arXiv research

Study on curves on specific arithmetic quotients of hyperbolic 2-ball.

New 2D complex hyperbolic structures found on sphere orbibundles.

We study the space $C(a_0,a_1,\dots,a_n)$ of hyperbolic 2-spheres with cone points of prescribed apex curvatures $2a_0,2a_1,\dots,2a_n\in]0,2π[$ and some related spaces. For $n=3$, we get a detailed description of such spaces. The euclidean 2-spheres were considered by W. P. Thurston: for $n=4$, the corresponding space…

We establish sharp Sobolev inequalities of order four on Euclidean d-balls for d greater than or equal to four. When d=4, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean 2-balls. Our method relies on the use of scattering theory on hyperbolic d-balls. As an application, we ch…

An ideal triangulation $\mathcal{T}$ of a hyperbolic 3-manifold $M$ with one cusp is non-peripheral if no edge of $\mathcal{T}$ is homotopic to a curve in the boundary torus of $M$. For such a triangulation, the gluing and completeness equations can be solved to recover the hyperbolic structure of $M$. A planar project…

We describe and study the loci equidistant from finitely many points in the so-called complex hyperbolic geometry, i.e., in the geometry of a holomorphic $2$-ball $\Bbb B$. In particular, we show that the bisectors (= the loci equidistant from $2$ points) containing the (smooth real algebraic) curve equidistant from gi…

In this paper we review the development and recent results of the Siu-Yang conjecture which is that every Kähler-Einstein compact complex manifold of complex dimension two with negative sectional curvature is biholomorphic to a compact quotient of the complex 2-ball.

K-means clustering improved for robustness to outliers and distribution shifts.

Unified algorithm for linear bandits with improved regret bound.

This paper contains a construction of a finite set X in the boundary of the unit 3-ball in R^3 whose minimal tree is knotted. The example answers Problem 5.17 in ''Problems in Low-dimensional Topology'' by Rob Kirby posed by Michael Freedman: ''Given a finite set of points X in the boundary of B^3, let T be a tree in B…

Fusion of robustness and uncertainty techniques improves adversarial defense.

We prove an equidistribution result for totally geodesic submanifolds in a compact locally symmetric space. In the case of Hermitian locally symmetric spaces, this gives a convergence theorem for currents of integration along totally geodesic subvarieties. As a corollary, we obtain that on a complex surface which is a …

New algorithm reduces online logistic regression regret without exponential constant.

Local robustness ensures that a model classifies all inputs within an $\ell_2$-ball consistently, which precludes various forms of adversarial inputs. In this paper, we present a fast procedure for checking local robustness in feed-forward neural networks with piecewise-linear activation functions. Such networks partit…

We consider a stochastic continuum armed bandit problem where the arms are indexed by the $\ell_2$ ball $B_{d}(1+ν)$ of radius $1+ν$ in $\mathbb{R}^d$. The reward functions $r :B_{d}(1+ν) \rightarrow \mathbb{R}$ are considered to intrinsically depend on $k \ll d$ unknown linear parameters so that $r(\mathbf{x}) = g(\ma…

We consider the problem of learning multi-ridge functions of the form f(x) = g(Ax) from point evaluations of f. We assume that the function f is defined on an l_2-ball in R^d, g is twice continuously differentiable almost everywhere, and A \in R^{k \times d} is a rank k matrix, where k << d. We propose a randomized, po…

Study magnetic Steklov eigenvalues on manifolds with boundary.

Batch normalization makes deep neural networks' representations increasingly orthogonal.

Paper studies robust MDPs, improving sample complexity and asymptotic performance.

As application demands for online convex optimization accelerate, the need for designing new methods that simultaneously cover a large class of convex functions and impose the lowest possible regret is highly rising. Known online optimization methods usually perform well only in specific settings, and their performance…

New model generates data on constrained sets without losing tractability.

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

Recursive training of generative models can lead to model collapse, and the recursion converges to a unique limiting distribution.

Paper proves convergence for private FL on non-Lipschitz convex objectives using normalization instead of clipping.

Unified framework for uniform signal recovery in nonlinear GCS with 1-bit/quantized measurements.

New hyperbolicity concepts expand manifold study.

The paper defines two types of hyperbolicity for complex manifolds and proves related results.

Introduces Möbius structures and hyperbolic ends for $k$-surfaces in hyperbolic space.

Introduces hyperbolic generalized framed surfaces and their properties.

Study on new hyperbolicity notions for non-Kähler manifolds and their deformations.

New formulas for hyperbolic mass using horospheres.

Abstract: Non-residually finite hyperbolic groups imply non-residually finite rigid hyperbolic groups.

Study complex hyperbolic lattices and their relation to strict hyperbolization.

Abstract reviews hyperbolic positive energy theorems.

We study hyperbolic bongles and find their volumes.

Paper finds inequalities for convex domains in hyperbolic space.

We study conservative partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We show that they are always accessible and deduce as a result that every conservative $C^{1+}$ partially hyperbolic in a hyperbolic 3-manifold must be ergodic, giving an afirmative answer to a conjecture of Hertz-Hertz-Ures in the co…

Study deformations of compact Kähler hyperbolic manifolds.

We show that a relatively hyperbolic graph with uniformly hyperbolic peripheral subgraphs is hyperbolic. As an application, we show that the disc graph and the electrified disc graph of a handlebody H of genus g>1 are hyperbolic, and we determine their Gromov boundaries.

Study shows hyperbolic subgroups can be free products of surface and free groups.

New 5D hyperbolic shapes that wrap around a circle found.

Study properties of hyperbolic Yamabe solitons in submanifolds.

Survey solves curvature problems with hyperbolic spaces.

Drilling hyperbolic groups to simplify complex conjectures.

HKConv learns hyperbolic features by aggregating kernel points.

Harmonic maps from hyperbolic planes to hyperbolic space exist with given boundary data.

We show the equivalence of several characterizations of relative hyperbolicity for metric spaces, and obtain extra information about geodesics in a relatively hyperbolic space. We apply this to characterize hyperbolically embedded subgroups in terms of nice actions on (relatively) hyperbolic spaces. We also study the d…

Characterizes boundaries of HHGs and their properties.