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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…
Error bounds, which refer to inequalities that bound the distance of vectors in a test set to a given set by a residual function, have proven to be extremely useful in analyzing the convergence rates of a host of iterative methods for…
The goal of ordinal embedding is to represent items as points in a low-dimensional Euclidean space given a set of constraints in the form of distance comparisons like "item $i$ is closer to item $j$ than item $k$". Ordinal constraints like…
We provide a novel characterization of semiparametric efficiency in a generic supervised learning setting where the outcome mean function -- defined as the conditional expectation of the outcome of interest given the other observed…
During the training of networks for distance metric learning, minimizers of the typical loss functions can be considered as "feasible points" satisfying a set of constraints imposed by the training data. To this end, we reformulate distance…
Boundary estimation in images and videos has been a very active topic of research, and organizing visual information into boundaries and segments is believed to be a corner stone of visual perception. While prior work has focused on…
A well-defined distance on the parameter space is key to evaluating estimators, ensuring consistency, and building confidence sets. While there are typically standard distances to adopt in a continuous space, this is not the case for…
Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued…
This paper presents a distance function between sets based on an average of distances between their elements. The distance function is a metric if the sets are non-empty finite subsets of a metric space. It can be applied to produce various…
We study constrained selection sets of random closed sets defined on a non-atomic probability space. Given a random interval $Y=[y_L,y_U]$ and scalar constraints on the expectation or the median of admissible selections, we characterize the…
The Hausdorff distance is a measure of (dis-)similarity between two sets which is widely used in various applications. Most of the applied literature is devoted to the computation for sets consisting of a finite number of points. This has…
This work proposes an algorithm to bound the minimum distance between points on trajectories of a dynamical system and points on an unsafe set. Prior work on certifying safety of trajectories includes barrier and density methods, which do…
The problem of searching for a model-based scene interpretation is analyzed within a probabilistic framework. Object models are formulated as generative models for range data of the scene. A new statistical criterion, the truncated object…
While conformal predictors reap the benefits of rigorous statistical guarantees on their error frequency, the size of their corresponding prediction sets is critical to their practical utility. Unfortunately, there is currently a lack of…
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…
The purpose of this paper is to give a survey on the notions of distance between subsets either of a metric space or of a measure space, including definitions, a classification, and a discussion of the best-known distance functions, which…
This paper revisits a fundamental problem in statistical inference from a non-asymptotic theoretical viewpoint $\unicode{x2013}$ the construction of confidence sets. We establish a finite-sample bound for the estimator, characterizing its…
The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step…
The geometric problem of estimating an unknown compact convex set from evaluations of its support function arises in a range of scientific and engineering applications. Traditional approaches typically rely on estimators that minimize the…
In this paper we focus on the linear functionals defining an approximate version of the gradient of a function. These functionals are often used when dealing with optimization problems where the computation of the gradient of the objective…