Related papers: Level Sets Based Distances for Probability Measure…

In this article, we study the properties of a class of functional spaces which arise from the investigation of nonlinear differential equations. We establish some integral inequalities then by applying these inequalities, we prove some…

We study distributional similarity measures for the purpose of improving probability estimation for unseen cooccurrences. Our contributions are three-fold: an empirical comparison of a broad range of measures; a classification of similarity…

Distribution function is essential in statistical inference, and connected with samples to form a directed closed loop by the correspondence theorem in measure theory and the Glivenko-Cantelli and Donsker properties. This connection creates…

Hypothesis testing in high dimensional data is a notoriously difficult problem without direct access to competing models' likelihood functions. This paper argues that statistical divergences can be used to quantify the difference between…

Image analysis frequently deals with shape estimation and image reconstruction. The ob jects of interest in these problems may be thought of as random sets, and one is interested in finding a representative, or expected, set. We consider a…

Distance function is a main metrics of measuring the affinity between two data points in machine learning. Extant distance functions often provide unreachable distance values in real applications. This can lead to incorrect measure of the…

We construct classifiers for multivariate and functional data. Our approach is based on a kind of distance between data points and classes. The distance measure needs to be robust to outliers and invariant to linear transformations of the…

In this paper, we study properties of certain risk measures associated with acceptance sets. These sets describe regulatory preconditions that have to be fulfilled by financial institutions to pass a given acceptance test. If the financial…

This paper gives a systematic account of various metrics on probability distributions (states) and on predicates. These metrics are described in a uniform manner using the validity relation between states and predicates. The standard…

Computing the reachability probability in infinite state probabilistic models has been the topic of numerous works. Here we introduce a new property called \emph{divergence} that when satisfied allows to compute reachability probabilities…

A functional data depth provides a center-outward ordering criterion which allows the definition of measures such as median, trimmed means, central regions or ranks in a functional framework. A functional data depth can be global or local.…

We develop a formalism to study the use of Level Set Method (LSM) in the investigation of evolution of observables in terms of parameters of the Hamiltonian, both of the system itself and the control part. A simple example with an analytic…

Machine learning models $-$ now commonly developed to screen, diagnose, or predict health conditions $-$ are evaluated with a variety of performance metrics. An important first step in assessing the practical utility of a model is to…

Though neural network models demonstrate impressive performance, we do not understand exactly how these black-box models make individual predictions. This drawback has led to substantial research devoted to understand these models in areas…

A new notion of metric differentiability of set-valued functions at a point is introduced in terms of right and left limits of special set-valued metric divided differences of first order. A local metric linear approximant of a metrically…

I explore the use of sets of probability measures as a representation of uncertainty.

When observations are organized into groups where commonalties exist amongst them, the dependent random measures can be an ideal choice for modeling. One of the propositions of the dependent random measures is that the atoms of the…

Modelling functions of sets, or equivalently, permutation-invariant functions, is a long-standing challenge in machine learning. Deep Sets is a popular method which is known to be a universal approximator for continuous set functions. We…

This work proposes a view of probability as a relative measure rather than an absolute one. To demonstrate this concept, we focus on finite outcome spaces and develop three fundamental axioms that establish requirements for relative…

Recently, a framework for application-oriented optimal experiment design has been introduced. In this context, the distance of the estimated system from the true one is measured in terms of a particular end-performance metric. This…