the…

The paper proves a hierarchy of Liouville theorems for polyharmonic functions on manifolds with nonnegative Ricci curvature.

Sharp stability threshold found for deep residual architectures.

For certain manifolds, nonnegative Ricci curvature limits dimension and forces almost abelian fundamental group.

Sublinear LSVI via LSH reduces runtime to sublinear in actions.

The aim of this paper is to introduce the sublinear Higson corona and show that the sublinear Higson corona of Euclidean cone of P and X is decomposed into the product of P and that of X. Here P is a compact metric space and X is unbounded proper metric space. For example, the sublinear Higson corona of n-dimensional E…

New model for Knightian uncertainty with jumps.

We give a proof of the sublinear tracking property for sample paths of random walks on various groups acting on spaces with hyperbolic-like properties. As an application, we prove sublinear tracking in Teichmueller distance for random walks on mapping class groups, and on Cayley graphs of a large class of finitely gene…

We provide a general construction of time-consistent sublinear expectations on the space of continuous paths. It yields the existence of the conditional G-expectation of a Borel-measurable (rather than quasi-continuous) random variable, a generalization of the random G-expectation, and an optional sampling theorem that…

Model financial market with fundraiser and stock, derive option prices.

New sublinear sketches improve ANN and KDE for massive data streams.

Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued functions (which form a nonlinear space), equivalently, on random closed sets. …

The paper develops methods for time-varying constrained online convex optimization.

Federated learning for combinatorial multi-agent bandits reduces regret and speeds up with fewer communications.

For $α\in (1,2)$, we present a generalized central limit theorem for $α$-stable random variables under sublinear expectation. The foundation of our proof is an interior regularity estimate for partial integro-differential equations (PIDEs). A classical generalized central limit theorem is recovered as a special case, p…

New algorithms for constrained online optimization with memory and predictions.

Memory-limited learning tackles adversarial bandits with reduced storage.

Many popular random partition models, such as the Chinese restaurant process and its two-parameter extension, fall in the class of exchangeable random partitions, and have found wide applicability in model-based clustering, population genetics, ecology or network analysis. While the exchangeability assumption is sensib…

The paper proves Liouville theorems for $V$-harmonic maps under specific curvature conditions.

New method connects CAT(0) spaces to hyperbolic spaces.

Study online learning in unknown Markov games with sublinear regret.

Let $(M^n, g, e^{-f}dv)$ be a smooth metric measure space of dimensional $n$. Suppose that $v$ is a positive weighted $p$-eigenfunctions associated to the eigenvalues $λ_{1,p}$ on $M$, namely $e^{f}div(e^{-f}|\nabla v|^{p-2}\nabla v)=-λ_{1,p}v^{p-1}.$ in the distribution sense. We first give a local gradient estimat…

In this work we consider adversarial contextual bandits with risk constraints. At each round, nature prepares a context, a cost for each arm, and additionally a risk for each arm. The learner leverages the context to pull an arm and then receives the corresponding cost and risk associated with the pulled arm. In additi…

Open problem seeks an online learning algorithm for binary classification.