Stable processes emerge as limits of deep neural networks with symmetric stable distributions.
problem Understanding the behavior of deep neural networks as they become infinitely wide.
method Analyzing fully connected feed-forward deep neural networks with symmetric stable distributions and showing the limit as a stable process.
result The infinite wide limit of the network is a stable process with multivariate stable distributions.
Deep neural networks with heavy-tailed weights converge to stable distributions.
problem Understanding the convergence of heavy-tailed weights in infinitely-wide neural networks.
method Analyzing infinitely-wide multi-layer perceptrons with i.i.d. symmetric α-stable weight distributions. result The vector of pre-activation values converges to i.i.d. symmetric α-stable distributions. Study examines kurtosis in heavy-tailed symmetric stable distributions.
problem Understanding kurtosis in heavy-tailed symmetric stable distributions.
method Empirical study focusing on symmetric stable distributions, investigating sample kurtosis behavior with sample size and tail index.
result Expected value of excess kurtosis divided by sample size is finite for any tail index, and sample estimate increases linearly with sample size and tail index.
Failure of the main argument for the use of heavy tailed distribution in Finance is given. More precisely, one cannot observe so many outliers for Cauchy or for symmetric stable distributions as we have in reality. keywords:outliers; financial indexes; heavy tails; Cauchy distribution; stable distributions
Paper explores Thompson Sampling for heavy-tailed distributions in sequential decision-making.
problem Sequential decision-making with heavy-tailed rewards.
method Revisit Thompson Sampling for symmetric α-stable distributions, presenting algorithms and proving regret bounds. result Thompson Sampling outperforms in heavy-tailed reward settings.
We propose a new blind source separation algorithm based on mixtures of alpha-stable distributions. Complex symmetric alpha-stable distributions have been recently showed to better model audio signals in the time-frequency domain than classical Gaussian distributions thanks to their larger dynamic range. However, infer…
Proposes vMF distribution for skewed elliptical distributions.
problem Skewed distributions not adequately modeled by symmetric distributions.
method Introduces von-Mises-Fisher (vMF) distribution to represent skewed elliptical distributions.
result vMF distribution provides an explicit and simple probability representation of skewed elliptical distributions.
New methods for estimating ARMA and GARCH models with stable noise.
problem Estimating parameters of ARMA and GARCH models with stable noise.
method Modified Hannan-Rissanen Method and Modified Empirical Characteristic Function for estimation.
result Efficiency, accuracy, and simplicity of proposed methods demonstrated through simulation.
A new distribution family extends the α-stable distribution with a degree of freedom parameter.
problem Lack of moments in the α-stable distribution. method Wright function framework to combine and extend distribution families.
result Generalized α-stable distribution with valid moments. We examine stable solutions of the following symmetric system on a complete, connected, smooth Riemannian manifold M without boundary, \begin{equation*} -Δ_g u_i = H_i(u_1,\cdots,u_m) \ \ \text{on} \ \ \mathbb{M}, \end{equation*} when Δg stands for the Laplace-Beltrami operator, $u_i:\mathbb{M}\to \mathbb…
MAFLA improves sampling from heavy-tailed distributions using MH-inspired corrections.
problem Sampling from heavy-tailed and multimodal distributions when neither target nor proposal densities can be evaluated.
method Metropolis-Adjusted Fractional Langevin Algorithm (MAFLA) with Score Balance Matching.
result MAFLA significantly improves finite-time sampling accuracy over unadjusted fractional Langevin dynamics.
New stable metric found on a complex space.
problem Stability of a non-symmetric metric on a complex space.
method Proving stability with respect to the Einstein-Hilbert action.
result First known example of a non-symmetric metric of positive scalar curvature.
Spaces of polynomials are shown to be Euclidean balls.
problem Understanding the geometry of Lorentzian and real stable polynomials.
method Refined connection between symmetric exclusion process and polynomial geometry.
result Spaces of Lorentzian and real stable polynomials are homeomorphic to closed Euclidean balls.
We show that the triply graded Khovanov-Rozansky homology of the torus link Tn,k stablizes as k→∞. We explicitly compute the stable homology (as a ring), which proves a conjecture of Gorsky-Oblomkov-Rasmussen-Shende. To accomplish this, we construct complexes Pn of Soergel bimodules which categorify t…
The paper analyzes return distribution of Chinese stock market indices over various time scales.
problem Understanding return distribution properties of Chinese stock markets.
method Systematic analysis of 1-min to 4000-min composite index datasets from 2005-2021.
result Return distribution properties are similar to mature markets, with distinct behavior at different time scales.
