New Fourier analysis method for non-uniform Boolean hypercube.
problem Non-uniform probability measures on the Boolean hypercube.
method ANOVA-based decomposition, explicit basis, least squares problem.
result Generalization of Fourier analysis for arbitrary probability measures.
Establishes a link between risk measures and uniform integrability in finance.
problem Understanding uniform integrability in the context of financial risk measures.
method Introduces the folding score of distortion risk measures to study uniform integrability directly with gains and losses.
result Obtains three sets of equivalent conditions for uniform integrability involving coherent risk measures.
Implementing k-NN classification using Gromov--Wasserstein distances
problem Comparing metric measure spaces
method Gromov--Wasserstein and fused Gromov--Wasserstein distances
result Universal consistency of k-NN classifiers Statistical performance bounds for reinforcement learning (RL) algorithms can be critical for high-stakes applications like healthcare. This paper introduces a new framework for theoretically measuring the performance of such algorithms called Uniform-PAC, which is a strengthening of the classical Probably Approximatel…
Active seriation recovers item order from noisy pairwise similarity measurements.
problem Recovering an unknown item ordering from noisy pairwise similarity measurements.
method Proposes an active seriation algorithm that provably recovers the latent ordering with high probability.
result Establishes optimal performance guarantees for successful recovery under a uniform separation condition.
For a sequence of nonnegative random variables, we provide simple necessary and sufficient conditions to ensure that each sequence of its forward convex combinations converges in probability to the same limit. These conditions correspond to an essentially measure-free version of the notion of uniform integrability.
For any family of measurable sets in a probability space, we show that either (i) the family has infinite Vapnik-Chervonenkis (VC) dimension or (ii) for every epsilon > 0 there is a finite partition pi such the pi-boundary of each set has measure at most epsilon. Immediate corollaries include the fact that a family wit…
The paper studies properties of Sliced Wasserstein energy for discrete measures.
problem Optimizing discrete probability measures using Sliced Wasserstein loss.
method Investigates the regularity and optimisation properties of the Sliced Wasserstein energy and its Monte-Carlo approximation.
result Convergence results on the critical points of Monte-Carlo approximations to the Sliced Wasserstein energy.
Three themes of general topology: quotient spaces; absolute retracts; and inverse limits - are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results include: 1) If f: A -> Y is a uniformly continuous map, where X and Y are metric spac…
In this paper, we develop the notion of entropy for uniform hypergraphs via tensor theory. We employ the probability distribution of the generalized singular values, calculated from the higher-order singular value decomposition of the Laplacian tensors, to fit into the Shannon entropy formula. We show that this tensor …
Novel groups exhibit contradictory behaviors with respect to Burnside laws.
problem Understanding probabilistic behaviors of groups under Burnside laws.
method Geometric analysis of relations, information-theoretic coding, combinatorial and probabilistic methods.
result Groups can satisfy Burnside laws with probability 1 for some generating sets and 0 for others.
Extends Langevin dynamics for constrained domains.
problem Optimization of constrained probability measures.
method Mirror mean-field Langevin dynamics (MMFLD).
result Linear convergence guarantees and propagation of chaos results.
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorit…
NUTS mixing time scales as d^(1/4) for Gaussian distributions.
problem Improving the efficiency of the No-U-Turn Sampler (NUTS) for Gaussian distributions.
method Coupling argument leveraging geometric structure of Gaussian concentration, uniformity analysis of NUTS transitions.
result The mixing time of NUTS scales as d^(1/4) for Gaussian distributions, up to logarithmic factors.
Unified high-probability regret bounds for online convex optimisation with randomised gradient estimators.
problem Online convex optimisation with randomised gradient estimators for ℓq-Lipschitz losses. method FTRL with randomised two-point finite-difference gradient estimators based on cone-measure sampling from ℓr-spheres. result Unified high-probability regret bounds for all p,q,r∈[1,∞]. Study uniform learnability of binary classification networks with communication.
problem Learning a network with communication between vertices from uniform ergodic Random Graph Process.
method Introduced structural Rademacher complexity and used martingale method and Marton's coupling.
result Uniform learnability as worst-case theoretical limits for binary classification problems.
