arXiv subjects

Alejandro Cholaquidis

Publications and source records attributed to Alejandro Cholaquidis.

At least 19 recordsLinked to original sources

arXiv2026

A bootstrap approach for testing invariance under parameterized group actions: orthogonal reflections and axial symmetry

Testing whether a multivariate distribution is invariant under an orthogonal transformation is a classical problem when the transformation is fixed in advance. We address a harder situation: the transformation is unknown and must be inferred from the data. We frame this as invariance under a group action whose representation is indexed by an unknown parameter. We work in $\mathbb{R}^d$ under a simple-spectrum assumption on the covariance matrix $Σ$. Under this assumption, any orthogonal transformation that preserves the distribution must commute with $Σ$, and therefore must be a {reflection through a subspace spanned by a subset of the principal directions. This reduces the search over the orthogonal group to a finite family of candidate reflections, one for each subset of principal axes.} For each candidate, we construct a Kolmogorov--Smirnov-type statistic based on projected data and sample splitting. We derive its asymptotic distribution in a triangular-array framework and establish bootstrap validity under suitable regularity conditions. Axial symmetry about an unspecified direction (that is, invariance under reflection across an unknown one-dimensional subspace) and hyperplane (Householder) symmetry about an unspecified normal direction are the two leading particular cases: we treat them in detail, the former driving the simulation study and the latter the real-data application, to show how a concrete problem involving an unknown group representation can be successfully addressed.

math.ST↗

arXiv2026

Conic reach and polynomial parallel volume in the plane

For a compact set $S\subset\R^2$, the local Steiner formula of Hug, Last and Weil expresses the parallel volume $V_S(t)$ through the proximal normal bundle and the truncated fiber lengths $\min\{t,δ_S\}$. We introduce conic reach, a geometric condition with two requirements: $S$ coincides with a cone near each point of a finite, well-separated singular set, and every proximal normal fiber away from those points has length at least $ρ$. In the plane, these requirements force every link to be a finite union of circular arcs whose complementary gaps have width bounded below. Computing the feet-localized tube of each cone and combining it with the local Steiner formula, we show that $V_S$ is a polynomial of degree at most two on $(0,ρ)$, with explicit coefficients; in particular $\polreach(S)\ge\conreach(S)$. A one-dimensional converse shows that a quadratic wall contribution forces a linear cut function. Compact domains with piecewise-$C^2$ boundary, uniformly wedge-like at their reentrant corners, have positive conic reach, and their volume coefficients are given by a Gauss--Bonnet-type formula. For the L-shaped polygon the three invariants separate: $\reach(L)=0$, $\conreach(L)=\tfrac13$, $\polreach(L)=1$. A cuspidal notch, two overlapping discs and a Cantor fan of segments show that tangential contact, curvature at a reentrant corner and degenerating link gaps each destroy polynomiality.

math.GT↗

arXiv2026

Weak convergence from projected laws on a positive-measure set of directions

The Cramér-Wold device characterises weak convergence of probability measures on $\mathbb{R}^d$ through convergence of all one-dimensional projected laws. We prove that, if the target projected laws are moment-determinate for surface-almost every direction, then weak convergence already follows from projected convergence on a positive-measure set of directions. This yields a simple probabilistic interpretation: if one samples a direction at random from any distribution on the sphere that is absolutely continuous with respect to surface measure, then, with probability one, convergence of the projected law along the sampled direction already forces global weak convergence under the same moment-determinacy assumption.

math.PR↗

arXiv2026

Concave Statistical Utility Maximization Bandits via Influence-Function Gradients

We study stochastic multi-armed bandits in which the objective is a statistical functional of the long-run reward distribution, rather than expected reward alone. Under mild continuity assumptions, we show that the infinite-horizon problem reduces to optimizing over stationary mixed policies: each weight vector $w$ on the simplex induces a mixture law $P^w$, and performance is measured by the concave utility $U(w)=\mathfrak U(P^w)$. For differentiable statistical utilities, we use influence-function calculus to derive stochastic gradient estimators from bandit feedback. This leads to an entropic mirror-ascent algorithm on a truncated simplex, implemented through multiplicative-weights updates and plug-in estimates of the influence function. We establish regret bounds that separate the mirror-ascent optimization error from the bias caused by estimating the influence function. The framework is developed for general concave distributional utilities and illustrated through variance and Wasserstein objectives, with numerical experiments comparing exact and plug-in influence-function implementations.