Study of tropical moduli spaces using symmetric Delta-complexes.
problem Understanding the fundamental groups and homology of tropical moduli spaces.
method Develop techniques for symmetric Delta-complexes, apply to moduli spaces of tropical curves.
result Delta_g and Delta_{g,n} are simply connected for positive g.
Stable capillary surfaces in weighted balls are disks.
problem Finding the shape of isoperimetric regions in weighted balls.
method Stability analysis and Hsiang symmetrization.
result Interior boundaries of isoperimetric regions in weighted balls are disks.
For a symmetric Hamiltonian system, lower bounds for the number of relative equilibria surrounding stable and formally unstable relative equilibria on nearby energy levels are given.
Please see the article for the abstract.
Geodesic spheres in certain symmetric spaces are quantitatively stable under small perturbations.
problem Stability of geodesic spheres in symmetric spaces under perturbations.
method Quantitative stability analysis using spectral gap of the Laplacian on geodesic spheres.
result Geodesic spheres are uniformly stable with respect to small C1-volume preserving perturbations. Elliptic curve governs Hopf linking in symmetric tensegrity.
problem Understanding Hopf linking in A4-symmetric tensegrity.
method Parametrized by points on the elliptic curve 30a2.
result Only one rational point corresponds to a stable configuration.
Following Cao-Hamilton-Ilmanen, in this paper we study the linear stability of Perelman's ν-entropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability restricted to the transversal traceless symmetric 2-tensors and the stability of Einstein manifolds with resp…
In this work we classify the stable regions (second order minima of perimeter under an area constraint) in tori of revolution with piecewise continuous decreasing Gauss curvature from the longest parallel and with a horizontal symmetry. Some applications to isoperimetric problems are also given.
New estimates show all stable Einstein manifolds are linear stable with respect to Perelman's ν-entropy.
problem Estimating the smallest eigenvalue of Laplace-Beltrami operator for stable Einstein manifolds.
method Estimating the smallest positive eigenvalue λ1 of the Laplace-Beltrami operator for standard Einstein manifolds (G/H,gst) and proving λ1>2E for all but 7 exceptions. result All stable Einstein manifolds found by Schwahn are linear stable with respect to Perelman's ν-entropy.
No stable discrete maps into certain curved spaces exist.
problem Stability of discrete maps into curved spaces.
method Analysis of weighted length or energy functionals on graphs.
result Non-existence of stable discrete minimal immersions or harmonic maps into specific homogeneous spaces.
Extends inequality for Rademacher complexities using p-stable variables.
problem Improving Rademacher complexity bounds using p-stable variables. method Extends contraction inequality to p-stable variables for 1<p<2. result New bounds for Rademacher complexities with p-stable variables. Investigates tempered stable distributions and processes, including density transformations and parameter estimation.
problem Understanding the properties and applications of tempered stable distributions and processes.
method Analysis of limit distributions, parameter estimation, density transformations, and computation of p-variation indices. result Computed p-variation indices for tempered stable processes and discussed exponential stock models driven by these processes. Proposes a symmetric graph autoencoder for unsupervised learning.
problem Graph representation learning without labeled data.
method Symmetric graph convolutional autoencoder with Laplacian sharpening and signed graphs.
result Outperforms state-of-the-art algorithms in clustering, link prediction, and visualization tasks.
In this paper, we discuss two topics: first, we show how to convert 1+1-topological quantum field theories valued in symmetric bimonoidal categories into stable homotopical data, using a machinery by Elmendorf and Mandell. Then, we discuss, in this framework, two recent results (independent of each other) on refinement…
The purpose of this paper is to investigate canonical metrics on a semi-stable vector bundle E over a compact Kahler manifold X. It is shown that, if E is semi-stable, then Donaldson's functional is bounded from below. This implies that E admits an approximate Hermitian-Einstein structure, generalizing a classic result…
The Martin boundary of certain groups is stable under specific conditions.
problem Stability of Martin boundaries in relatively hyperbolic groups.
method Extending Floyd-Ancona inequalities, defining spectral degenerescence, and proving stability criteria.
result The Martin boundary of admissible symmetric finitely supported probability measures on geometrically finite Kleinian groups of dimension at most 5 is always strongly stable.
Finite index solutions to Bernoulli problem are always axially symmetric.
problem Entire solutions to the Bernoulli free boundary problem with finite Morse index in 3D.
method Proof of axial symmetry for finite index solutions.
result Finite index solutions to the Bernoulli problem in 3D are axially symmetric.
