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The assumption of elliptical symmetry has an important role in many theoretical developments and applications, hence it is of primary importance to be able to test whether that assumption actually holds true or not. Various tests have been…
We study random points on the real line generated by the eigenvalues in unitary invariant random matrix ensembles or by more general repulsive particle systems. As the number of points tends to infinity, we prove convergence of the…
This paper considers the problem of regression analysis with random covariance matrix as outcome and Euclidean covariates in the framework of Fr\'echet regression on the Bures-Wasserstein manifold. Such regression problems have many…
This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a…
We consider a fixed angle inverse scattering problem in the presence of a known Riemannian metric. First, assuming a no caustics condition, we study the direct problem by utilizing the progressing wave expansion. Under a symmetry assumption…
We propose a novel method for testing isotropy of a three-dimensional distribution using Shannon entropy. We test the method on some Monte Carlo simulations of isotropic and anisotropic distributions and find that the method can effectively…
The density ratio of two probability distributions is one of the fundamental tools in mathematical and computational statistics and machine learning, and it has a variety of known applications. Therefore, density ratio estimation from…
We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the…
Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraints, and hence retraction maps are used to…
Nowadays stochastic computer simulations with both numeral and distribution inputs are widely used to mimic complex systems which contain a great deal of uncertainty. This paper studies the design and analysis issues of such computer…
Normalizing flows have shown great promise for modelling flexible probability distributions in a computationally tractable way. However, whilst data is often naturally described on Riemannian manifolds such as spheres, torii, and hyperbolic…
In this article, we establish the mathematical foundations for modeling the randomness of shapes and conducting statistical inference on shapes using the smooth Euler characteristic transform. Based on these foundations, we propose two…
In statistics on manifolds, the notion of the mean of a probability distribution becomes more involved than in a linear space. Several location statistics have been proposed, which reduce to the ordinary mean in Euclidean space. A…
In this paper we consider adaptive sampling's local-feature size, used in surface reconstruction and geometric inference, with respect to an arbitrary landmark set rather than the medial axis and relate it to a path-based adaptive metric on…
Monotonicity is a key qualitative prediction of a wide array of economic models derived via robust comparative statics. It is therefore important to design effective and practical econometric methods for testing this prediction in empirical…
We study a wide class of metrics in a Lebesgue space with a standard measure, the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the…
Non-Euclidean data that are indexed with a scalar predictor such as time are increasingly encountered in data applications, while statistical methodology and theory for such random objects are not well developed yet. To address the need for…
Geometric frameworks for analyzing curves are common in applications as they focus on invariant features and provide visually satisfying solutions to standard problems such as computing invariant distances, averaging curves, or registering…
We study barycenters in the space of probability measures on a Riemannian manifold, equipped with the Wasserstein metric. Under reasonable assumptions, we establish absolute continuity of the barycenter of general measures $\Omega \in…
We study a generalization of the Fr\'echet mean on metric spaces, which we call $\phi$-means. Our generalization is indexed by a convex function $\phi$. We find necessary and sufficient conditions for $\phi$-means to be finite and provide a…