The Shadow of Vietoris--Rips Complexes in Limits

Abstract

The Vietoris-Rips complex, denoted $R_\beta(X)$, of a metric space $(X,d)$ at scale $\beta$ is an abstract simplicial complex where each $k$-simplex corresponds to $(k+1)$ points of $X$ within diameter $\beta$. For any abstract simplicial complex $K$ with the vertex set $K^{(0)}$ a Euclidean subset, its shadow, denoted $S(K)$, is the union of the convex hulls of simplices of $K$. This article centers on the homotopy properties of the shadow of Vietoris-Rips complexes $K=R_\beta(X)$ with vertices from $\mathbb{R}^N$, along with the canonical projection map $ p\colon R_\beta (X) \to S(R_\beta(X))$. The study of the geometric/topological behavior of $p$ is a natural yet non-trivial problem. The map $p$ may have many ``singularities'', which have been partially resolved only in low dimensions $N\leq 3$. The obstacle naturally leads us to study systems of these complexes $\{S(R_{\beta}(S)) \mid \beta > 0, S\subset X\}$. We address the challenge posed by singularities in the shadow projection map by studying systems of the shadow complex using inverse system techniques from shape theory, showing that the limit map exhibits favorable homotopy-theoretic properties. More specifically, leveraging ideas and frameworks from Shape Theory, we show that in the limit ``$\beta \to 0$ and $S \to X$'', the limit map ``$\lim p$'' behaves well with respect to homotopy/homology groups when $X$ is an ANR (Absolute Neighborhood Retract) and admits a metric that satisfies some regularity conditions. This results in limit theorems concerning the homotopy properties of systems of these complexes as the proximity scale parameter approaches zero and the sample set approaches the underlying space (e.g., a submanifold or Euclidean graph). The paper concludes by discussing the potential of these results for finite reconstruction problems in one-dimensional submanifolds.