arXiv subjects

Omer Bobrowski

Publications and source records attributed to Omer Bobrowski.

At least 19 recordsLinked to original sources

arXiv2026

Universal topological statistics on triangulated singular spaces

We prove a universality theorem for random persistent homology over a class of triangulable spaces. More precisely, let $M \subset \mathbb{R}^D$ be a compact $C^2$-triangulable space satisfying a geometric quality condition and let $f: M \to \mathbb{R}$ be a probability density. Then the expected persistence ratio measure computed from the \v{C}ech or Vietoris-Rips complex of a Poisson point process with intensity $nf$ has a universal limit independent of $(M, f)$. Since smooth manifolds, algebraic varieties, semialgebraic sets and Whitney stratified spaces are all triangulable spaces, our theorem applies to a large class of non-Euclidean spaces. Beyond persistent homology, our proof covers a general class of scale-invariant functionals. It relies on a geometric transfer method that adapts constructions in Euclidean space to triangulable spaces through successive approximations by Freudenthal-Kuhn triangulations, and control of interference across singular strata.

math.PR↗

arXiv2026

Sharp Phase Transition for the Formation of Infinite Tubes

Classical bond percolation theory studies the conditions for a given point in a random graph to be connected to infinity, or "escape" to infinity, via a sequence of random edges. In this work, we present a higher-dimensional generalization of this question, asking whether a fixed loop (or, more generally, a topological sphere) can escape to infinity via a tube formed by random plaquettes. We refer to this phenomenon as tube percolation. We first compare tube percolation with previously studied higher-dimensional percolation phenomena, including face and cycle percolation. For tubes of codimension one, we further relate the critical probability for tube percolation to those for percolation of finite clusters and shielded percolation in the dual bond percolation model. Next, we introduce a tubular analogue of the classical one-arm event, the tubular one-arm event, and prove that it exhibits a sharp threshold at criticality: below criticality, its probability decays exponentially in scale, whereas above criticality, it admits a mean-field-type lower bound. The proof relies on the O'Donnell-Saks-Schramm-Servedio (OSSS) inequality together with an exploration algorithm adapted to the topology of tubes. Finally, we study the tubular box-crossing property. Unlike ordinary path connectedness, "tube connectedness" is not transitive, and thus there is no natural notion of clusters. Nevertheless, we establish an analogue of the uniqueness of the infinite cluster from classical bond percolation. Combining this result with the sharp threshold for the tubular one-arm event, we prove that the existence of a box-crossing tube also exhibits a sharp threshold.

math.PR↗

arXiv2026

A Universal Nearest-Neighbor Estimator for Intrinsic Dimensionality

Estimating the intrinsic dimensionality (ID) of data is a fundamental problem in machine learning and computer vision, providing insight into the true degrees of freedom underlying high-dimensional observations. Existing methods often rely on geometric or distributional assumptions and can significantly fail when these assumptions are violated. In this paper, we introduce a novel ID estimator based on nearest-neighbor distance ratios that involves simple calculations and achieves state-of-the-art results. Most importantly, we provide a theoretical analysis proving that our estimator is \emph{universal}, namely, it converges to the true ID independently of the distribution generating the data. We present experimental results on benchmark manifolds and real-world datasets to demonstrate the performance of our estimator.

cs.LG↗

arXiv2025

Sharp Phase Transitions for k-Fold Coverage Using Morse Theory

We introduce a novel approach for studying random k-coverage, using Morse theory for the k-nearest neighbor (k-NN) distance function. We prove a sharp phase transition for the number of critical points of the k-NN distance function, from which we conclude a phase transition for k-coverage. In addition, in the critical window our new framework enables us to prove a Poisson process approximation (in both location and size) for the last uncovered regions.

math.PR↗

arXiv2024

Universality in Random Persistent Homology and Scale-Invariant Functionals

In this paper, we prove a universality result for the limiting distribution of persistence diagrams arising from geometric filtrations over random point processes. Specifically, we consider the distribution of the ratio of persistence values (death/birth), and show that for fixed dimension, homological degree and filtration type (Cech or Vietoris-Rips), the limiting distribution is independent of the underlying point process distribution, i.e., universal. In proving this result, we present a novel general framework for universality in scale-invariant functionals on point processes. Finally, we also provide a number of new results related to Morse theory in random geometric complexes, which may be of an independent interest.

