arXiv subjects

Wolfgang Polonik

Publications and source records attributed to Wolfgang Polonik.

At least 19 recordsLinked to original sources

arXiv2026

Common Drivers in Sparsely Interacting Hawkes Processes

We study a multivariate Hawkes process as a model for time-continuous relational event networks. The model does not assume the network to be known, it includes covariates, and it allows for both common drivers, parameters common to all the actors in the network, and also local parameters specific for each actor. We derive rates of convergence for all of the model parameters when both the number of actors and the time horizon tends to infinity. To prevent an exploding network, sparseness is assumed. We also discuss numerical aspects.

math.ST↗

arXiv2026

First-order asymptotic expansions for spectral convergence of compact self-adjoint operators on general spectral subsets, with application to kernel Gram matrices

We study the spectral convergence of compact, self-adjoint operators on a separable Hilbert space, and derive the first-order asymptotic expansions for their eigenvalues, eigenvectors and eigenprojections, along with remainder bounds expressed in terms of weighted perturbation quantities. Our analysis focuses on eigenvalues indexed by a general subset, with minimal restrictions on their selection. In particular, when the eigenvalues of interest are clustered, i.e., the inner spectral gaps are small, we provide different types of expansions for the eigenvalues, with remainder bounds that are robust to the inner spectral gaps; this contrasts with the existing literature, which mainly focuses on isolated eigenvalues. The usefulness of the provided expansions is illustrated by an application to kernel Gram matrices, deriving concentration inequalities as well as weak convergence results, which, in contrast to existing literature, primarily rely on assumptions on the kernel that are easy to check.

math.ST↗

arXiv2026

Antimodes and Graphical Anomaly Exploration via Adaptive Depth Quantile Functions

This work proposes and investigates a novel method for anomaly detection and shows it to be competitive in a variety of Euclidean and non-Euclidean situations. It is based on an extension of the depth quantile functions (DQF) approach. The DQF approach encodes geometric information about a point cloud via functions of a single variable, whereas each observation in a data set is associated with a single such function. Plotting these functions provides a very beneficial visualization aspect. This technique can be applied to any data lying in a Hilbert space. The proposed anomaly detection approach is motivated by the geometric insight of the presence of anomalies in data being tied to the existence of antimodes in the data generating distribution. Coupling this insight with novel theoretical understanding into the shape of the DQFs gives rise to the proposed adaptive DQF (aDQF) methodology. Applications to various data sets illustrate the DQF and aDQF's strong anomaly detection performance, and the benefits of its visualization aspects.

stat.ME↗

arXiv2026

Kernel smoothing on manifolds

Under the assumption that data lie on a compact (unknown) manifold without boundary, we derive finite sample bounds for kernel smoothing and its (first and second) derivatives, and we establish asymptotic normality through Berry-Esseen type bounds. Special cases include kernel density estimation, kernel regression and the heat kernel signature. Connections to the graph Laplacian are also discussed.

math.ST↗

arXiv2025

Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmaps

We develop nonparametric regression methods for the case when the true regression function is not necessarily smooth. More specifically, our approach is using the fractional Laplacian and is designed to handle the case when the true regression function lies in an $L_2$-fractional Sobolev space with order $s\in (0,1)$. This function class is a Hilbert space lying between the space of square-integrable functions and the first-order Sobolev space consisting of differentiable functions. It contains fractional power functions, piecewise constant or polynomial functions and bump function as canonical examples. For the proposed approach, we prove upper bounds on the in-sample mean-squared estimation error of order $n^{-\frac{2s}{2s+d}}$, where $d$ is the dimension, $s$ is the aforementioned order parameter and $n$ is the number of observations. We also provide preliminary empirical results validating the practical performance of the developed estimators.

math.ST↗

arXiv2025

Multivariate Gaussian Approximation for Random Forest via Region-based Stabilization

We derive Gaussian approximation bounds for $k$-Potential Nearest Neighbor ($k$-PNN) based random forest predictions based on a set of training points given by a Poisson process under fairly mild regularity assumptions on the data generating process. Our approach is based on the key observation that $k$-PNN based random forest predictions satisfy a certain geometric property called region-based stabilization. We also compare the rates with those of $k$-nearest neighbor-based random forests, highlighting a form of universality in our result. In the process of developing our results, we also establish a probabilistic result on multivariate Gaussian approximation bounds for general functionals of Poisson process that are region-based stabilizing. This general result makes use of the Malliavin-Stein method, and is potentially applicable to various related statistical problems.

math.ST↗

arXiv2024

Algorithms for ridge estimation with convergence guarantees

The extraction of filamentary structure from a point cloud is discussed. The filaments are modeled as ridge lines or higher dimensional ridges of an underlying density. We propose two novel algorithms, and provide theoretical guarantees for their convergences, by which we mean that the algorithms can asymptotically recover the full ridge set. We consider the new algorithms as alternatives to the Subspace Constrained Mean Shift (SCMS) algorithm for which no such theoretical guarantees are known.