stat.ML↗

arXiv2026

Conformal Robust Set Estimation

Conformal prediction provides finite-sample, distribution-free coverage under exchangeability, but standard constructions may lack robustness in the presence of outliers or heavy tails. We propose a robust conformal method based on a non-conformity score defined as the half-mass radius around a point, equivalently the distance to its $(\lfloor n/2\rfloor+1)$-nearest neighbour. We show that the resulting conformal regions are marginally valid for any sample size and converge in probability to a robust population central set defined through a distance-to-a-measure functional. Under mild regularity conditions, we establish exponential concentration and tail bounds that quantify the deviation between the empirical conformal region and its population counterpart. These results provide a probabilistic justification for using robust geometric scores in conformal prediction, even for heavy-tailed or multi-modal distributions.

math.ST↗

arXiv2025

Estimating axial symmetry using random projections

This paper studies the problem of identifying directions of axial symmetry in multivariate distributions. Theoretical results are derived on how the measure or cardinality of the set of symmetry directions relates to spherical symmetry. The problem is framed using random projections, leading to a proof that in $\RR^2$, agreement on two random projections is enough to identify the true axes of symmetry. A corresponding result for higher dimensions is conjectured. An estimator for the symmetry directions is proposed and proved to be consistent in the plane.

math.ST↗

arXiv2025

Conformal inference for regression on Riemannian Manifolds

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 space of covariance matrices, data on the Grassmannian manifold obtained as a result of principal component analysis, among many others. In this work we investigate prediction sets for regression scenarios when the response variable, denoted by $Y$, resides in a manifold, and the covariable, denoted by $X$, lies in an Euclidean space. This extends the concepts delineated in \cite{waser14} to this novel context. Aligning with traditional principles in conformal inference, these prediction sets are distribution-free, indicating that no specific assumptions are imposed on the joint distribution of $(X,Y)$, and they maintain a non-parametric character. We prove the asymptotic almost sure convergence of the empirical version of these regions on the manifold to their population counterparts. The efficiency of this method is shown through a comprehensive simulation study and an analysis involving real-world data.

stat.ML↗

arXiv2025

On consistent estimation of dimension values

The problem of estimating, from a random sample of points, the dimension of a compact subset $S$ of the Euclidean space is considered. The emphasis is put on consistency results in the statistical sense. That is, statements of convergence to the true dimension value when the sample size grows to infinity. Among the many available definitions of dimension, we have focused (on the grounds of its statistical tractability) on three notions: the Minkowski dimension, the correlation dimension and the, perhaps less popular, concept of pointwise dimension. We prove the statistical consistency of some natural estimators of these quantities. Our proofs partially rely on the use of an instrumental estimator formulated in terms of the empirical volume function $V_n(r)$, defined as the Lebesgue measure of the set of points whose distance to the sample is at most $r$. In particular, we explore the case in which the true volume function $V(r)$ of the target set $S$ is a polynomial on some interval starting at zero. An empirical study is also included. Our study aims to provide some theoretical support, and some practical insights, for the problem of deciding whether or not the set $S$ has a dimension smaller than that of the ambient space. This is a major statistical motivation of the dimension studies, in connection with the so-called ``Manifold Hypothesis''.

math.ST↗

arXiv2025

Depth based trimmed means

Robust estimation of location is a fundamental problem in statistics, particularly in scenarios where data contamination by outliers or model misspecification is a concern. In univariate settings, methods such as the sample median and trimmed means balance robustness and efficiency by mitigating the influence of extreme observations. This paper extends these robust techniques to the multivariate context through the use of data depth functions, which provide a natural means to order and rank multidimensional data. We review several depth measures and discuss their role in generalizing trimmed mean estimators beyond one dimension. Our main contributions are twofold: first, we prove the almost sure consistency of the multivariate trimmed mean estimator under mixing conditions; second, we establish a general limit distribution theorem for a broad family of depth-based estimators, encompassing popular examples such as Tukey's and projection depth. These theoretical advancements not only enhance the understanding of robust location estimation in high-dimensional settings but also offer practical guidelines for applications in areas such as machine learning, economic analysis, and financial risk assessment. A small example with simulated data is performed, varying the depth measure used and the percentage of trimmed data.

