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We propose a novel Riemannian geometric framework for variational inference in Bayesian models based on the nonparametric Fisher-Rao metric on the manifold of probability density functions. Under the square-root density representation, the…

Mean density of lower dimensional random closed sets, as well as the mean boundary density of full dimensional random sets, and their estimation are of great interest in many real applications. Only partial results are available so far in…

We address the problem of testing hypotheses about a specific value of the Fr\'echet mean in metric spaces, extending classical mean testing from Euclidean spaces to more general settings. We extend an Euclidean testing procedure…

This article develops nonparametric inference procedures for estimation and testing problems for means on manifolds. A central limit theorem for Frechet sample means is derived leading to an asymptotic distribution theory of intrinsic…

The Central Limit Theorem (CLT) is one of the most fundamental results in statistics. It states that the standardized sample mean of a sequence of $n$ mutually independent and identically distributed random variables with finite first and…

We prove a central limit theorem (CLT) for the Frechet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Frechet mean is unique. Previous general CLT results in this…

This paper introduces a novel extension of Fr\'{e}chet means, called \textit{generalized Fr\'{e}chet means} as a comprehensive framework for characterizing features in probability distributions in general topological spaces. The generalized…

The Fr\'echet distance is a popular distance measure for curves which naturally lends itself to fundamental computational tasks, such as clustering, nearest-neighbor searching, and spherical range searching in the corresponding metric…

In this paper, we study fluctuations of conditionally centered statistics of the form $$N^{-1/2}\sum_{i=1}^N c_i(g(\sigma_i)-\mathbb{E}_N[g(\sigma_i)|\sigma_j,j\neq i])$$ where $(\sigma_1,\ldots ,\sigma_N)$ are sampled from a dependent…

We discuss the generic properties of a general, smoothly varying, spherically symmetric mass distribution $\mathcal{D}(r,\theta)$, with no cosmological term ($\theta$ is a length scale parameter). Observing these constraints, we show that…

Non-Gaussianities of dynamical origin are disentangled from primordial ones using the formalism of large deviation statistics with spherical collapse dynamics. This is achieved by relying on accurate analytical predictions for the one-point…

Variation of empirical Fr\'echet means on a metric space with curvature bounded above is encoded via random fields indexed by unit tangent vectors. A central limit theorem shows these random tangent fields converge to a Gaussian such field…

We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus $g$, and are non-compact…

Fr\'echet regression extends the principles of linear regression to accommodate responses valued in generic metric spaces. While this approach has primarily focused on exploring relationships between Euclidean predictors and non-Euclidean…

In Euclidean spaces, the empirical mean vector as an estimator of the population mean is known to have polynomial concentration unless a strong tail assumption is imposed on the underlying probability measure. The idea of median-of-means…

Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit $d$-sphere ($d\ge 2$). We investigate the distribution of their defect i.e., the difference between the measure of positive and negative regions. Marinucci and…

We study a random partial covering model on the $(d-1)$-dimensional unit sphere, where $N$ spherical caps are placed independently and uniformly at random, each covering a surface fraction of $1/N$. This model provides a continuous…

It has recently been shown that there are substantial differences in the regularity behavior of the empirical process based on scalar diffusions as compared to the classical empirical process, due to the existence of diffusion local time.…

In 2010, Shiffman and Zelditch proved a central limit theorem (CLT) for smooth statistics of Gaussian random zeros in codimension one over compact K\"ahler manifolds. They raised the question of whether this result admits a two-fold…

In this article we define new Fr\`Echet features for random cumulative distribution functions using contrast. These contrasts allow to construct Wasserstein costs and our new features minimize the average costs as the Fr\`Echet mean…