Study geodesics on K3 surfaces near orbifold limit.
problem Understanding geodesics on K3 surfaces near the orbifold limit.
method Improves metric estimates for K3 surfaces, uses hyperkähler identities.
result Restrictions and existence conditions for stable geodesics.
A definition for elliptical tempered stable distribution, based on the characteristic function, have been explained which involve a unique spectral measure. This definition provides a framework for creating a connection between infinite divisible distribution, and particularly elliptical tempered stable distribution, w…
Heavy-tailed distributions are widely used in robust mixture modelling due to possessing thick tails. As a computationally tractable subclass of the stable distributions, sub-Gaussian α-stable distribution received much interest in the literature. Here, we introduce a type of expectation maximization algorithm that e…
The main aim of this paper is to study existence and stability properties of rotationally symmetric proper biharmonic maps between two m-dimensional models (in the sense of Greene and Wu). We obtain a complete classification of rotationally symmetric, proper biharmonic conformal diffeomorphisms in the special case th…
The study classifies equivariant biharmonic maps and proves stability results for certain maps.
problem Classifying and analyzing equivariant biharmonic maps and their stability.
method Generalized biharmonic equation for equivariant maps, improved second variation formula for biharmonic maps.
result No stable proper biharmonic maps with constant square norm of tension field exist from a compact Riemannian manifold into a space form of positive sectional curvature.
We study stability and local minimizing properties of Lp- norms of Riemannian curvature tensor denoted by Rp by variational methods. We compute the Hessian of Rp at compact rank 1 symmetric spaces and prove that they are stable for Rp for certain values of p > 2. A similar resu…
The paper uses FRFT to fit GTS distribution to asset returns.
problem Modeling asset returns with GTS distribution.
method Fractional Fourier Transform (FRFT) for fitting.
result GTS distribution fits SPY ETF and Bitcoin BTC returns.
Study non-integrable distributions with various affine connections.
problem Characterize non-integrable distributions in Riemannian manifolds with different connections.
method Obtain Gauss, Codazzi, and Ricci equations for non-integrable distributions with semi-symmetric metric, non-metric, and statistical connections.
result Find new examples of Einstein and distributions with constant scalar curvature.
We study the statistics of records of a one-dimensional random walk of n steps, starting from the origin, and in presence of a constant bias c. At each time-step the walker makes a random jump of length ηdrawn from a continuous distribution f(η) which is symmetric around a constant drift c. We focus in particular on th…
When looking at Bott's original proof of his periodicity theorem for the stable homotopy groups of the orthogonal and unitary groups, one sees in the background a differential geometric periodicity phenomenon. We show that this geometric phenomenon extends to the standard inclusion of the orthogonal group into the unit…
In recent years, correntropy and its applications in machine learning have been drawing continuous attention owing to its merits in dealing with non-Gaussian noise and outliers. However, theoretical understanding of correntropy, especially in the statistical learning context, is still limited. In this study, within the…
Researchers compute the full spectrum of Laplace operator on distance spheres in symmetric spaces.
problem Computing the full Laplace spectrum on distance spheres in symmetric spaces.
method Lie-theoretic methods to explicitly compute the spectrum.
result Unified formula for the full spectrum of Laplace operator on distance spheres in symmetric spaces of rank one.
The Mean-Variance Criterion is equivalent to Second-order Stochastic Dominance under symmetric Elliptical distributions.
problem Determining the equivalence of Mean-Variance Criterion and Stochastic Dominance Criteria.
method Analyzing under symmetric and Skew-Elliptical distributions using Monte Carlo simulations.
result The Mean-Variance Criterion does not coincide with Second-order Stochastic Dominance for some types of risk-averse investors.
This paper models cryptocurrencies using α-stable distributions, outperforming other models.
problem Modeling the highly speculative and leptokurtic nature of cryptocurrencies.
method Used α-stable distribution and compared it with other heavy tailed distributions. Employed maximum likelihood method for estimation. result The α-stable distribution fits cryptocurrency return data better than other models. New concept of coarse medians for higher rank symmetric spaces.
problem Understanding medians in higher rank symmetric spaces.
method Introducing coarse r-median spaces and proving their existence. result Existence of coarse higher medians on divisible and quasi-homogeneous convex domains.
The study classifies stable submanifolds in product spaces of projective spaces.
problem Classifying stable submanifolds in product spaces of projective spaces.
method Provided a classification theorem for compact stable minimal immersions in product spaces of projective spaces.
result Characterized complex minimal immersions in the product of two complex projective spaces.