Source-linked AI summary
Towards Persistence-Based Reconstruction in Euclidean Spaces
Frédéric Chazal, Steve Oudot
TL;DR
Existing reconstruction methods become impractical in high-dimensional ambient spaces, so the paper develops a persistence-based algorithm whose complexity depends on intrinsic dimension and provides topological guarantees for sampled shapes.
Problem
Existing reconstruction methods can scale exponentially with ambient dimension, limiting their practicality for low-dimensional structures in medium- or high-dimensional spaces.
Method
The algorithm iteratively selects landmarks while maintaining nested Rips or witness complexes and tracking their persistent Betti numbers.
Results
The approach provides efficient, provably good topological estimation in arbitrary dimensions, with complexity scaling with the intrinsic dimension for smooth submanifolds.
Takeaways & Limitations
The framework offers a practical route toward reconstructing low-dimensional manifolds in high-dimensional spaces with topological guarantees.
Takeaways & Limitations
For higher-dimensional manifolds, positive weights may be required because the unweighted restricted Delaunay triangulation can fail to capture the topology.
Abstract
from arXiv · showhide
Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes of compact subsets of Euclidean space $\R^d$. However, the complexities of these methods scale up exponentially with d, which makes them impractical in medium or high dimensions, even for handling low-dimensional submanifolds. In this paper, we introduce a novel approach that stands in-between classical reconstruction and topological estimation, and whose complexity scales up with the intrinsic dimension of the data. Specifically, when the data points are sufficiently densely sampled from a smooth $m$-submanifold of $\R^d$, our method retrieves the homology of the submanifold in time at most $c(m)n^5$, where $n$ is the size of the input and $c(m)$ is a constant depending solely on $m$. It can also provably well handle a wide range of compact subsets of $\R^d$, though with worse complexities. Along the way to proving the correctness of our algorithm, we obtain new results on Čech, Rips, and witness complex filtrations in Euclidean spaces.
Vers une reconstruction bas´ee sur la persistance dans les espaces euclidiens
L’article propose une approche de reconstruction située entre reconstruction classique et inférence topologique, dont la complexité dépend de la dimension intrinsèque plutôt que de la dimension ambiante. Elle combine un raffinement glouton de type maxmin avec la persistance topologique et s’appuie sur de nouveaux résultats concernant plusieurs filtrations.
- Motivation: Les méthodes récentes pour les sous-variétés lisses de dimension arbitraire deviennent impraticables car leur complexité croît exponentiellement avec la dimension ambiante d.
- Approche: L’approche proposée fait croître la complexité avec la dimension intrinsèque des données et combine raffinement glouton type maxmin et persistance topologique.L’algorithme construit itérativement un sous-ensemble de landmarks à partir d’un nuage de points dans R^d.
- Fondements théoriques: Les garanties théoriques reposent sur l’étude des filtrations de Čech, de Rips et de complexes de témoins dans R^d.Les auteurs transfèrent des résultats existants sur les unions de boules vers ces filtrations et présentent de nouveaux résultats.
1 Introduction
The paper addresses the impractical dimension dependence of manifold reconstruction by combining greedy landmark refinement with topological persistence. It establishes structural results linking Čech, Rips, and witness filtrations to this reconstruction approach.
- 2O(d2)n2 remains too large for recent reconstruction methods, even when samples lie on or near a low-dimensional submanifold.
- Persistent homology of the α-shape filtration does not by itself establish agreement with the underlying shape because the relevant nerve equivalences may not commute with inclusions.
- Commuting homotopy equivalences at homology and homotopy levels extend prior results from α-shapes to Čech, Rips, and witness complex filtrations.
- The witness complex uses non-landmark samples to drive construction and may make topological noise depend on the density of W rather than the landmark set L.
- The proposed algorithm combines greedy refinement and topological persistence while iteratively building landmarks and maintaining a nested pair of simplicial complexes.
2 Various complexes and their relationships
This section defines Čech, Rips, and witness complexes in Euclidean spaces and establishes their relationships through inclusion bounds. In particular, witness complexes lie between Čech complexes under sampling and coverage conditions.
- Scope and tightness: The bounds are stated for arbitrary metric spaces, so they are not tightest in Euclidean spaces but retain generality and simpler statements.Tighter Euclidean bounds are possible, including prior results for Rips complexes and a combined approach for witness complexes.
- Čech complex: A Čech simplex exists exactly when the corresponding open balls have a nonempty common intersection.The Čech complex is the nerve of the open-ball cover centered at the finite point set.
- Rips complex: A Rips simplex consists of points that are pairwise within Euclidean distance α, yielding the standard Čech–Rips relationship.The section proves the relationship using the common-intersection and pairwise-distance conditions.
- Witness complex: An α-witness complex contains simplices whose vertices are α-witnessed by points of a witness set, generalizing the standard witness complex at α = 0.Witnessing uses distances to the (k + 1)th nearest landmark, and the complex is maximal among complexes with witnessed faces.
- Witness–Čech relationships: Under the stated sampling assumptions, the witness complex is bounded between Čech complexes: C_{α−2(ε+δ)}(L) ⊆ C_α^W(L) ⊆ C_{2α+6(ε+δ)}(L).The assumptions include dH(X,W) ≤ δ, dH(W,L) ≤ ε, and ε + δ < 1/2; when δ ≤ ε, the corollary gives a specialized bound.
