arXiv research

Liouville theorems extended to graphs with bounded geometry.

We present the notion of asymptotically large depth for a metric space which is (a priory) weaker than having subexponential asymptotic dimension growth and (a priory) stronger than property A.

We present a new, more elementary proof of the Freedman-Teichner result that the geometric classification techniques (surgery, s-cobordism, and pseudoisotopy) hold for topological 4-manifolds with groups of subexponential growth. In an appendix Freedman and Teichner give a correction to their original proof, and reform…

Study shows torsion grows subexponentially in book of I-bundles but can grow exponentially in non-regular covers.

We prove several cases of Zimmer's conjecture for actions of higher-rank cocompact lattices on low dimensional manifolds. For example, if $Γ$ is a cocompact lattice in $\mathrm{Sl}(n, \mathbb R)$, $M$ is a compact manifold, and $ω$ a volume form on $M$ we show that any homomorphism $ρ\colon Γ\rightarrow \mathrm{Diff}(M…

Graphs with bounded degrees and non-negative Ollivier-Ricci curvature have subexponential growth and diffusive random walk.

Study on hyperbolic groups, focusing on separability and splittings.

We introduce a new quasi-isometry invariant $\subcorank X$ of a metric space $X$ called {\it subexponential corank}. A metric space $X$ has subexponential corank $k$ if roughly speaking there exists a continuous map $g:X\to T$ such that for each $t\in T$ the set $g^{-1}(t)$ has subexponential growth rate in $X$ and the…

New insights into natural exponential families improve regret bounds for bandit problems.

The study classifies translating and self-expanding solitons in 3D space.

Growth of monetary assets and debts is commonly described by the formula of compound interest which for the case of continuous compounding is the exponential growth law. Its differential form is dc/dt = i c where dc/dt describes the rate of monetary growth, i the compounded interest rate and c the actual principal. Exp…

Study of linear classifiers in infinite imbalance scenarios.

This is a sequel to the paper [Cas]. Here, we extend the methods of Farb-Wolfson using the theory of FI_G-modules to obtain stability of equivariant Galois representations of the etale cohomology of orbit configuration spaces. We establish subexponential bounds on the growth of unstable cohomology, and then use the Gro…

We continue the study of a general class of spaces of 0-cycles on a manifold defined and begun by Farb-Wolfson-Wood. Using work of Gadish on linear subspace arrangements, we obtain representation stability for the cohomology of the ordered version of these spaces. We establish subexponential bounds on the growth of uns…

Let $M^m$ be a minimal properly immersed submanifold in an ambient space close, in a suitable sense, to the space form $\mathbb{N}^n_k$ of curvature $-k\le 0$. In this paper, we are interested in the relation between the density function $Θ(r)$ of $M^m$ and the spectrum of the Laplace-Beltrami operator. In particular, …

Study improves error bounds for sparse regression with heavy-tailed covariates.

We contribute to the arithmetic/topology dictionary by relating asymptotic point counts and arithmetic statistics over finite fields to homological stability and representation stability over $\Cb$ in the example of configuration spaces of $n$ points in smooth varieties. To do this, we import the method of homological …

In this paper, we obtain the finite-horizon and infinite-horizon ruin probability asymptotics for risk processes with claims of subexponential tails for non-stationary arrival processes that satisfy a large deviation principle. As a result, the arrival process can be dependent, non-stationary and non-renewal. We give t…

To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The Volume Conjecture for small angles states that the value of the $n$-th colored Jones polynomial at $e^{\a/n}$ is a sequence of complex numbers that grows subexponentially, for a fixed s…

Single gradient step finds adversarial examples in random neural networks.

Paper studies second order tail probabilities in risk models.

Study of Alexander polynomials of torus knots and links, showing zeros equidistribute on unit circle.

We address a long-standing and long-investigated problem in combinatorial topology, and break the exponential barrier for triangulations of real projective space, constructing a trianglation of $\mathbb{RP}^n$ of size $e^{(\frac{1}{2}+o(1))\sqrt{n}{\log n}}$.

Estimates change point in high dimensional time series models.

