Asymptotics of Lower Dimensional Zero-Density Regions

Abstract

Topological data analysis (TDA) allows us to explore the topological features of a dataset. Among topological features, lower dimensional ones have recently drawn the attention of practitioners in mathematics and statistics due to their potential to aid the discovery of low dimensional structure in a data set. However, lower dimensional features are usually challenging to detect from a probabilistic perspective.In this paper, lower dimensional topological features occurring as zero-density regions of density functions are introduced and thoroughly investigated. Specifically, we consider sequences of coverings for the support of a density function in which the coverings are comprised of balls with shrinking radii. We show that, when these coverings satisfy certain sufficient conditions as the sample size goes to infinity, we can detect lower dimensional, zero-density regions with increasingly higher probability while guarding against false detection. We supplement the theoretical developments with the discussion of simulated experiments that elucidate the behavior of the methodology for different choices of the tuning parameters that govern the construction of the covering sequences and characterize the asymptotic results.

Spherical Rotation Dimension Reduction with Geometric Loss Functions

“…SPCA works well for dataset living on a low-dimensional sphere S ∼ = S d ⊂ R d where the sphere dimension d ≤ d − 1. Intuitively, this problem is well-defined as an estimation problem of the (center, radius and dimension of) sphere S. The geometric constraints imposed by the SPCA method helps us to identify low-dimensional structures in the reduced dataset, which turns out to be exceedingly challenging (Luo et al, 2020). The objective of SPCA is not to minimize the geometric loss defined by the mean square distance.…”

Topological Learning for Motion Data via Mixed Coordinates

“…Motions are then represented as a discrete sequence of points (v(t), θ(t)) = f (t), t ∈ T = 1, • • • , N in C, joining these sequence of points f (t) in C we yield a piecewise linear curve f : [0, 2π] → C as a lowdimensional topological object in C [8] shown in Figure 1. In this way, even relatively simple motions can form geometrically complex curves in the space C. Human gaits are periodic motions, since humans would repeat actions over a period of time.…”