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Linear-Size Approximations to the Vietoris-Rips Filtration

Donald R. Sheehy

arXiv:1203.6786v2cs.CGcs.DS

TL;DR

Vietoris–Rips filtrations encode useful topology but are too large to construct fully. The paper uses hierarchical net-trees to build sparse zigzag and non-zigzag filtrations, obtaining linear size, near-linear construction, and persistence diagrams that approximate the original.

  • Problem

    Vietoris–Rips filtrations are useful for topological data analysis but can be too large, with even their k-skeleton having size O(n^(k+1)).

  • Method

    The paper constructs sparse zigzag and non-zigzag Vietoris–Rips filtrations using a hierarchical net-tree and topology-preserving point removal.

  • Results

    The sparse filtrations have size O(n), can be computed in O(n log n) time, and yield c-approximations with c = 1/(1−2ε), where ε ≤ 1/3.

  • Takeaways & Limitations

    The method provides an efficient way to approximate persistent homology across all scales for larger metric-space inputs.

  • Takeaways & Limitations

    The sparse zigzag filtration is initially underspecified until the point deletion times are computed from the net-tree.

Abstract

from arXiv · show

The Vietoris-Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is often extracted from its so-called persistence diagram. Unfortunately, this filtration is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an $n$-point metric space such that its persistence diagram is a good approximation to that of the Vietoris-Rips filtration. This new filtration can be constructed in $O(n\log n)$ time. The constant factors in both the size and the running time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets. We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guarantees.

1 Introduction

The paper addresses the tension between Vietoris–Rips filtrations’ useful topological information and their rapid size growth by constructing sparse filtrations with approximation guarantees.

  • Vietoris–Rips filtrations encode useful topology but can grow exponentially in the number of input points.Even the k-skeleton has size O(n^(k+1)).
  • The paper proposes a sparse Vietoris–Rips filtration with size O(n) and construction time O(n log n).The constants depend on the metric’s doubling dimension and user-defined approximation parameter ε.
  • The construction uses a hierarchical net-tree to obtain sparse metric approximations with topological guarantees.The net-tree is the main construction tool and is related to hierarchical metric spanners and nearest-neighbor data structures.
  • The paper develops a sparse zigzag filtration in which simplices can be added and removed as scale increases.Removing points is shown not to change topology, so the zigzag does not zigzag at the homology level.
  • The zigzag construction can be converted into an ordinary filtration that also approximates the Vietoris–Rips filtration.Both constructions are analyzed through their persistence diagrams.

2 Related Work

Prior work includes persistent homology, sparse metric and Vietoris–Rips approximations, and combinatorial simplifications, but this paper targets topologically guaranteed sparsification of an entire filtration.

  • Persistent homology has been applied across domains, while zigzag persistence allows complexes to alternate between growing and shrinking.Zigzag persistence extends the usual inclusion-only setting.
  • Existing Vietoris–Rips approximations include witness complexes and mesh-based methods for Euclidean spaces.
  • Chazal and Oudot studied persistence diagrams from increasingly dense subsamples but lacked a single diagram with a provable guarantee.They also lacked general filtration-size guarantees except under strict data assumptions.
  • Combinatorial simplification methods had simplified static complexes rather than filtrations.This paper instead exploits geometry to sparsify an entire filtration while preserving topology.

3 Background

This section introduces doubling metrics, simplicial complexes, homology, filtrations, persistence diagrams, zigzag modules, and the equivalence tools used to compare them.

  • A doubling metric has bounded doubling dimension, defined from the number of radius-r balls needed to cover any radius-2r ball.The doubling dimension is d = ⌈lg λ⌉.
  • A metric with doubling dimension d and spread Δ contains at most Δ^O(d) points.This follows by recursively covering larger balls with smaller balls until each contains at most one point.
  • A simplicial complex is a collection of vertices and subsets closed under taking subsets, with simplex dimension |σ| − 1.
  • A filtration is a nested sequence of spaces, and a filtered simplicial complex consists of nested simplicial complexes.Applying homology to the filtration produces a persistence module.
  • A persistence diagram records feature birth and death times, with distant-from-diagonal points representing longer-lived features.The persistence algorithm computes the diagram from a filtration, including zigzag filtrations.
  • A c-approximation matches diagram points while allowing birth and death times to differ by at most a factor c.The paper relates this definition to bottleneck-distance stability.
  • Contiguous simplicial maps induce identical homology maps, enabling retractions to establish homology isomorphisms.The persistence equivalence theorem also gives identical diagrams when commuting sequences have vertical isomorphisms.

