arXiv subjects

Chunyin Siu

Publications and source records attributed to Chunyin Siu.

4 recordsLinked to original sources

arXiv2026

The Global Topology of Orthogonally Decomposable Tensor Landscapes

The homogeneous form associated with a symmetric tensor, restricted to the sphere, is the objective of the best rank-one approximation problem and, with random coefficients, the energy of a mean-field spin glass. Its critical points have been studied extensively, but the global organization of the landscape they form is far less understood. We study this global structure for positive orthogonally decomposable tensors. We determine the persistent homology of the sublevel and superlevel filtrations in closed form, for every homological dimension $q$, every ambient dimension $D$ and every tensor order $k$, via a recurrence in the ambient dimension. Taking the coefficients to be random, we further prove laws of large numbers for the resulting persistence diagrams at dimension $0$ and $D-2$, giving an exact description of the typical global topology of a spin-glass-like energy landscape. We illustrate our results with numerical computations and simulations.

math.OC↗

arXiv2025

The Topological Behavior of Preferential Attachment Graphs

We investigate the higher-order connectivity of scale-free networks using algebraic topology. We model scale-free networks as preferential attachment graphs, and we study the algebraic-topological properties of their clique complexes. We focus on the Betti numbers and the homotopy-connectedness of these complexes. We determine the asymptotic almost sure orders of magnitude of the Betti numbers. We also establish the occurence of homotopical phase transitions for the infinite complexes, and we determine the critical thresholds at which the homotopy-connectivity changes. This partially verifies Weinberger's conjecture on the homotopy type of the infinite complexes. We conjecture that the mean-normalized Betti numbers converge to power-law distributions, and we present numerical evidence. Our results also highlight the subtlety of the scaling limit of topology, which arises from the tension between topological operations and analytical limiting process. We discuss such tension at the end of the Introduction.

math.AT↗

arXiv2024

The Asymptotics of the Expected Betti Numbers of Preferential Attachment Clique Complexes

The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ``hubs''. We study the higher-order connectivity of such a network by investigating the topological properties of its clique complex. We concentrate on the expected Betti numbers, a sequence of topological invariants of the complex related to the numbers of holes of different dimensions. We determine the asymptotic growth rates of the expected Betti numbers, and prove that the expected Betti number at dimension 1 grows linearly fast, while those at higher dimensions grow sublinearly fast. Our theoretical results are illustrated by simulations. (Changes are made in this version to generalize Proposition 14 and to streamline proofs. These changes are shown in blue.)

math.PR↗

arXiv2024

Detection of Small Holes by the Scale-Invariant Robust Density-Aware Distance (RDAD) Filtration

A novel topological-data-analytical (TDA) method is proposed to distinguish, from noise, small holes surrounded by high-density regions of a probability density function. The proposed method is robust against additive noise and outliers. Traditional TDA tools, like those based on the distance filtration, often struggle to distinguish small features from noise, because both have short persistences. An alternative filtration, called the Robust Density-Aware Distance (RDAD) filtration, is proposed to prolong the persistences of small holes of high-density regions. This is achieved by weighting the distance function by the density in the sense of Bell et al. The concept of distance-to-measure is incorporated to enhance stability and mitigate noise. The persistence-prolonging property and robustness of the proposed filtration are rigorously established, and numerical experiments are presented to demonstrate the proposed filtration's utility in identifying small holes.

math.ST↗