Topological reconstruction of compact supports of dependent stationary random variables

TL;DR

This work develops a unified framework for topological reconstruction of the common support of dependent stationary vector sequences with compact, positive-reach supports. By combining a deterministic reconstruction theorem based on offsets and positive reach with probabilistic blocking techniques, the authors derive conditions under which finite samples are asymptotically (ε,α)-dense for a broad class of dependence structures, including m-dependent, m-approximable, β-mixing, and weakly dependent sequences, and apply these results to stationary Markov chains. In particular, explicit density bounds and sample-size thresholds are obtained for random difference equations and the Möbius Markov chain on the circle, with simulations confirming the topological recovery of the support (e.g., the circle). The framework thus extends manifold-learning-type reconstruction to dependent data, offering practical criteria for topology inference from correlated observations in diverse applications. The results bridge probabilistic dependence, geometric regularity via positive reach, and topological inference, enabling robust reconstruction of complex supports from real-world time series.

Abstract

In this paper we extend results on reconstruction of probabilistic supports of random i.i.d variables to supports of dependent stationary $\mathbb R^d$-valued random variables. All supports are assumed to be compact of positive reach in Euclidean space. Our main results involve the study of the convergence in the Hausdorff sense of a cloud of stationary dependent random vectors to their common support. A novel topological reconstruction result is stated, and a number of illustrative examples are presented. The example of the Möbius Markov chain on the circle is treated at the end with simulations.

Figures (5)

• Figure 1: This space has positive reach $\tau$ in ${\mathbb R}^2$ but a neighborhood of point $A$ indicates it is not a submanifold (with boundary).
• Figure 2: An extreme disposition of points, $x,q\in S$ and $p,q\in\partial M$. The points $q,p$ are on a circle tangent to $T_p(M)$, of radius $\tau$ and center on the vertical dashed line representing the normal direction $T_p^{\perp,+}(M)$, pointing in the exterior region, while $x$ is anywhere in $M\cap S$ at a distance at most ${\epsilon\over 2}$ from $p$.
• Figure 3: The ball $B(u,\epsilon)$ intersects the unit circle at two points $A$ and $B$.
• Figure 4: Illustrations of the set $\{x_1,\cdots,x_{n}\}$ which is a realisation of the stationary random variables ${\mathbb X}_n = \{X_1,\ldots, X_n\}$ for different values of $n$ and with $\varphi=0$.
• Figure 5: In the above graphics, the points of ${\mathbb X}_n$ are in red. Each of these points is the center of the circle with radius $r=0.1$. This is an illustration of the reconstruction result $\bigcup_{x\in{\mathbb X}_n}B(x,r)\simeq M$, with different values of $n$ and with $r=0.1$. In the above, there is reconstruction when $n=140$ and $r=0.1$. The density $\epsilon\over 2$ is at least ${2\pi\over 280} = 0.0224$, and so $\epsilon$ is at least $0.0448$. For this value of $\epsilon=0.0448$, $\epsilon <r=0.1$ and this reconstruction is consistent with Theorem \ref{['mainrecon']}.

Theorems & Definitions (39)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Theorem 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Definition 2.5