4 The Relaxed Vietoris–Rips Filtration

The relaxed Vietoris–Rips filtration perturbs distances with scale-dependent point weights, allowing local coverage and later topology-preserving deletions while remaining multiplicatively close to the original filtration.

  • The relaxed distance adds scale-dependent weights to points, effectively shrinking metric balls so nearby balls can cover them.This supports removing points that contribute little to the union of balls.
  • Each point receives a nonnegative deletion time, and its weight is defined piecewise around that time.The weight is zero before the transition, then grows with slope 1/2 and later with slope ε.
  • The relaxed distance is monotonically nondecreasing in α and remains at least the original distance.Its controlled growth supports the subsequent comparison arguments.
  • A Vietoris–Rips complex contains simplices whose pairwise distances under the chosen distance function are at most α.The original and relaxed filtrations use the same construction with d and the relaxed distance, respectively.
  • The original and relaxed filtrations are related by a multiplicative interleaving, with c = 1/(1−2ε).The displayed inclusion bounds the relaxed construction between scale-shifted versions of the original one.

5 The Sparse Zigzag Vietoris–Rips Filtration

The sparse zigzag Vietoris–Rips filtration constructs a linear-size subcomplex by deleting points as scale increases, while preserving the relaxed filtration’s persistence diagram.

  • A sparse subcomplex of the relaxed Vietoris–Rips complex is constructed with linear size at every scale.
  • The resulting zigzag filtration has only a linear total number of simplices and an identical persistence diagram to the relaxed Vietoris–Rips filtration.
  • The open net Nα contains points whose deletion times exceed α.
  • At scale α, the sparse zigzag complex Qα is the subcomplex of the relaxed complex induced by the vertices in Nα.
  • The filtration changes only at discrete scales containing point deletions and edge insertions.
  • The deletion-time construction is deferred until the net-tree section, leaving Q underspecified at this stage.

6 Hierarchical Net-Trees

Hierarchical net-trees provide the scale-dependent sparse nets used by the filtration, with packing and covering guarantees that support efficient construction and controlled point deletion.

  • A net-tree is a rooted tree whose vertices represent input points, with leaves corresponding bijectively to the metric-space points.
  • Each net-tree level gives a sparse approximation at a different scale, while larger disks cover points and smaller disks satisfy disjoint packing.
  • Net-tree nodes have radii that grow by a constant factor across levels, yielding λ^O(1) children per node.
  • The net-tree can be constructed in O(n log n) randomized time or O(n log ∆) deterministic time without knowing the doubling dimension in advance.
  • Each point p receives deletion time tp = 1/[ε(1 − 2ε)] rad(par(vp)), and is removed when α reaches tp.
  • For each scale α, the induced net Nα satisfies packing and covering conditions.
  • Distinct points in Nα are separated by at least Kpε(1 − 2ε)α.
  • Every input point lies within ε(1 − 2ε)α of some point in Nα.

7 Topology-preserving sparsification

The sparsification preserves homology by projecting deleted points onto surviving net points without increasing relaxed distances, making the sparse complexes topologically equivalent to the corresponding relaxed complexes.

  • Deleting a vertex p and its incident simplices from the relaxed complex at scale tp does not change its topology.
  • The projection πα maps each point onto a point in Nα, the surviving net at scale α.
  • The distance from a point to its projection is bounded by the difference between their weights.
  • Projection distances do not increase: ˆdα(πα(p), q) ≤ ˆdα(p, q).
  • For Nα ⊆ X ⊆ P, inclusion of Qα into VR(X, ˆdα, α) induces a homology isomorphism.
  • The inclusions among Qα, Q̄α, ˆRα, and R̂α induce homology isomorphisms at every discrete scale.

