arXiv research

Introduces bounded scale measure and generalizes property A.

Study risk-sensitive reinforcement learning with Lipschitz dynamic risk measures, establishing regret bounds.

The paper calculates extreme measures in continuous time conic finance.

A new framework tightens risk measure confidence bounds.

The paper bounds payoffs and option prices in discrete models.

Paper develops a new generalization bound using PAC-Bayes theory and Gibbs distributions.

In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD*(K,N) we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of CD*(K,N) also for intermediate times and in addition the measures along these geode…

New PAC-Bayes bounds derived using Legendre transform and f-divergences.

New risk measures control subgroup imbalances, improving PAC-Bayesian bounds.

This paper presents a unified approach based on Wasserstein distance to derive concentration bounds for empirical estimates for two broad classes of risk measures defined in the paper. The classes of risk measures introduced include as special cases well known risk measures from the finance literature such as condition…

We estimate risk measures in Markov cost processes with lower and upper bounds.

In the celebrated book entitled Metric Structures for Riemannian and Non-Riemannian Spaces, so-called Green Book, Gromov presented a problem regarding a metric measure space. Gromov posed the question Bound the expansion coefficient from below in terms of the observable diameter. The overall aim of the current study is…

We study generalizations of Reifenberg's Theorem for measures in $\mathbb R^n$ under assumptions on the Jones' $β$-numbers, which appropriately measure how close the support is to being contained in a subspace. Our main results, which holds for general measures without density assumptions, give effective measure bounds…

Bounds on factual and counterfactual distributions under measurement error in discrete models.

Sharp upper bounds on inscribed radius for metric spaces with convex boundary.

Study approximates probability measures using structured classes of functions.

This paper fills in local bounds for Spearman's footrule and Gini's gamma measures of association.

Upper bound for max-sliced 2-Wasserstein distance between measures.

If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, not necessarily altern…

Paper finds lower bounds for eigenvalues of Bi-drifted Laplacian on smooth metric measure spaces.

Innovative inequalities for divergences with applications in PAC-Bayesian bounds and Monte Carlo.

Study shows infinite-dimensional third bounded cohomology for non-orientable surfaces.

We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abe…

We study convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no-arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized …

We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embedded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal quantization, and learning theory, we derive new probabilistic bounds for the per…

A local cut point is by definition a point that disconnectes its sufficiently small neighborhood. We show that there exists an upper bound for the degree of a local cut point in a metric measure space satisfying the generalized Bishop--Gromov inequality. As a corollary, we obtain an upper bound for the number of ends o…

We construct geodesics in the Wasserstein space of probability measure along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic. Using these geodesics we show that a local Poincaré inequality and the measure contraction property follow from t…

The paper proves geometric comparisons on metric measure spaces with integral Bakry-Émery Ricci tensor bounds.

On a Riemannian metric-measure space, we establish an Alexandrov-Bakelman-Pucci type measure estimate connecting Bakry-Émery Ricci curvature lower bound, modified Laplacian and the measure of certain special sets. We apply this estimate to prove Harnack inequalities for the modified Laplacian operator and fully non-lin…

New algorithms reduce regret in online MDPs by adapting to data and variance.

Proposes a learned Bayesian Cramér-Rao bound for unknown measurement models.

We prove that if $(X,\mathsf d,\mathfrak m)$ is an essentially non-branching metric measure space with $\mathfrak m(X)=1$, having Ricci curvature bounded from below by $K$ and dimension bounded from above by $N \in (1,\infty)$, understood as a synthetic condition called Measure-Contraction property, then a sharp isoper…

In an $L_\infty$-framework, we present a few extension theorems for linear operators. We focus the attention on majorant preserving and sandwich preserving types of extensions. These results are then applied to the study of price systems derived by a reasonable restriction of the class of equivalent martingale measures…

Causal discovery algorithms infer causal relations from data based on several assumptions, including notably the absence of measurement error. However, this assumption is most likely violated in practical applications, which may result in erroneous, irreproducible results. In this work we show how to obtain an upper bo…

Uniform convergence of metrics on surfaces with bounded curvature measures proved.

New algorithm recovers sparse binary vectors from generalized linear measurements efficiently.

A new method optimizes robustness measures under input uncertainty using randomized Gaussian process upper confidence bound.

Sharp bounds found for various risk measures using generalized FGM copulas.

Study connects curvature bounds to map existence and flow solutions.

Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy…

It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a…

Study bounds the measure of zero sets of Neumann Laplace eigenfunctions.

Study shows volume constraints lead to isoperimetric constant bounds in specific metric spaces.

This paper, focusing on the growth rate of the measure, gives pointwise bounds of solutions of eigenvalue equations of the Laplace-Beltrami operator on noncompact Riemannian manifolds.

In this paper we present some bounds of Hausdorff measures of objects definable in o-minimal structures: sets, fibers of maps, inverse images of curves of maps, etc. Moreover, we also give some explicit bounds for semi-algebraic or semi-Pfaffian cases, which depend only on the combinatoric data representing the objects…

We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence…

Paper investigates conditions for independence of weak gradients on metric spaces.

Measure contraction properties $MCP(K,N)$ are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension $N$, then $MCP(K,N)$ is equivalent to Ricci curvature bounded below by $K$. On the other hand, it was ob…