arXiv subjects

Taegyu Kang

Publications and source records attributed to Taegyu Kang.

4 recordsLinked to original sources

arXiv2026

Giants through higher-order paths in random simplicial complexes

We investigate the giant component formed via high-dimensional paths in the multi-parameter random simplicial complex (MRSC) model. For a $d$-dimensional simplicial complex, we define $d$-dimensional connectivity through incidence between $(d-1)$- and $d$-dimensional simplices. The phase transition of the largest $d$-dimensional connected component is determined in terms of the parameter $λ$ that governs the number of $d$-simplices incident to a typical $(d-1)$-simplex. In the subcritical regime, we show that the largest component contains $Θ(\log n)$ many $(d-1)$-simplices with high probability in the MRSC model. In the supercritical regime, we determine the asymptotic proportion of $1$-simplices in the giant component in dimension $2$, for $λ_c < λ< \barλ$, where $\barλ > 4$ is an explicit constant. In particular, for Linial-Meshulam complexes, this result holds throughout the entire supercritical regime. Additionally, we show that the number of vertices in the giant component undergoes a discontinuous phase transition in $d$-dimensional Linial-Meshulam complexes, in the sense that the asymptotic proportion of vertices in the giant jumps from $0$ to $1$. Our approach is based on local-weak convergence. We establish local-weak convergence in probability for the MRSC model and prove the concentration result via a refined analysis of the breadth-first exploration process, which tracks contributions from newly discovered and previously explored vertices.

math.PR↗

arXiv2026

Layered Hill estimator for extreme data in clusters

A new estimator is proposed for estimating the tail exponent of a heavy-tailed distribution. This estimator, referred to as the layered Hill estimator, is a generalization of the traditional Hill estimator, building upon a layered structure formed by clusters of extreme values. We argue that the layered Hill estimator provides a robust alternative to the traditional approach, exhibiting desirable asymptotic properties such as consistency and asymptotic normality for the tail exponent. Both theoretical analysis and simulation studies demonstrate that the layered Hill estimator shows significantly better and more robust performance, particularly when a portion of the extreme data is missing.

math.ST↗

arXiv2023

Differentially Private Topological Data Analysis

This paper is the first to attempt differentially private (DP) topological data analysis (TDA), producing near-optimal private persistence diagrams. We analyze the sensitivity of persistence diagrams in terms of the bottleneck distance, and we show that the commonly used Čech complex has sensitivity that does not decrease as the sample size $n$ increases. This makes it challenging for the persistence diagrams of Čech complexes to be privatized. As an alternative, we show that the persistence diagram obtained by the $L^1$-distance to measure (DTM) has sensitivity $O(1/n)$. Based on the sensitivity analysis, we propose using the exponential mechanism whose utility function is defined in terms of the bottleneck distance of the $L^1$-DTM persistence diagrams. We also derive upper and lower bounds of the accuracy of our privacy mechanism; the obtained bounds indicate that the privacy error of our mechanism is near-optimal. We demonstrate the performance of our privatized persistence diagrams through simulations as well as on a real dataset tracking human movement.

stat.ML↗

arXiv2022

Large deviations for the volume of $k$-nearest neighbor balls

This paper develops the large deviations theory for the point process associated with the Euclidean volume of $k$-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of $k$-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of $\mathcal M_0$-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most $k$ in a geometric graph in the dense regime.

math.PR↗