Foster and Hart proposed an operational measure of riskiness for discrete random variables. We show that their defining equation has no solution for many common continuous distributions including many uniform distributions, e.g. We show how to extend consistently the definition of riskiness to continuous random variabl…
Uniform Closure Method and Bayes classifier perform similarly in classifying open knots.
problem Classifying knots in open macromolecular chains.
method Used the Bayes MAP classifier and compared it to the Uniform Closure Method.
result Both methods have comparable accuracy and positive predictive value.
Unified framework for uniform signal recovery in nonlinear GCS with 1-bit/quantized measurements.
problem Uniform recovery guarantees for nonlinear generative compressed sensing.
method Unified framework using generalized Lasso and Lipschitz approximation.
result Uniform recovery of all signals in the ball up to an error of ε using approximately O(k/ε^2) samples.
Uniform measures have played a fundamental role in geometric measure theory since they naturally appear as tangent objects. For instance, they were essential in the groundbreaking work of Preiss on the rectifiability of Radon measures. However, relatively little is understood about the structure of general uniform meas…
The study of the geometry of n-uniform measures in Rd has been an important question in many fields of analysis since Preiss' seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is on…
Develops European power option pricing under correlated interest rate and asset processes.
problem Pricing European power options under correlated interest rate and asset processes.
method Martingale method and Girsannov transform.
result Derives European power option pricing formulae under two market assumptions.
The paper develops bounds for predictive values in binary classification.
problem Lack of confidence intervals for positive and negative predictive values.
method Bi-criterion framework and distribution-free large deviation and uniform convergence bounds.
result New bounds for predictive values without relying on concentration inequalities.
New method proves absolute continuity of Wasserstein barycenters on manifolds with lower Ricci curvature bound.
problem Proving absolute continuity of Wasserstein barycenters on manifolds with lower Ricci curvature bound.
method Introducing new displacement functionals exploiting Hessian equality and revisiting Souslin space theory, Dunford-Pettis theorem, and de la Vallée Poussin criterion for uniform integrability.
result Absolute continuity of Wasserstein barycenters is established for a general class of manifolds with lower Ricci curvature bound.
Identifying statistical dependence between the features and the label is a fundamental problem in supervised learning. This paper presents a framework for estimating dependence between numerical features and a categorical label using generalized Gini distance, an energy distance in reproducing kernel Hilbert spaces (RK…
Estimate arrival times in random recursive trees using iterated Jordan centralities.
problem Estimate arrival times in random recursive trees.
method Pointwise approach using iterated Jordan centralities.
result Tail bounds for relative estimation error.
Algorithm removes leaves to find root in uniform trees.
problem Finding the root in large uniform attachment trees.
method Leaf-stripping algorithm recursively removes leaves.
result Set of remaining vertices contains the root with high probability.
New algorithm FLUTE achieves uniform-PAC convergence in RL with linear approx.
problem RL with linear function approximation lacks uniform-PAC guarantees.
method FLUTE algorithm with minimax value function estimator and multi-level partition scheme.
result Uniform-PAC convergence to optimal policy with high probability.
Develops non-standard analysis for coherent risk estimation.
problem Estimating coherent risk measures in financial contexts.
method Non-standard analysis, hyperfinite representations, discrete Kusuoka formulae, plug-in asymptotics.
result Uniform almost sure consistency and asymptotic normality of spectral plug-in estimators.
Study critical exponents on hyperbolic surfaces with long boundaries using Weil-Petersson measures.
problem Analyzing critical exponents on hyperbolic surfaces with long boundaries.
method Using spine graph construction and comparing normalized Weil-Petersson and Kontsevich measures.
result Asymptotic convergence-in-mean result of normalized Weil-Petersson measures to normalized Kontsevich measures.
Starting with the work of Preiss on the geometry of measures, the classification of uniform measures in Rd has remained open, except for d=1 and for compactly supported measures in d=2, and for codimension 1. In this paper we study 1-dimensional measures in Rd for all d and classify unif…
New algorithm for estimating multivariate quantiles using stochastic optimal transport.
problem Estimating multivariate quantiles from data.
method Stochastic algorithm for entropic optimal transport in Banach spaces, using Fourier coefficients.
result Almost sure convergence of the stochastic algorithm in infinite-dimensional Banach spaces.