math.PR↗

arXiv2024

Morse Theory for the k-NN Distance Function

We study the $k$-th nearest neighbor distance function from a finite point-set in $\mathbb{R}^d$. We provide a Morse theoretic framework to analyze the sub-level set topology. In particular, we present a simple combinatorial-geometric characterization for critical points and their indices, along with detailed information about the possible changes in homology at the critical levels. We conclude by computing the expected number of critical points for a homogeneous Poisson process. Our results deliver significant insights and tools for the analysis of persistent homology in order-$k$ Delaunay mosaics, and random $k$-fold coverage.

cs.CG↗

arXiv2023

Central Limit Theorems for Local Functionals of Dynamic Point Processes

We establish finite-dimensional central limit theorems for local, additive, interaction functions of temporally evolving point processes. The dynamics are those of a spatial Poisson process on the flat torus with points subject to a birth-death mechanism, and which move according to Brownian motion while alive. The results reveal the existence of a phase diagram describing at least three distinct structures for the limiting processes, depending on the extent of the local interactions and the speed of the Brownian motions. The proofs, which identify three different limits, rely heavily on Malliavin-Stein type CLTs for $U$-statistics on a representation of the dynamic point process via a distributionally equivalent marked point process.

math.PR↗

arXiv2023

Cluster-Persistence for Weighted Graphs

Persistent homology is a natural tool for probing the topological characteristics of weighted graphs, essentially focusing on their $0$-dimensional homology. While this area has been substantially studied, we present a new approach to constructing a filtration for cluster analysis via persistent homology. The key advantages of the new filtration is that (a) it provides richer signatures for connected components by introducing non-trivial birth times, and (b) it is robust to outliers. The key idea is that nodes are ignored until they belong to sufficiently large clusters. We demonstrate the computational efficiency of our filtration, its practical effectiveness, and explore into its properties when applied to random graphs.

math.AT↗

arXiv2022

Functional Central Limit Theorems for Local Statistics of Spatial Birth-Death Processes in the Thermodynamic Regime

We present normal approximation results at the process level for local functionals defined on dynamic Poisson processes in $\mathbb{R}^d$. The dynamics we study here are those of a Markov birth-death process. We prove functional limit theorems in the so-called thermodynamic regime. Our results are applicable to several functionals of interest in the stochastic geometry literature, including subgraph and component counts in the random geometric graphs.

math.PR↗

arXiv2022

On the Universality of Random Persistence Diagrams

One of the most elusive challenges within the area of topological data analysis is understanding the distribution of persistence diagrams. Despite much effort, this is still largely an open problem. In this paper, we present a series of novel conjectures regarding the behavior of persistence diagrams arising from random point-clouds. We claim that these diagrams obey a universal probability law, and include an explicit expression as a candidate for what this law is. We back these conjectures with an exhaustive set of experiments, including both simulated and real data. We demonstrate the power of these conjectures by proposing a new hypothesis testing framework for individual features within persistence diagrams.

math.ST↗

arXiv2022

Random Simplicial Complexes: Models and Phenomena

We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results have been recently established rigorously in mathematics, especially in the context of algebraic topology. In application to real-world systems, the reviewed models are typically used as null models, so that we take a statistical stance, emphasizing, where applicable, the entropic properties of the reviewed models. We also review a collection of phenomena and features observed in these models, and split the presented results into two classes: phase transitions and distributional limits. We conclude with an outline of interesting future research directions.

math.PR↗

arXiv2021

A Coupled Alpha Complex

The alpha complex is a subset of the Delaunay triangulation and is often used in computational geometry and topology. One of the main drawbacks of using the alpha complex is that it is non-monotone, in the sense that if ${\cal X}\subset{\cal X}'$ it is not necessarily (and generically not) the case that the corresponding alpha complexes satisfy ${\cal A}_r({\cal X})\subset{\cal A}_r({\cal X}')$. The lack of monotonicity may introduce significant computational costs when using the alpha complex, and in some cases even render it unusable. In this work we present a new construction based on the alpha complex, that is homotopy equivalent to the alpha complex while maintaining monotonicity. We provide the formal definitions and algorithms required to construct this complex, and to compute its homology. In addition, we analyze the size of this complex in order to argue that it is not significantly more costly to use than the standard alpha complex.

