Related papers: Necessary and sufficient condition for the existen…

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…

Fr\'echet means, conceptually appealing, generalize the Euclidean expectation to general metric spaces. We explore how well Fr\'echet means can be estimated from independent and identically distributed samples and uncover a fundamental…

For $1\le p \le \infty$, the Fr\'echet $p$-mean of a probability measure on a metric space is an important notion of central tendency that generalizes the usual notions in the real line of mean ($p=2$) and median ($p=1$). In this work we…

In this paper, we study the Dvoretzky covering problem with non-uniformly distributed centers. When the probability law of the centers admits an absolutely continuous density which satisfies a regular condition on the set of essential…

We derive the necessary and sufficient condition for almost sure convergence of the sequence of measurable functions, and consider some applications in the theory of Fourier series and in the theory of random fields.

Fr\'echet means are a popular type of average for non-Euclidean datasets, defined as those points which minimise the average squared distance to a set of data points. We consider the behaviour of sample Fr\'echet means on normed spaces…

In the paper, we first prove a sufficient condition for the Riemann hypothesis which involves the order of magnitude of the partial sum of the Liouville function. Then we show a formula which is curiously related to the proved sufficient…

For sets $\mathcal Q$ and $\mathcal Y$, the generalized Fr\'echet mean $m \in \mathcal Q$ of a random variable $Y$, which has values in $\mathcal Y$, is any minimizer of $q\mapsto \mathbb E[\mathfrak c(q,Y)]$, where $\mathfrak c \colon…

A measure $\mu$ on the unit circle $\mathbb{T}$ belongs to Steklov class $\mathcal{S}$ if its density $w$ with respect to the Lebesgue measure on $\mathbb{T}$ is strictly positive: $\inf_{\mathbb{T}} w > 0$. Let $\mu$, $\mu_{-1}$ be…

We compute higher derivatives of the Fr\'{e}chet function on spheres with an absolutely continuous and rotationally symmetric probability distribution. Consequences include (i)~a practical condition to test if the mode of the symmetric…

For most optimisation methods an essential assumption is the vector space structure of the feasible set. This condition is not fulfilled if we consider optimisation problems over the sphere. We present an algorithm for solving a special…

When dealing with certain mathematical problems, it is sometimes necessary to show that some function induces a metric on a certain space. When this function is not a well renowned example of a distance, one has to develop very particular…

We provide new necessary and sufficient conditions for the convergence of positive series developing Bertran-De Morgan and Cauchy type tests given in [M. Martin, Bull. Amer. Math. Soc. 47(1941), 452-457] and [L. Bourchtein et al, Int. J.…

The Fr\'echet mean is an important statistical summary and measure of centrality of data; it has been defined and studied for persistent homology captured by persistence diagrams. However, the complicated geometry of the space of…

We study the stochastic convergence of the Ces\`{a}ro mean of a sequence of random variables. These arise naturally in statistical problems that have a sequential component, where the sequence of random variables is typically derived from a…

In this paper, we study the exterior Dirichlet problem for the fully nonlinear elliptic equation $f(\lambda(D^{2}u))=1$. We obtain the necessary and sufficient conditions of existence of radial solutions with prescribed asymptotic behavior…

A new class of statistical deformable models is introduced to study high-dimensional curves or images. In addition to the standard measurement error term, these deformable models include an extra error term modeling the individual…

We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an $\ell^p$ condition and a generalized bounded variation condition. This latter condition requires that a sequence can be…

The Fr\'echet distance is a popular similarity measure that is well-understood for polygonal curves in $\mathbb{R}^d$: near-quadratic time algorithms exist, and conditional lower bounds suggest that these results cannot be improved…

To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Fr\'echet mean. In this…