$(M^n,g)$ be a complete Riemannian manifold without conjugate points. In this paper, we show that if $M$ is also simply connected, then $M$ is flat, provided that $M$ is also asymptotically harmonic manifold with minimal horospheres (AHM). The (first order) flatness of $M$ is shown by using the strongest criterion: $\{…

We study tilting subweibull distributions and their tail behavior.

Introduces new hyperbolicity concepts for complex manifolds.

Study partially hyperbolic flows on flat bundles, proving equivalence for complete affine manifolds.

The paper proposes a new method to measure risk with fine-grained tail sensitivity.

Trace norm regularization is a popular method of multitask learning. We give excess risk bounds with explicit dependence on the number of tasks, the number of examples per task and properties of the data distribution. The bounds are independent of the dimension of the input space, which may be infinite as in the case o…

This work improves bounds on Bayesian coreset quality.

The quantification of diversification benefits due to risk aggregation plays a prominent role in the (regulatory) capital management of large firms within the financial industry. However, the complexity of today's risk landscape makes a quantifiable reduction of risk concentration a challenging task. In the present pap…

This paper extends the standard chaining technique to prove excess risk upper bounds for empirical risk minimization with random design settings even if the magnitude of the noise and the estimates is unbounded. The bound applies to many loss functions besides the squared loss, and scales only with the sub-Gaussian or …

We study the crossing number of links that are formed by edges of a triangulation T of the 3-sphere with n tetrahedra. We show that the crossing number is bounded from above by an exponential function of n^2. In general, this bound can not be replaced by a subexponential bound. However, if T is polytopal (resp. shellab…

We investigate the average-case complexity of decision problems for finitely generated groups, in particular the word and membership problems. Using our recent results on ``generic-case complexity'' we show that if a finitely generated group $G$ has the word problem solvable in subexponential time and has a subgroup of…

We study the computational cost of recovering a unit-norm sparse principal component $x \in \mathbb{R}^n$ planted in a random matrix, in either the Wigner or Wishart spiked model (observing either $W + λxx^\top$ with $W$ drawn from the Gaussian orthogonal ensemble, or $N$ independent samples from $\mathcal{N}(0, I_n + …

Optimal data-driven formulations are found for learning and decision-making with historical data.

This article shows that counting or computing the small eigenvalues of the Witten Laplacian in the semi-classical limit can be done without assuming that the potential is a Morse function as the authors did in [LNV]. In connection with persistent cohomology, we prove that the rescaled logarithms of these small eigenval…

This paper investigates the average-case time complexity of certifying RIP matrices.

New PAC-Bayes bounds for heavy-tailed losses using supermartingales.

Let $G$ be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to 1, e.g. $G= \SL_2 (\C) \times \SL_2 (\C)$ or $\SL_3 (\C)$. Then the fundamental rank of $G$ is $2,$ and according to the conjecture made in \cite{BV}, lattices in $G$ should have 'little' --- in the very weak sense…

Ridge regression performs optimally in noisy environments with heavy-tailed distributions.

Historical economic growth in countries of the former USSR is analysed. It is shown that Unified Growth Theory is contradicted by the data, which were used, but not analysed, during the formulation of this theory. Unified Growth Theory does not explain the mechanism of economic growth. It explains the mechanism of Malt…

Study finds the minimum number of finite Gaussian mixtures for best approximation.

The Unified Growth Theory is a puzzling collection of myths based on illusions created by hyperbolic distributions. Some of these myths are discussed. The examination of data shows that the three stages of growth (Malthusian Regime, Post-Malthusian Regime and Modern Growth Regime) did not exist and that Industrial Revo…

Historical economic growth in Asia (excluding Japan) is analysed. It is shown that Unified Growth Theory is contradicted by the data, which were used (but not analysed) during the formulation of this theory. Unified Growth Theory does not explain the mechanism of economic growth. It explains the mechanism of Malthusian…

Study confined subgroups in groups with contracting elements, showing their growth rate is strictly greater than half of the ambient growth rate.

Growth rate of the world Growth Domestic Product (GDP) is analysed to determine possible pathways of the future economic growth. The analysis is based on using the latest data of the World Bank and it reveals that the growth rate between 1960 and 2014 was following a trajectory approaching asymptotically a constant val…