math.ST↗

arXiv2025

On standardness and the non-estimability of certain functionals of a set

Standardness is a popular assumption in the literature on set estimation. It also appears in statistical approaches to topological data analysis, where it is common to assume that the data were sampled from a probability measure that satisfies the standard assumption. Relevant results in this field, such as rates of convergence and confidence sets, depend on the standardness parameter, which in practice may be unknown. In this paper, we review the notion of standardness and its connection to other geometrical restrictions. We prove the almost sure consistency of a plug-in type estimator for the so-called standardness constant, already studied in the literature. We propose a method to correct the bias of the plug-in estimator and corroborate our theoretical findings through a small simulation study. We also show that it is not possible to determine, based on a finite sample, whether a probability measure satisfies the standard assumption.

math.ST↗

arXiv2024

GROS: A General Robust Aggregation Strategy

A new, very general, robust procedure for combining estimators in metric spaces is introduced GROS. The method is reminiscent of the well-known median of means, as described in \cite{devroye2016sub}. Initially, the sample is divided into $K$ groups. Subsequently, an estimator is computed for each group. Finally, these $K$ estimators are combined using a robust procedure. We prove that this estimator is sub-Gaussian and we get its break-down point, in the sense of Donoho. The robust procedure involves a minimization problem on a general metric space, but we show that the same (up to a constant) sub-Gaussianity is obtained if the minimization is taken over the sample, making GROS feasible in practice. The performance of GROS is evaluated through five simulation studies: the first one focuses on classification using $k$-means, the second one on the multi-armed bandit problem, the third one on the regression problem. The fourth one is the set estimation problem under a noisy model. Lastly, we apply GROS to get a robust persistent diagram.

math.ST↗

arXiv2024

On the notion of polynomial reach: a statistical application

The volume function V(t) of a compact set S\in R^d is just the Lebesgue measure of the set of points within a distance to S not larger than t. According to some classical results in geometric measure theory, the volume function turns out to be a polynomial, at least in a finite interval, under a quite intuitive, easy to interpret, sufficient condition (called ``positive reach'') which can be seen as an extension of the notion of convexity. However, many other simple sets, not fulfilling the positive reach condition, have also a polynomial volume function. To our knowledge, there is no general, simple geometric description of such sets. Still, the polynomial character of $V(t)$ has some relevant consequences since the polynomial coefficients carry some useful geometric information. In particular, the constant term is the volume of S and the first order coefficient is the boundary measure (in Minkowski's sense). This paper is focused on sets whose volume function is polynomial on some interval starting at zero, whose length (that we call ``polynomial reach'') might be unknown. Our main goal is to approximate such polynomial reach by statistical means, using only a large enough random sample of points inside S. The practical motivation is simple: when the value of the polynomial reach , or rather a lower bound for it, is approximately known, the polynomial coefficients can be estimated from the sample points by using standard methods in polynomial approximation. As a result, we get a quite general method to estimate the volume and boundary measure of the set, relying only on an inner sample of points and not requiring the use any smoothing parameter. This paper explores the theoretical and practical aspects of this idea.

math.ST↗

arXiv2023

Home range estimation under a restricted sampling scheme

The analysis of animal movement has gained attention recently. New continuous-time models and statistical methods have been developed to estimate some sets related to their movements, such as the home-range and the core-area among others, when the information of the trajectory is provided by a GPS. Because data transfer costs and GPS battery life are practical constraints in ecological studies, the experimental designer must make critical sampling decisions in order to maximize information. To capture fine-scale motion, long-term behavior must be sacrificed, and vice versa. To overcome this limitation, we introduce the on--off sampling scheme, where the GPS is alternately on and off. This scheme is already used in practice but with insufficient statistical theoretical support. We prove the consistency of home-range estimators with an underlying reflected diffusion model under this sampling method (in terms of the Hausdorff distance). The same rate of convergence is achieved as in the case where the GPS is always on for the whole experiment. This is illustrated by a simulation study and real data. We also provide estimators of the stationary distribution, its level sets (which give estimators of the core area), and the drift function.