cs.CG↗

arXiv2021

Poisson process approximation under stabilization and Palm coupling

We present new Poisson process approximation results for stabilizing functionals of Poisson and binomial point processes. These functionals are allowed to have an unbounded range of interaction and encompass many examples in stochastic geometry. Our bounds are derived for the Kantorovich-Rubinstein distance using the generator approach to Stein's method. We give different types of bounds for different point processes. While some of our bounds are given in terms of coupling of the point process with its Palm version, the others are in terms of the local dependence structure formalized via the notion of stabilization. We provide two supporting examples for our new framework - one is for Morse critical points of the distance function, and the other is for large k-nearest neighbor balls. Our bounds considerably extend the results in Barbour and Brown (1992), Decreusefond, Schulte and Thale (2016) and Otto (2020).

math.PR↗

arXiv2021

Joint Geometric and Topological Analysis of Hierarchical Datasets

In a world abundant with diverse data arising from complex acquisition techniques, there is a growing need for new data analysis methods. In this paper we focus on high-dimensional data that are organized into several hierarchical datasets. We assume that each dataset consists of complex samples, and every sample has a distinct irregular structure modeled by a graph. The main novelty in this work lies in the combination of two complementing powerful data-analytic approaches: topological data analysis (TDA) and geometric manifold learning. Geometry primarily contains local information, while topology inherently provides global descriptors. Based on this combination, we present a method for building an informative representation of hierarchical datasets. At the finer (sample) level, we devise a new metric between samples based on manifold learning that facilitates quantitative structural analysis. At the coarser (dataset) level, we employ TDA to extract qualitative structural information from the datasets. We showcase the applicability and advantages of our method on simulated data and on a corpus of hyper-spectral images. We show that an ensemble of hyper-spectral images exhibits a hierarchical structure that fits well the considered setting. In addition, we show that our new method gives rise to superior classification results compared to state-of-the-art methods.

cs.LG↗

arXiv2021

Cycle Registration in Persistent Homology with Applications in Topological Bootstrap

In this article we propose a novel approach for comparing the persistent homology representations of two spaces (filtrations). Commonly used methods are based on numerical summaries such as persistence diagrams and persistence landscapes, along with suitable metrics (e.g. Wasserstein). These summaries are useful for computational purposes, but they are merely a marginal of the actual topological information that persistent homology can provide. Instead, our approach compares between two topological representations directly in the data space. We do so by defining a correspondence relation between individual persistent cycles of two different spaces, and devising a method for computing this correspondence. Our matching of cycles is based on both the persistence intervals and the spatial placement of each feature. We demonstrate our new framework in the context of topological inference, where we use statistical bootstrap methods in order to differentiate between real features and noise in point cloud data.

cs.LG↗

arXiv2020

Homological Percolation: The Formation of Giant k-Cycles

In this paper we introduce and study a higher-dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant k-dimensional cycles (with 0-cycles being connected components). Considering a continuum percolation model in the flat d-dimensional torus, we show that all the giant k-cycles (k=1,...,d-1) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant k-cycles are increasing in k and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing.

math.PR↗

arXiv2020

On the Spectrum of Dense Random Geometric Graphs

In this paper we study the spectrum of the random geometric graph $G(n,r)$, in a regime where the graph is dense and highly connected. In the \erdren $G(n,p)$ random graph it is well known that upon connectivity the spectrum of the normalized graph Laplacian is concentrated around $1$. We show that such concentration does not occur in the $G(n,r)$ case, even when the graph is dense and almost a complete graph. In particular, we show that the limiting spectral gap is strictly smaller than $1$. In the special case where the vertices are distributed uniformly in the unit cube and $r=1$, we show that for every $0\le k \le d$ there are at least $\binom{d}{k}$ eigenvalues near $1-2^{-k}$, and the limiting spectral gap is exactly $1/2$. We also show that the corresponding eigenfunctions in this case are tightly related to the geometric configuration of the points.

math.PR↗

arXiv2019

Homological Percolation and the Euler Characteristic

In this paper we study the connection between the phenomenon of homological percolation (the formation of "giant" cycles in persistent homology), and the zeros of the expected Euler characteristic curve. We perform an experimental study that covers four different models: site-percolation on the cubical and permutahedral lattices, the Poisson-Boolean model, and Gaussian random fields. All the models are generated on the flat torus $T^d$, for $d=2,3,4$. The simulation results strongly indicate that the zeros of the expected Euler characteristic curve approximate the critical values for homological-percolation. Our results also provide some insight about the approximation error. Further study of this connection could have powerful implications both in the study of percolation theory, and in the field of Topological Data Analysis.

math-ph↗