stat.ML↗

arXiv2024

Gaussian and Bootstrap Approximation for Matching-based Average Treatment Effect Estimators

We establish Gaussian approximation bounds for covariate and rank-matching-based Average Treatment Effect (ATE) estimators. By analyzing these estimators through the lens of stabilization theory, we employ the Malliavin-Stein method to derive our results. Our bounds precisely quantify the impact of key problem parameters, including the number of matches and treatment balance, on the accuracy of the Gaussian approximation. Additionally, we develop multiplier bootstrap procedures to estimate the limiting distribution in a fully data-driven manner, and we leverage the derived Gaussian approximation results to further obtain bootstrap approximation bounds. Our work not only introduces a novel theoretical framework for commonly used ATE estimators, but also provides data-driven methods for constructing non-asymptotically valid confidence intervals.

math.ST↗

arXiv2024

On the asymptotic normality of persistent Betti numbers

Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process $ (r,s) \mapsto n^{-1/2} (β^{r,s}_q ( \mathcal{K}(n^{1/d} \mathcal{X}_n))-\mathbb{E}[β^{r,s}_q ( \mathcal{K}( n^{1/d} \mathcal{X}_n))])$. So far, pointwise limit theorems have been established in different settings. In particular, the pointwise asymptotic normality of (persistent) Betti numbers has been established for stationary Poisson processes and binomial processes with constant intensity function in the so-called critical (or thermodynamic) regime, see Yogeshwaran et al. [2017] and Hiraoka et al. [2018]. In this contribution, we derive a strong stabilization property (in the spirit of Penrose and Yukich [2001] of persistent Betti numbers and generalize the existing results on the asymptotic normality to the multivariate case and to a broader class of underlying Poisson and binomial processes. Most importantly, we show that the multivariate asymptotic normality holds for all pairs $(r,s)$, $0\le r\le s<\infty$, and that it is not affected by percolation effects in the underlying random geometric graph.

math.PR↗

arXiv2023

Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression

We show both adaptive and non-adaptive minimax rates of convergence for a family of weighted Laplacian-Eigenmap based nonparametric regression methods, when the true regression function belongs to a Sobolev space and the sampling density is bounded from above and below. The adaptation methodology is based on extensions of Lepski's method and is over both the smoothness parameter ($s\in\mathbb{N}_{+}$) and the norm parameter ($M>0$) determining the constraints on the Sobolev space. Our results extend the non-adaptive result in \cite{green2021minimax}, established for a specific normalized graph Laplacian, to a wide class of weighted Laplacian matrices used in practice, including the unnormalized Laplacian and random walk Laplacian.

math.ST↗

arXiv2023

Testing For Global Covariate Effects in Dynamic Interaction Event Networks

In statistical network analysis it is common to observe so called interaction data. Such data is characterized by actors forming the vertices and interacting along edges of the network, where edges are randomly formed and dissolved over the observation horizon. In addition covariates are observed and the goal is to model the impact of the covariates on the interactions. We distinguish two types of covariates: global, system-wide covariates (i.e. covariates taking the same value for all individuals, such as seasonality) and local, dyadic covariates modeling interactions between two individuals in the network. Existing continuous time network models are extended to allow for comparing a completely parametric model and a model that is parametric only in the local covariates but has a global non-parametric time component. This allows, for instance, to test whether global time dynamics can be explained by simple global covariates like weather, seasonality etc. The procedure is applied to a bike-sharing network by using weather and weekdays as global covariates and distances between the bike stations as local covariates.

stat.ME↗

arXiv2022

A Flexible Approach for Normal Approximation of Geometric and Topological Statistics

We derive normal approximation results for a class of stabilizing functionals of binomial or Poisson point process, that are not necessarily expressible as sums of certain score functions. Our approach is based on a flexible notion of the add-one cost operator, which helps one to deal with the second-order cost operator via suitably appropriate first-order operators. We combine this flexible notion with the theory of strong stabilization to establish our results. We illustrate the applicability of our results by establishing normal approximation results for certain geometric and topological statistics arising frequently in practice. Several existing results also emerge as special cases of our approach.

math.PR↗

arXiv2022

Topologically penalized regression on manifolds

We study a regression problem on a compact manifold M. In order to take advantage of the underlying geometry and topology of the data, the regression task is performed on the basis of the first several eigenfunctions of the Laplace-Beltrami operator of the manifold, that are regularized with topological penalties. The proposed penalties are based on the topology of the sub-level sets of either the eigenfunctions or the estimated function. The overall approach is shown to yield promising and competitive performance on various applications to both synthetic and real data sets. We also provide theoretical guarantees on the regression function estimates, on both its prediction error and its smoothness (in a topological sense). Taken together, these results support the relevance of our approach in the case where the targeted function is ''topologically smooth''.

