arXiv research

Surveying Ricci flow for weak lower scalar curvature bounds.

The paper establishes bounds on scalar curvature on asymptotically flat manifolds.

Paper develops proper, lower-bounded losses for weakly supervised classification.

Weak base-point freeness leads to Kähler-Ricci flow diameter bounds.

Sharp bounds on weak convergence rate for rough volatility models.

Boosting is a celebrated machine learning approach which is based on the idea of combining weak and moderately inaccurate hypotheses to a strong and accurate one. We study boosting under the assumption that the weak hypotheses belong to a class of bounded capacity. This assumption is inspired by the common convention t…

In this paper we propose a class of local definitions of weak lower scalar curvature bounds that is well defined for $C^0$ metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from $C^0$…

Variational inference (VI) is a widely used framework in Bayesian estimation. For most of the non-Gaussian statistical models, it is infeasible to find an analytically tractable solution to estimate the posterior distributions of the parameters. Recently, an improved framework, namely the extended variational inference…

Proposes efficient bounds for causal effect estimation under weak confounding.

Proves curvature tensor convergence for smoothable spaces.

The study explores the strengths and weaknesses of models that generalize from weak to strong supervision.

Study compares synthetic and distributional Ricci curvature bounds.

The weak splitting number $wsp(L)$ of a link $L$ is the minimal number of crossing changes needed to turn $L$ into a split union of knots. We describe conditions under which certain $\mathbb{R}$-valued link invariants give lower bounds on $wsp(L)$. This result is used both to obtain new bounds on $wsp(L)$ in terms of t…

Survey on preserving curvature bounds for non-smooth Ricci flow.

Optimal algorithm converts weak to strong learner with less data.

An algorithm learns from multiple models to match an oracle's risk.

Preserves scalar curvature bounds under weak convergence of 3-manifolds.

We are concerned with the global weak continuity of the Cartan structural system -- or equivalently, the Gauss--Codazzi--Ricci system -- on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian m…

Paper reconciles different Ricci flow approaches and proves weak solutions.

New principles prove precompactness of domains with lower Ricci curvature bound.

Study on when smooth Ricci flow remains smooth at the start.

Develops ELBD for efficient feature selection in VAE latent variables.

Relying on the recent work of Liu-Székelyhidi we give a weak asymptotic estimate for the Bergman kernels of polarized Kähler manifolds with Ricci lower bound and Sobolev constant upper bound. We will also give a simple proof for the partial $C^0$ estimate along the (generalized) Kähler-Ricci flow on Fano manifolds.

The paper proves scalar curvature lower bounds along Ricci flow on compact manifolds.

We develop a general framework on Dirichlet spaces to prove a weak form of the Bakry-Émery estimate and study its consequences. This estimate may be satisfied in situations, like metric graphs, where generalized notions of Ricci curvature lower bounds are not available.

Suppose a sequence $M_j$ of Alexandrov spaces collapses to a space $X$ with only weak singularities. Yamaguchi constructed a map $f_j:M_j\to X$ called an almost Lipschitz submersion for large $j$. We prove that if $M_j$ has a uniform positive lower bound for the volumes of spaces of directions, which is sufficiently la…

Study improves weak error estimates for rough volatility models.

In this paper, we consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold $M$ isometrically immersed into another Riemannian manifold $\bar M$ for arbitrary codimension. We first assume the pull back Weitzenböck operator (defined in Section 2) of $\bar M$ bounded from below, and obtain an extrinsic…

Algorithm minimizes regret in non-stationary dueling bandits with unknown parameters.

New graph feedback model for bandits with improved regret bounds.

The report explores conditions for positive or non-negative scalar curvature in 3-manifolds and weak Ricci curvature bounds.

The paper develops bounds and regularity for minimal boundaries in non-smooth spaces with Ricci curvature.

The Wasserstein metric is an important measure of distance between probability distributions, with applications in machine learning, statistics, probability theory, and data analysis. This paper provides upper and lower bounds on statistical minimax rates for the problem of estimating a probability distribution under W…

Study on tradeoffs between mistakes and ERM oracle calls in online and transductive learning.

Labeling training data is a key bottleneck in the modern machine learning pipeline. Recent weak supervision approaches combine labels from multiple noisy sources by estimating their accuracies without access to ground truth labels; however, estimating the dependencies among these sources is a critical challenge. We foc…

Study examines surfaces with bounded fractional mean curvature, proving control over local parametrization.

We prove that complete submanifolds, on which the Omori-Yau weak maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the la…

We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isom…

In the first part Busemann concavity as non-negative curvature is introduced and a bi-Lipschitz splitting theorem is shown. Furthermore, if the Hausdorff measure of a Busemann concave space is non-trivial then the space is doubling and satisfies a Poincaré condition and the measure contraction property. Using a compari…

Estimates for $p$-capacities on symmetric manifolds.

The small-ball method was introduced as a way of obtaining a high probability, isomorphic lower bound on the quadratic empirical process, under weak assumptions on the indexing class. The key assumption was that class members satisfy a uniform small-ball estimate: that $Pr(|f| \geq κ\|f\|_{L_2}) \geq δ$ for given const…

We consider the problem of providing nonparametric confidence guarantees for undirected graphs under weak assumptions. In particular, we do not assume sparsity, incoherence or Normality. We allow the dimension $D$ to increase with the sample size $n$. First, we prove lower bounds that show that if we want accurate infe…

Two new algorithms solve privacy-constrained SVI and SSP problems.

New margin bound improves generalization for voting classifiers.

In this note we discuss the fundamental groups and diameters of positively Ricci curved $n$-manifolds. We use a method combining the results about equivarient Hausdorff convergence developed by Fukaya and Yamaguchi with the Ricci version of splitting theorem by Cheeger and Colding to give new information on the topolog…

In this paper we show the existence of weak solutions $w : M \rightarrow \mathbb{R}$ of the inverse mean curvature flow starting from a relatively compact set (possibly, a point) on a large class of manifolds satisfying Ricci lower bounds. Under natural assumptions, we obtain sharp estimates for the growth of $w$ and f…

In the past decade, sparse principal component analysis has emerged as an archetypal problem for illustrating statistical-computational tradeoffs. This trend has largely been driven by a line of research aiming to characterize the average-case complexity of sparse PCA through reductions from the planted clique (PC) con…

Boosting improves accuracy with fewer calls to weak learners for certain concept classes.