8 Straightening out the Zigzags

The paper removes zigzag behavior while preserving the sparse filtration’s persistence information, yielding a non-zigzag filtration with the same persistence diagram. Homology isomorphisms and extended-persistence arguments establish this equivalence.

  • 8 Straightening out the Zigzags: The sparse zigzag filtration can be modified into an ordinary filtration without changing its persistence diagram.The construction reverses homology isomorphisms and uses closed sparse Vietoris–Rips complexes.
  • 8 Straightening out the Zigzags: The zigzag is sparse at the complex level but computes a persistence module that does not zigzag.The zigzagging is interpreted as sparsifying the complex without changing topology.
  • 8.2 A Sparse Filtration without the Zigzag: For each index, the inclusion from Q_a_k to S_a_k induces a homology isomorphism.The proof constructs intermediate complexes and simplicial retractions whose contiguous maps establish the isomorphism.
  • 8.2 A Sparse Filtration without the Zigzag: The persistence diagrams of Q and S are identical.The result follows from commutative diagrams whose maps induce homology isomorphisms and the Persistence Equivalence Theorem.
  • 8.3 The Connection with Extended Persistence: The sparse zigzag can be related to an extended filtration whose first half is the sparse Vietoris–Rips filtration.The second half contributes no extended or relative pairs, so extended persistence need not be computed.

9 Theoretical Guarantees

The sparse filtrations provide multiplicative persistence-diagram approximations to Vietoris–Rips while retaining linear total size. Their guarantees depend on the approximation parameter, doubling dimension, and net-tree packing behavior.

  • 9.2 Size Complexity: The sparse filtrations have linear total size in the number of input points.The size proof begins by bounding the number of filtration edges and then controls higher-order simplices.
  • 9.1 The Approximation Guarantee: Both sparse filtrations yield c-approximations to the Vietoris–Rips persistence diagram, with c = 1/(1−2ε) and ε ≤ 1/3.The proof compares the classical and relaxed filtrations, then identifies the sparse and relaxed persistence diagrams.
  • 9.2 Size Complexity: The number of neighbors retained for each point is bounded by a quantity exponential only in the doubling dimension.The bound follows from controlling the spread of each neighbor set using net-tree separation and distance bounds.
  • 9.2 Size Complexity: The filtration graph is governed by the doubling dimension, the net-tree packing constant, and the approximation tightness ε.These variables determine the neighbor-set bound and therefore the number of higher-order simplices.

10 An algorithm to construct the sparse filtration

Construction uses a net-tree to define deletions, find nearby points, and enumerate cliques efficiently. Neighbor searches take constant work per point, leaving net-tree construction as the only superlinear step.

  • 10 An algorithm to construct the sparse filtration: The net-tree defines point deletion times and supplies the data structure for finding neighbors in the perturbed metric.After neighbor discovery, the filtration is obtained by finding cliques.
  • 10 An algorithm to construct the sparse filtration: Relative-tree searches find the nearby points by ascending only a constant number of net-tree levels and searching a constant number of subtrees.The relatives of a node are same-level nodes represented by geometrically close points, and their number is constant.
  • 10 An algorithm to construct the sparse filtration: The net-tree’s relative edges support a range search for the enlarged neighbor set E′(p).The search starts at a suitable ancestor and descends through relative subtrees to the level of the point.
  • 10 An algorithm to construct the sparse filtration: The total construction time is O(n log n), because neighbor searches and clique computation take constant time per point.The net-tree computation is the only superlinear component.

11 Conclusions and Directions for Future Work

The paper presents an efficient approximation to persistent homology for the Vietoris–Rips filtration and identifies directions for extending the approach to other filtrations and hierarchical structures.

  • Computing approximate persistence diagrams at all scales may make persistence-based methods on metric spaces tractable for much larger inputs.
  • The same sparsification proofs may extend to the Čech filtration and possibly to complexes using alternative distance-like functions.
  • A more general class of hierarchical structures could support similar constructions beyond the net-tree used here.
  • The paper’s analytic technique bounds differences between persistence diagrams by embedding a zigzag filtration into a homology-equivalent non-zigzag filtration.
  • This technique is closely related to the relationship between levelset zigzag and extended persistence and may apply to other zigzag filtrations.
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