arXiv research

New algorithm tackles subgroup fairness in AI with multiple sensitive attributes.

The paper bounds the expectation of empirical processes indexed by Hölder classes.

Derives integral representations for a Lévy process and its extremum, hitting time, with fast evaluation.

Develops a new essential supremum concept for financial models.

The paper establishes conditions for Bayesian consistency in supremum metric.

Study optimal control of diffusion processes with infimum or supremum costs.

We study the pointwise supremum of convex integral functionals $\mathcal{I}_{f,γ}(ξ)= \sup_{Q} \left( \int_Ωf(ω,ξ(ω))Q(dω)-γ(Q)\right)$ on $L^\infty(Ω,\mathcal{F},\mathbb{P})$ where $f:Ω\times\mathbb{R}\rightarrow\overline{\mathbb{R}}$ is a proper normal convex integrand, $γ$ is a proper convex function on the set of p…

Deep belief networks can approximate any multivariate density with binary hidden units.

The paper evaluates functions of stable Lévy processes and their extrema efficiently.

In this note we find a formula for the supremum distribution of spectrally positive or negative Lévy processes with a broken linear drift. This gives formulas for ruin probabilities in the case when two insurance companies (or two branches of the same company) divide between them both claims and premia in some specifie…

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consis…

We obtain supremum of the k-th normalized Steklov eigenvalues of all rotational symmetric conformal metrics on the cylinder with k>1. The case k=1 for all conformal metrics has been completely solved by Fraser and Schoen. We give geometric description in terms of minimal surfaces for metrics attaining the supremum. We …

On a Fano manifold M we study the supremum of the possible t such that there is a Kähler metric in c_1(M) with Ricci curvature bounded below by t. This is shown to be the same as the maximum existence time of Aubin's continuity path for finding Kähler-Einstein metrics. We show that on P^2 blown up in one point this sup…

Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.

Let $L_g$ be the subcritical GJMS operator on an even-dimensional compact manifold $(X, g)$ and consider the zeta-regularized trace $\mathrm{Tr}_ζ(L_g^{-1})$ of its inverse. We show that if $\ker L_g = 0$, then the supremum of this quantity, taken over all metrics $g$ of fixed volume in the conformal class, is always g…

New algorithms for efficient return distribution approximation in reinforcement learning.

This work introduces a new metric for comparing imprecise probability models.

Let $M$ be a compact manifold of dimension $n$. In this paper, we introduce the {\em Mass Function} $a \geq 0 \mapsto \xp{M}{a}$ (resp. $a \geq 0 \mapsto \xm{M}{a}$) which is defined as the supremum (resp. infimum) of the masses of all metrics on $M$ whose Yamabe constant is larger than $a$ and which are flat on a ball…

Paper relaxes triangle inequality for KL divergence between Gaussian distributions.

Paper examines stability of Bayesian posterior measures using integral probability metrics.

We study the minimax optimal rates for estimating a range of Integral Probability Metrics (IPMs) between two unknown probability measures, based on $n$ independent samples from them. Curiously, we show that estimating the IPM itself between probability measures, is not significantly easier than estimating the probabili…

We study the set of volumes of constant scalar curvature one metrics on an atoroidal three-manifold.The infinum of this set is believed to be attained at a hyperbolic metric. We prove that the supremum of this set is always infinity. The technique is: minimal surfaces, Thurston norm in homology and new conformal invari…

A nonparametric two-sample test using a parametric integral probability metric

We prove that for a so-called sticky process $S$ there exists an equivalent probability $Q$ and a $Q$-martingale $\tilde{S}$ that is arbitrarily close to $S$ in $L^p(Q)$ norm. For continuous $S$, $\tilde{S}$ can be chosen arbitrarily close to $S$ in supremum norm. In the case where $S$ is a local martingale we may choo…

New bounds use IPMs to improve generalization in machine learning.

We study the systole of a random surface, where by a random surface we mean a surface constructed by randomly gluing together an even number of triangles. We study two types of metrics on these surfaces, the first one coming from using ideal hyperbolic triangles and the second one using triangles that carry a given Rie…

The paper provides statistical guarantees for generative models using dimension reduction.

A new IPM uses ReLU networks to measure probability discrepancies.

Quantum probability metrics improve distribution comparison in high dimensions.

We study coherent risk measures which are time-consistent for multiple filtrations. We show that a coherent risk measure is time-consistent for every filtration if and only if it is one of four main types. Furthermore, if the risk measure is strictly monotone it is linear, and if the reference probability space is not …

Develops a new divergence framework that combines $f$-divergences and IPMs.

We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding ve…

The paper proposes a new method for covariate balancing using IPM to improve causal inference.

In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincaré metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the p…

Gaussian process (GP) regression is a powerful interpolation technique due to its flexibility in capturing non-linearity. In this paper, we provide a general framework for understanding the frequentist coverage of point-wise and simultaneous Bayesian credible sets in GP regression. As an intermediate result, we develop…

Bayesian approach to robust risk measures under model uncertainty.

Consider a financial market in which an agent trades with utility-induced restrictions on wealth. By introducing a general convex-analytic framework which includes the class of umbrella wedges in certain Riesz spaces and faces of convex sets (consisting of probability measures), together with a duality theory for polar…

Let $M$ be a compact connected manifold of dimension $n$ endowed with a conformal class $C$ of Riemannian metrics of volume one. For any integer $k\geq0$, we consider the conformal invariant $λ_k ^c (C)$ defined as the supremum of the $k$-th eigenvalue $λ_k (g)$ of the Laplace-Beltrami operator $Δ_g$, where $g$ runs ov…

We show how to compute lower bounds for the supremum Bayes error if the class-conditional distributions must satisfy moment constraints, where the supremum is with respect to the unknown class-conditional distributions. Our approach makes use of Curto and Fialkow's solutions for the truncated moment problem. The lower …

Upper bound on expected supremum of Bernoulli process.

We consider the equivariant Yamabe problem, i.e. the Yamabe problem on the space of G-invariant metrics for a compact Lie group G. The G-Yamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume G-invariant metrics minimizing the total scalar curvature functional in their …

We consider the optimal prediction problem of stopping a spectrally negative Lévy process as close as possible to a given distance $b \geq 0$ from its ultimate supremum, under a squared error penalty function. Under some mild conditions, the solution is fully and explicitly characterised in terms of scale functions. We…

The paper explores statistical and topological properties of sliced probability divergences.

For every smooth del Pezzo surface $S$, smooth curve $C\in|-K_{S}|$ and $β\in(0,1]$, we compute the $α$-invariant of Tian $α(S,(1-β)C)$ and prove the existence of Kähler--Einstein metrics on $S$ with edge singularities along $C$ of angle $2πβ$ for $β$ in certain interval. In particular we give lower bounds for the inva…

The systole of a compact non simply connected Riemannian manifold is the smallest length of a non-contractible closed curve ; the systolic ratio is the quotient $(\mathrm{systole})^n/\mathrm{volume}$. Its supremum on the set of all the riemannian metrics, is known to be finite for a large class of manifolds, including …

The Yamabe invariant Y(M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar-curvature Riemannian metrics g on M. (To be precise, one only considers those constant-scalar-curvature metrics which are Yamabe minimizers, but this technicality does not, e.g. affect …

We propose a robust risk measurement approach that minimizes the expectation of overestimation plus underestimation costs. We consider uncertainty by taking the supremum over a collection of probability measures, relating our approach to dual sets in the representation of coherent risk measures. We provide results that…

New framework improves experimental design using integral probability metrics.