arXiv subjects

Andrea Aveni

Publications and source records attributed to Andrea Aveni.

4 recordsLinked to original sources

arXiv2026

Uniform Consistency of Generalized Fréchet Means

Loss-based notions of centre on nonlinear spaces range from the Fréchet mean and power means to the geometric median and, in a limiting sense, the Chebyshev centre. To use such summaries statistically, one first needs a law of large numbers that remains valid beyond smooth manifolds and beyond a fixed choice of loss. We study generalized Fréchet means on metric spaces with the Heine--Borel property, obtained by replacing squared distance with a convex loss under a mild exponential-growth condition. We prove existence and compactness of the population mean set, establish a sharp diameter bound, obtain almost-sure consistency of empirical $ϕ$-means, and derive a uniform strong law over compact classes of losses. The analysis is driven by a deterministic argmin principle together with a Glivenko--Cantelli theorem for monotone classes. For isotropic densities on Riemannian symmetric spaces, we identify the population $ϕ$-mean for every strictly increasing loss for which the objective is finite, including bounded robust losses. We also illustrate the framework on spheres and on the polyhedral space of ultrametric phylogenetic trees.

math.ST↗

arXiv2024

On the Weighted Top-Difference Distance: Axioms, Aggregation, and Approximation

We study a family of distance functions on rankings that allow for asymmetric treatments of alternatives and consider the distinct relevance of the top and bottom positions for ordered lists. We provide a full axiomatic characterization of our distance. In doing so, we retrieve new characterizations of existing axioms and show how to effectively weaken them for our purposes. This analysis highlights the generality of our distance as it embeds many (semi)metrics previously proposed in the literature. Subsequently, we show that, notwithstanding its level of generality, our distance is still readily applicable. We apply it to preference aggregation, studying the features of the associated median voting rule. It is shown how the derived preference function satisfies many desirable features in the context of voting rules, ranging from fairness to majority and Pareto-related properties. We show how to compute consensus rankings exactly, and provide generalized Diaconis-Graham inequalities that can be leveraged to obtain approximation algorithms. Finally, we propose some truncation ideas for our distances inspired by Lu and Boutilier (2010). These can be leveraged to devise a Polynomial-Time-Approximation Scheme for the corresponding rank aggregation problem.

cs.GT↗

arXiv2022

Most numbers are not normal

We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers $x \in (0,1]$ with the following property is comeager: for all integers $b\ge 2$ and $k\ge 1$, the sequence of vectors made by the frequencies of all possibile strings of length $k$ in the $b$-adic representation of $x$ has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen in [Math. Proc. Cambridge Philos. Soc. 137 (2004), 43--53]. We provide analogues in the context of analytic P-ideals and regular matrices.

math.NT↗

arXiv2021

Concerning three classes of non-Diophantine arithmetics

We present three classes of abstract prearithmetics, $\{\mathbf{A}_M\}_{M \geq 1}$, $\{\mathbf{A}_{-M,M}\}_{M \geq 1}$, and $\{\mathbf{B}_M\}_{M > 0}$. The first one is weakly projective with respect to the nonnegative real Diophantine arithmetic $\mathbf{R_+}=(\mathbb{R}_+,+,\times,\leq_{\mathbb{R}_+})$, the second one is weakly projective with respect to the real Diophantine arithmetic $\mathbf{R}=(\mathbb{R},+,\times,\leq_{\mathbb{R}})$, while the third one is projective with respect to the extended real Diophantine arithmetic $\overline{\mathbf{R}}=(\overline{\mathbb{R}},+,\times,\leq_{\overline{\mathbb{R}}})$. In addition, we have that every $\mathbf{A}_M$ and every $\mathbf{B}_M$ are a complete totally ordered semiring, while every $\mathbf{A}_{-M,M}$ is not. We show that the projection of any series of elements of $\mathbb{R}_+$ converges in $\mathbf{A}_M$, for any $M \geq 1$, and that the projection of any non-oscillating series series of elements of $\mathbb{R}$ converges in $\mathbf{A}_{-M,M}$, for any $M \geq 1$, and in $\mathbf{B}_M$, for all $M > 0$. We also prove that working in $\mathbf{A}_M$ and in $\mathbf{A}_{-M,M}$, for any $M \geq 1$, and in $\mathbf{B}_M$, for all $M>0$, allows to overcome a version of the paradox of the heap.

math.LO↗