Study approximates probability measures using structured classes of functions.
problem Approximating probability measures in Wasserstein-p distance. method Structured classes of approximators for functions in Lp(Ω), transferring to measures in Wp(Ω). result Linear rate approximation for measures with densities bounded away from zero.
Geodesics found in spacetime satisfy curvature conditions.
problem Finding geodesics in spacetime satisfying specific curvature conditions.
method Proving existence of geodesics with entropic semiconvexity and uniform L∞ densities. result Existence of geodesics satisfying the timelike curvature-dimension condition.
PAC learning sample complexity is decidable with finite support bounds.
problem Determining the exact sample complexity for PAC learning concepts.
method Observation and proof of decidability with a-priori bounds.
result Sample complexity can be exactly determined for various concepts with finite support bounds.
A new test evaluates risk estimation accuracy using probability integral transform.
problem Measuring the accuracy of financial market risk estimations.
method Probability Integral Transform (PIT) of ex post realized returns against ex ante probability distributions.
result The new test shows the importance of capturing the dynamic of financial markets.
Recently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly rather high-dimensional partial differential equations (PDEs). Very recently, there have also been a number of rigorous mathematical results …
New method uses graphene transistors for efficient non-uniform random number generation.
problem Generating non-uniform random variates efficiently.
method GFET-based hardware non-uniform random number generator.
result Demonstrated speedup of Monte Carlo integration by up to 2x.
We show that for any weakly convergent sequence of ergodic SL2(R)-invariant probability measures on a stratum of unit-area translation surfaces, the corresponding Siegel-Veech constants converge to the Siegel-Veech constant of the limit measure. Together with a measure equidistribution result due to Eskin-M…
Study finds root vertex in large networks with high probability.
problem Finding the root vertex in large growing networks.
method Constructs confidence sets for the root vertex in various random network models.
result Confidence sets of size independent of the number of vertices contain the root vertex with high probability.
Improved matrix completion for non-uniformly sampled data.
problem Estimating unobserved entries in a matrix with varying sampling probabilities.
method Developed entry-specific bounds for low-rank matrix completion under structured non-uniform sampling.
result Error bounds for each entry match minimax lower bounds under certain conditions.
The Statistical Learning Theory (SLT) provides the theoretical guarantees for supervised machine learning based on the Empirical Risk Minimization Principle (ERMP). Such principle defines an upper bound to ensure the uniform convergence of the empirical risk Remp(f), i.e., the error measured on a given data sample, to …
Study sets limits for detecting a subhypergraph in uniform hypergraphs.
problem Recovering a subhypergraph from a uniform hypergraph with different edge probabilities.
method Information-theoretic analysis for weak and exact recovery.
result Sharp conditions for weak or exact recovery of the subhypergraph.
One fundamental goal in any learning algorithm is to mitigate its risk for overfitting. Mathematically, this requires that the learning algorithm enjoys a small generalization risk, which is defined either in expectation or in probability. Both types of generalization are commonly used in the literature. For instance, …
According to a classical result of E.~Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the "hyperbolic" toric Kähler-Einstein equation eΦ=detD2Φ on proper convex cones. We…
The paper analyzes greedy algorithms for MMD minimization, showing their efficiency and approximation error.
problem Minimizing Maximum Mean Discrepancy (MMD) for probability measure quantization.
method Iterative algorithms including kernel herding, greedy MMD minimization, and Sequential Bayesian Quadrature (SBQ).
result The greedy algorithms have a lower approximation error than SBQ, but are significantly faster.
Uniform convergence of metrics on surfaces with bounded curvature measures proved.
problem Proving uniform convergence of metrics on Alexandrov surfaces with bounded integral curvature.
method Weak convergence of measures and analytic approximation of metrics.
result Uniform convergence of metrics on Alexandrov surfaces proved.
A novel kernel-based test detects equality versus singularity of two probability measures.
problem Detecting equality versus singularity of two probability distributions.
method Combines kernel mean and kernel covariance embeddings to construct a likelihood ratio test statistic.
result The test statistic satisfies a '0/\infty' law, vanishing under the null and diverging under the alternative.