cs.LG↗

arXiv2021

On approximation theorems for the Euler characteristic with applications to the bootstrap

We study approximation theorems for the Euler characteristic of the Vietoris-Rips and Cech filtration. The filtration is obtained from a Poisson or binomial sampling scheme in the critical regime. We apply our results to the smooth bootstrap of the Euler characteristic and determine its rate of convergence in the Kantorovich-Wasserstein distance and in the Kolmogorov distance.

math.PR↗

arXiv2021

High Order Adjusted Block-wise Empirical Likelihood For Weakly Dependent Data

It is well known that the empirical likelihood ratio confidence region suffers finite sample under-coverage issue, and this severely hampers its application in statistical inferences.} The root cause of this under-coverage is an upper limit imposed by the convex hull of the estimating functions that is used in the construction of the profile empirical likelihood. For i.i.d data, various methods have been proposed to solve this issue by modifying the convex hull, but it is not clear how well these methods perform when the data are no longer independent. In this paper, we propose an adjusted blockwise empirical likelihood that is designed for weakly dependent multivariate data. We show that our method not only preserves the much celebrated asymptotic $χ^2-$distribution, but also improves the finite sample coverage probability by removing the upper limit imposed by the convex hull. Further, we show that our method is also Bartlett correctable, thus is able to achieve high order asymptotic coverage accuracy.

stat.ME↗

arXiv2021

Bootstrapping Persistent Betti Numbers and Other Stabilizing Statistics

The present contribution investigates multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. Existing limit theorems for topological statistics prove difficult to use in practice for the construction of confidence intervals, motivating the use of the bootstrap in this capacity. However, the standard nonparametric bootstrap does not directly provide for asymptotically valid confidence intervals in some situations. A smoothed bootstrap procedure, instead, is shown to give consistent estimation in these settings. The present work relates to other general results in the area of stabilizing statistics, including central limit theorems for functionals of Poisson and Binomial processes in the critical regime. Specific statistics considered include the persistent Betti numbers of Čech and Vietoris-Rips complexes over point sets in $\mathbb R^d$, along with Euler characteristics, and the total edge length of the $k$-nearest neighbor graph. Special emphasis is made throughout to weakening the necessary conditions needed to establish bootstrap consistency. In particular, the assumption of a continuous underlying density is not required. A simulation study is provided to assess the performance of the smoothed bootstrap for finite sample sizes, and the method is further applied to the cosmic web dataset from the Sloan Digital Sky Survey (SDSS). Source code is available at github.com/btroycraft/stabilizing_statistics_bootstrap.

math.ST↗

arXiv2020

On the choice of weight functions for linear representations of persistence diagrams

Persistence diagrams are efficient descriptors of the topology of a point cloud. As they do not naturally belong to a Hilbert space, standard statistical methods cannot be directly applied to them. Instead, feature maps (or representations) are commonly used for the analysis. A large class of feature maps, which we call linear, depends on some weight functions, the choice of which is a critical issue. An important criterion to choose a weight function is to ensure stability of the feature maps with respect to Wasserstein distances on diagrams. We improve known results on the stability of such maps, and extend it to general weight functions. We also address the choice of the weight function by considering an asymptotic setting; assume that $\mathbb{X}_n$ is an i.i.d. sample from a density on $[0,1]^d$. For the Čech and Rips filtrations, we characterize the weight functions for which the corresponding feature maps converge as $n$ approaches infinity, and by doing so, we prove laws of large numbers for the total persistences of such diagrams. Those two approaches (stability and convergence) lead to the same simple heuristic for tuning weight functions: if the data lies near a $d$-dimensional manifold, then a sensible choice of weight function is the persistence to the power $α$ with $α\geq d$.

math.PR↗

arXiv2019

Multiscale geometric feature extraction for high-dimensional and non-Euclidean data with application

A method for extracting multiscale geometric features from a data cloud is proposed and analyzed. The basic idea is to map each pair of data points into a real-valued feature function defined on $[0,1]$. The construction of these feature functions is heavily based on geometric considerations, which has the benefits of enhancing interpretability. Further statistical analysis is then based on the collection of the feature functions. The potential of the method is illustrated by different applications, including classification of high-dimensional and non-Euclidean data. For continuous data in Euclidean space, our feature functions contain information about the underlying density at a given base point (small scale features), and also about the depth of the base point (large scale feature). As shown by our theoretical investigations, the method combats the curse of dimensionality, and also shows some adaptiveness towards sparsity. Connections to other concepts, such as random set theory, localized depth measures and nonlinear multidimensional scaling, are also explored.

math.ST↗