arXiv research

Polyak step size GD reaches final radius of convergence after log iterations.

Improved computational complexity in statistical models using second-order information.

We introduce RSE to measure robustness in estimation problems.

Paper optimizes hyperparameters for high-dimensional regression models.

Random translation surfaces converge to a Poisson plane as genus grows.

ELU algorithm improves on EM for over-specified Gaussian mixtures.

A method to fix radius distortion in generative models on curved spaces.

We prove near-tight concentration of measure for polynomial functions of the Ising model under high temperature. For any degree $d$, we show that a degree-$d$ polynomial of a $n$-spin Ising model exhibits exponential tails that scale as $\exp(-r^{2/d})$ at radius $r=\tildeΩ_d(n^{d/2})$. Our concentration radius is opti…

In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$, $\operatorname{conv}(p)\ge \min\{\frac{1}{2}\operatorname{inj}(p),\operatorname{foc}(B_{\operatorname…

This guide simplifies high-probability regret bounds in empirical risk minimization.

Positive injectivity radius for manifolds with Lie structure at infinity.

The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with focal points but no conjugate points. The approach is suggested by a characteriz…

Uniform curvature bounds for regularized metrics with bounds on Ricci tensor and injectivity radius.

Upper bound for conjugate radius in open manifolds with scalar curvature and spectrum constraints.

Proves upper bound on filling radius for manifolds with positive scalar curvature.

Compact theorem for minimal surfaces with lower injectivity radius.

Injectivity radius on Stiefel manifold is π.

Lower bound on boundary injectivity radius for specific tubes.

Study gives bounds on filling radius for Riemannian manifolds.

The Hidden Markov Model (HMM) is one of the mainstays of statistical modeling of discrete time series, with applications including speech recognition, computational biology, computer vision and econometrics. Estimating an HMM from its observation process is often addressed via the Baum-Welch algorithm, which is known t…

The paper proves estimates and theorems for Kähler manifolds.

Study finds the covering radius of RM(4,8) is 26.

Clarifies definition of polarized canonical radius in Kahler Ricci flow.

The paper bounds bandwidth and focal radius for manifolds with positive isotropic curvature.

Upper bound on Stiefel manifold's injectivity radius found.

Upper estimate of Dirac eigenvalue linked to hyperspherical radius.

The study proves triviality and rigidity results for Ricci solitons and estimates their conjugate radius.

Paper optimizes statistical estimation for randomized smoothing to reduce adversarial robustness certification time.

Estimating signals with linear recurrence relations under Gaussian noise is nearly as hard as sparse signals.

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

A translation structure equips a Riemann surface with a singular flat metric. Not much is known about the shape of a random translation surface. We compute an upper bound on the expected value of the covering radius of a translation surface in any stratum H_1(kappa). The covering radius of a translation surface is the …

This paper considers metric balls $B(p,R)$ in two dimensional Riemannian manifolds when $R$ is less than half the convexity radius. We prove that $Area(B(p,R)) \geq \frac{8}πR^2$. This inequality has long been conjectured for $R$ less than half the injectivity radius. This result also yields the upper bound $μ_2(B(p,R)…

We derive an explicit lower bound on the radius of a ball embedded in a quaternionic hyperbolic manifold.

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

Paper generalizes inscribed radius estimate to hyperbolic Einstein manifolds.

We prove in a direct, geometric way that for any compatible Riemannian metric on a Lie manifold the injectivity radius is positive

Study finds shortest geodesic loops on Stiefel manifold and calculates its injectivity radius.

We prove that spherical spectral analysis and synthesis hold in Damek-Ricci spaces and derive two-radius theorems.

Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.

The paper improves bounds on injectivity radius for manifolds with positive scalar curvature.

In this note we relate the geometric notion of fill radius with the fundamental group of the manifold. We prove: ''Suppose that a closed Riemannian manifold M satisfies the property that its universal cover has bounded fill radius. Then the fundamental group of M is virtually free.'' We explain the relevance of this th…

Characterizes submanifolds with minimum ratio of diameter to focal radius.

Generalizes a soul-bound for noncompact Alexandrov spaces.

We define a capacity which measures the size of Weinstein tubular neighbourhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Moreover, we prove a generalization of Gromov's packing…

We prove a sharp estimate for the inscribed radius under certain fully nonlinear curvature flows. This estimate is asymptotically sharp on cylinders.

We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold $M = (0,\infty) \times Y$ whose rotation radius is constant outside some compact interval. The Laplacian on $M$ is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potenti…

A canal surface is the envelope of a moving sphere with varying radius, defined by the trajectory C(t) (spine curve) of its center and a radius function r(t). In this paper, we investigate when parameter curves of the canal surface are also lines of curvature. Last of all, for special spine curves we obtain the radius …

Given a positive function $u\in W^{1,n}$, we define its John-Nirenberg radius at point $x$ to be the supreme of the radius such that $\int_{B_t}|\nabla\log u|^n<ε_0^n$ when $n>2$, and $\int_{B_t}|\nabla u|^2<ε_0^2$ when $n=2$. We will show that for a collapsing sequence in a fixed conformal class under some curvature c…