3 Structural properties of filtrations over compact subsets of Rd
Below the weak feature size, all offsets of a compact set share the same topology, enabling persistent homology and homotopy information from sampled Čech and Rips-type filtrations to recover the topology of offsets. For smooth submanifolds, these offset invariants recover the original set’s homology.
- Offset topology: Offsets X_α and X_α′ are isotopic when no critical distance value lies in [α, α′], and every offset below wfs(X) has the same topology.The larger offset deformation retracts onto the smaller one.
- Sampled offsets: If dH(X,L)<ε<1/4 wfs(X), then homology of X_λ is the image of H_k(L_α) → H_k(L_α′) for α,α′∈[ε,wfs(X)−ε] with α′−α≥2ε.The analogous statement holds for based homotopy groups.
- Manifold case: For a smooth submanifold X, sufficiently small offsets X_λ are homotopy equivalent to X, so the sampled-offset theorem retrieves the homology of X itself.For general compact sets, it retrieves the homology of sufficiently small offsets, which need not be homotopy equivalent to X.
- Čech filtration: For the Čech filtration, the homology of X_λ is obtained from im(H_k(C_α(L)) → H_k(C_α′(L))) under the same sampling conditions, with barcode noise bounded by 2ε.Intervals of length at least 2ε represent the homology of X_λ.
- Rips and witness filtrations: Theorems 3.6–3.9 extend the image characterization to Rips and witness-type filtrations, recovering homology and based homotopy groups from inclusions between appropriately scaled complexes.More generally, any filtration intertwined with the Čech filtration can recover offset homology, although its barcode noise may grow linearly with α.
4 The case of smooth submanifolds of Rd
For smooth positive-reach submanifolds, the paper shows that witness-complex persistence can reveal homology earlier than the Čech and Rips scales under suitable sampling. The result relies on weighted restricted Delaunay triangulations, which are homeomorphic to the manifold and inject into appropriate witness complexes.
- 4 The case of smooth submanifolds of Rd: The analysis assumes a positive-reach submanifold and landmark samples that are both sufficiently dense and sufficiently separated.An ε-sample covers the manifold within ε, while an ε-sparse sample keeps points at least ε apart.
- 4 The case of smooth submanifolds of Rd: Theorem 4.1 supports the conjecture that witness filtrations produce cleaner barcodes by showing that long homology intervals can appear earlier than in Čech and Rips filtrations.The theorem addresses earlier appearance of long intervals, while the passage frames barcode cleanliness in terms of reduced noise and earlier long intervals.
- 4 The case of smooth submanifolds of Rd: Theorem 4.1 guarantees a subcomplex homeomorphic to X whose homology injects into the α-witness complex under the stated sampling and α conditions.This identifies a range of α values in which the topology of X is captured inside the witness complex.
- 4 The case of smooth submanifolds of Rd: A weighted restricted Delaunay triangulation provides the proof’s geometric intermediary because it is homeomorphic to X and relates directly to the α-witness complex.The argument uses weighted power diagrams, restricted nerves, and the inclusion of the restricted Delaunay complex into a witness complex.
- 4 The case of smooth submanifolds of Rd: For curves and surfaces, zero weights suffice, whereas higher-dimensional manifolds may require positive weights because unweighted restricted Delaunay complexes can miss topological invariants.The required weights become smaller as the landmark set becomes denser.
5 Application to reconstruction
The section presents a greedy landmark algorithm that maintains nested Rips complexes and uses persistent Betti-number plateaus to recover homological information about compact sets. Its Euclidean complexity is O(33d|W|^5) generally and O(35m|W|^5) for sufficiently sampled smooth m-submanifolds.
- Algorithm: The algorithm greedily adds the witness farthest from the current landmarks, updates ε, rebuilds R4ε(L) and R16ε(L), and computes their persistent homology.It terminates when all witnesses are landmarks and outputs the evolution of persistent Betti numbers versus ε.
- Theoretical guarantee: When W is a sufficiently dense δ-sample of compact X, persistent homology matches the homology of Xλ whenever δ < ε(i) < 1/18wfs(X).This creates a plateau of persistent Betti numbers whose width is at least 1/18wfs(X) −δ.
- Theoretical guarantee: The output is a nested pair of abstract complexes whose Euclidean images lie at Hausdorff distance O(ε) of X while their persistent homology matches Xλ.Users or software agents can select a relevant scale by detecting one or more plateaus in the Betti-number diagram.
- Complexity: O(|W||L| + |R16ε(L)||L|3 + |R16ε(L)|3) is the running time of one algorithm iteration.The terms account for landmark-distance updates, rebuilding the complexes, and computing persistence.
- Complexity: O(33d|W|^5) is the running time for arbitrary Euclidean point clouds, improving to O(35m|W|^5) for sufficiently sampled smooth m-submanifolds.The dimension-dependent bound follows from the general simplex-size estimate, while the manifold bound depends on intrinsic dimension m.
6 Conclusion
The paper presents an efficient, provably good, and easy-to-implement algorithm for topological estimation of general shapes in any dimension. It also provides a framework for analyzing other persistence-based methods and outputs a nested pair of complexes at a user-defined scale for homology estimation.
- The approach yields an efficient, provably good, and easy-to-implement algorithm for topological estimation of general shapes in any dimensions.
- The theoretical framework can also support analysis of other persistence-based methods.
- The algorithm addresses a weaker version of classical reconstruction by outputting a nested pair of complexes at a user-defined scale.