math.ST↗

arXiv2022

Universally consistent estimation of the reach

The reach of a set $M \subset \mathbb R^d$, also known as condition number when $M$ is a manifold, was introduced by Federer in 1959. The reach is a central concept in geometric measure theory, set estimation, manifold learning, among others areas. We introduce a universally consistent estimate of the reach, just assuming that the reach is positive. Under an additional assumption we provide rates of convergence. We also show that it is not possible to determine, based on a finite sample, if the reach of the support of a density is zero or not. We provide a small simulation study and a bias correction method for the case when $M$ is a manifold.

math.ST↗

arXiv2022

Statistical analysis of measures of non-convexity

Several measures of non-convexity (departures from convexity) have been introduced in the literature, both for sets and functions. Some of them are of geometric nature, while others are more of topological nature. We address the statistical analysis of some of these measures of non-convexity of a set $S$, by dealing with their estimation based on a sample of points in $S$. We introduce also a new measure of non-convexity. We discuss briefly about these different notions of non-convexity, prove consistency and find the asymptotic distribution for the proposed estimators. We also consider the practical implementation of these estimators and illustrate their applicability to a real data example.

math.ST↗

arXiv2022

An $r$-convex set which is not locally contractible

The study of shape restrictions of subsets of $\mathbb{R}^d$ have several applications in many areas, being convexity, $r$-convexity, and positive reach, some of the most famous, and typically imposed in set estimation. The following problem was attributed to K. Borsuk, by J. Perkal in 1956: find an $r$-convex set which is not locally contractible. Stated in that way is trivial to find such a set. However, if we ask the set to be equal to the closure of its interior (a condition fulfilled for instance if the set is the support of a probability distribution absolutely continuous with respect to the $d$-dimensional Lebesgue measure), the problem is much more difficult. We present a counter example of a not-locally contractible set, which is $r$-convex. This also proves that the class of supports with positive reach of absolutely continuous distributions includes strictly the class of $r$-convex supports.

math.GT↗

arXiv2022

Estimation of surface area

We study the problem of estimating the surface area of the boundary $\partial S$ of a sufficiently smooth set $S\subset\mathbb{R}^d$ when the available information is only a finite subset $\X\subset S$. We propose two estimators. The first makes use of the Devroye--Wise support estimator and is based on Crofton's formula, which, roughly speaking, states that the $(d-1)$-dimensional surface area of a smooth enough set is the mean number of intersections of randomly chosen lines. For that purpose, we propose an estimator of the number of intersections of such lines with support based on the Devroye--Wise support estimators. The second surface area estimator makes use of the $α$-convex hull of $\X$, which is denoted by $C_α(\X)$. More precisely, it is the $(d-1)$-dimensional surface area of $C_α(\X)$, as denoted by $|C_α(\X)|_{d-1}$, which is proven to converge to the $(d-1)$-dimensional surface area of $\partial S$. Moreover, $|C_α(\X)|_{d-1}$ can be computed using Crofton's formula. Our results depend on the Hausdorff distance between $S$ and $\X$ for the Devroye--Wise estimator, and the Hausdorff distance between $\partial S$ and $\partial C_α(\X)$ for the second estimator.

math.ST↗

arXiv2021

Level set and density estimation on manifolds

We tackle the problem of the estimation of the level sets L_f(λ) of the density f of a random vector X supported on a smooth manifold M\subsetR^d , from an iid sample of X. To do that we introduce a kernel-based estimator f^n,h , which is a slightly modified version of the one proposed in [45], and proves its a.s. uniform convergence to f . Then, we propose two estimators of L f (λ), the first one is a plug-in: L f^n,h (λ), which is proven to be a.s. consistent in Hausdorff distance and distance in measure, if L f(λ) does not meet the boundary of M . While the second one assumes that L f(λ) is r-convex, and is estimated by means of the r-convex hull of L f^n,h(λ). The performance of our proposal is illustrated through some simulated examples. In a real data example we analyze the intensity and direction of strong and moderate winds.

math.ST↗