Related papers: Asymptotic data analysis on manifolds

M-estimation, aka empirical risk minimization, is at the heart of statistics and machine learning: Classification, regression, location estimation, etc. Asymptotic theory is well understood when the loss satisfies some smoothness…

Classical mathematical statistics deals with models that are parametrized by a Euclidean, i.e. finite dimensional, parameter. Quite often such models have been and still are chosen in practical situations for their mathematical simplicity…

This work considers the asymptotic behavior of the distance between two sample covariance matrices (SCM). A general result is provided for a class of functionals that can be expressed as sums of traces of functions that are separately…

This paper introduces a novel approach to statistics and data analysis, departing from the conventional assumption of data residing in Euclidean space to consider a Riemannian Manifold. The challenge lies in the absence of vector space…

We study the asymptotic properties of geodesically convex $M$-estimation on non-linear spaces. Namely, we prove that under very minimal assumptions besides geodesic convexity of the cost function, one can obtain consistency and asymptotic…

In time series analysis, statistics based on collections of estimators computed from sub-samples play a crucial role in an increasing variety of important applications. Proving results about the joint asymptotic distribution of such…

For functional data lying on an unknown nonlinear low-dimensional space, we study manifold learning and introduce the notions of manifold mean, manifold modes of functional variation and of functional manifold components. These constitute…

We consider an estimation problem of expected functionals of a general random element that values in a metric space. If the functional forms an explicit function of some unknown parameters, we can estimate it by plugging-in a suitable…

Regression on manifolds, and, more broadly, statistics on manifolds, has garnered significant importance in recent years due to the vast number of applications for non Euclidean data. Circular data is a classic example, but so is data in…

The paper aims at finding widely and smoothly defined nonparametric location and scatter functionals. As a convenient vehicle, maximum likelihood estimation of the location vector m and scatter matrix S of an elliptically symmetric t…

Many functionals of interest in statistics and machine learning can be written as minimizers of expected loss functions. Such functionals are called $M$-estimands, and can be estimated by $M$-estimators -- minimizers of empirical average…

For data living in a manifold $M\subseteq \mathbb{R}^m$ and a point $p\in M$ we consider a statistic $U_{k,n}$ which estimates the variance of the angle between pairs of vectors $X_i-p$ and $X_j-p$, for data points $X_i$, $X_j$, near $p$,…

We consider the consensual distributed optimization problem in the Riemannian context. Specifically, the minimization of a sum of functions form is studied where each individual function in the sum is located at the node of a network. An…

We introduce a regression model for data on non-linear manifolds. The model describes the relation between a set of manifold valued observations, such as shapes of anatomical objects, and Euclidean explanatory variables. The approach is…

Asymptotic mean value properties, their converse and some other related results are considered for solutions to the $m$-dimensional Helmholtz equation (metaharmonic functions) and solutions to its modified counterpart (panharmonic…

We analytically compute asymptotic expansions of a 1-dimensional sub-manifold of stable and unstable manifolds in a 4-dimensional symplectic mapping by using the method called asymptotic expansions beyond all orders. This method enables us…

Let $\alpha_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)$ be the predictive distributions of a sequence $(X_1,X_2,\ldots)$ of $p$-dimensional random vectors. Suppose $$\alpha_n= \mathcal{N} _p (M_n,Q_n)$$ where…

The object of study is the problem of testing for uniformity of the multinomial distribution. We consider tests based on symmetric statistics, defined as the sum of some function of cell-frequencies. Mainly, attention is focused on the…

Asymptotic expansions are derived for the tail distribution of the product of two correlated normal random variables with non-zero means and arbitrary variances, and more generally the sum of independent copies of such random variables.…

Geometric quantiles are popular location functionals to build rank-based statistical procedures in multivariate settings. They are obtained through the minimization of a non-smooth convex objective function. As a result, the singularity of…