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Metrics for generalized persistence modules
Peter Bubenik, Vin de Silva, Jonathan Scott
TL;DR
The paper asks how to define interleaving metrics for generalized persistence modules over arbitrary preordered sets. It develops a functorial categorical framework covering arbitrary target categories and standard persistence examples, and shows that suitable additional indexing structure yields an extended pseudometric with soft stability. The framework is broad but uses “module” loosely rather than in the strict abelian-category sense.
Problem
Existing persistence theory motivates extending interleaving metrics beyond standard indexing sets and target categories while retaining stability.
Method
The paper models generalized persistence modules as functors P → D and uses interleavings with sublinear projections or superlinear families to define metrics.
Results
An added sublinear projection or superlinear family induces an extended pseudometric on functors P → D, including the usual interleaving distance over R.
Takeaways & Limitations
The framework encompasses standard persistence theories, including classical sublevelset persistent homology, while extending the range of indexing and target categories.
Takeaways & Limitations
The paper uses “module” in a loose sense, despite the term’s association with abelian categories.
Abstract
from arXiv · showhide
We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of examples, inverse-image persistence modules, which occur whenever a topological space is mapped to a metric space. Several standard theories of persistence and their stability can be described in this framework. This includes the classical case of sublevelset persistent homology. We introduce a distinction between `soft' and `hard' stability theorems. While our treatment is direct and elementary, the approach can be explained abstractly in terms of monoidal functors.
Introduction
Persistence replaces unstable fixed-scale topological calculations with systems of invariants, motivating an algebraic and categorical framework for generalized persistence and stability metrics.
- Persistence evaluates homology across all scales, making quantities unstable at one scale stable across the full range.
- Persistence diagrams encode algebraic structure rather than depending directly on topology, freeing persistence theory from its topological origins.
- Functoriality explains how simplicial-complex diagrams become vector-space diagrams while preserving identities and composition.
- The paper develops metrics and stability for generalized persistence modules over arbitrary posets or preordered sets, with interleavings quantified by additional structure.
- The framework connects classical persistence variations, including multidimensional, zigzag, extended, levelset, angular, image, kernel, and cokernel persistence.
1. Persistence Modules as Functors
The paper represents persistence constructions as functors and natural transformations, then generalizes persistence modules from standard indexing and target categories to arbitrary preordered sets and categories.
- Categories: A category comprises objects, morphisms, associative composition, and identity morphisms.
- Functors and natural transformations: Functors map objects and morphisms compatibly with categorical structure, while natural transformations provide maps between functors.
- Persistence diagrams as functors: Homology converts simplicial-complex diagrams into vector-space diagrams because homology is a functor.
- Persistence diagrams as functors: A nested simplicial-complex family is a functor from an indexing category whose morphisms represent inclusions between complexes.
- Generalized persistence modules: Generalized persistence modules are functors P → D, where P may be any preordered set and D any category.
- Scope: The paper uses “module” loosely, despite the term’s strong association with abelian categories.
- Examples: The framework includes sublevelset modules, formed by taking subspaces f^-1(-∞, t] and inclusion maps for s ≤ t.
R Top VectF
The supplied passages introduce homology-valued constructions beyond the basic generalized persistence-module setup, including merge trees and homology with field coefficients.
- Homology in this construction uses coefficients in a field F.
- The merge tree of f is obtained as a composite construction.
R Top Set
The paper frames stability for generalized persistence modules through functorial constructions, covering inverse-image modules and standard persistence theories. It distinguishes soft from hard stability and emphasizes formal, reusable stability arguments.
- Examples: The framework includes standard persistence theories, including sublevelset persistent homology and merge-tree constructions.Connected components of sublevelsets produce the persistence data underlying a traditional merge tree.
- Metric structure: A preordered set with a sublinear projection or superlinear family supplies the additional structure needed for interleaving metrics.The paper presents these as two alternative proposals for structuring the indexing category.
- Stability: The interleaving distance is an extended pseudometric, and post-composition with a functor is 1-Lipschitz.This formalizes stability for arbitrary categories and functors.
- Inverse-image persistence: Inverse-image persistence modules arise when a topological space is equipped with a function into a metric space.The construction yields persistence modules in topological spaces and supports an inverse-image stability theorem under a translation condition.
- Stability perspective: The paper treats these stability theorems as formal consequences whose straightforward proofs isolate reusable aspects of stability.The authors present the approach as extending classical results while retaining essentially the same arguments.
- Soft and hard stability: Soft stability concerns Lipschitz or uniformly continuous composite constructions, whereas hard stability concerns Lipschitz or uniformly continuous intermediate operations.The paper focuses on soft stability and the high-level arguments producing it.
2. Interleaving metrics.
The paper develops interleavings for persistence modules indexed by preordered sets and equips them with metrics using translation-based structure. These metrics are extended pseudometrics, functorial under post-composition, and recover the classical real-line distance.
- Motivation: Interleaving provides the approximate-isomorphism relation needed to measure proximity between persistence modules.The paper develops interleavings qualitatively before adding sublinear projections or superlinear families to define metrics.
- Translations: A translation is a monotone self-map Γ of a preordered set satisfying x ≤ Γ(x), and it induces a shift map F ⇒ FΓ.The shift map is obtained by applying the persistence-module functor to each relation x ≤ Γ(x).
- Interleavings: A (Γ,K)-interleaving consists of natural transformations F ⇒ GΓ and G ⇒ FK satisfying two compatibility equations.The symmetric case Γ=K is called a Γ-interleaving.
- Classical case: For real-indexed persistence modules, an Ωε-interleaving is the classical ε-interleaving, and the resulting metric recovers the usual interleaving distance.The paper explicitly identifies d with the usual interleaving distance over the real line.
- Metric properties: The interleaving distance is an extended pseudometric and satisfies d(HF,HG) ≤ d(F,G) for every functor H.Thus post-composition is nonexpansive with respect to the induced metrics.
- Two metric constructions: Under the stated maximality condition, a sublinear projection and its superlinear family define equal interleaving distances.The relationship can be interpreted through a co-unit of an adjunction.
- Functoriality: With suitable structure on P, the construction is functorial from small categories to extended pseudometric spaces and 1-Lipschitz maps.This packages metric persistence-module constructions categorically.
3. Functions into a metric space
The paper derives interleaving metrics for persistence modules indexed by subsets of a metric space using asymmetric Hausdorff distance and metric offsets. Inverse images then transfer these constructions to persistence modules built from maps into the metric space.
- Subset-indexed modules: A metric space induces a poset of subsets whose generalized persistence modules recover several standard persistence theories.The construction works over subposets of the subset poset.
- Hausdorff metrics: The asymmetric Hausdorff distance measures how far one subset escapes from another and supplies the relevant Lawvere-metric structure.Its reflexivity and triangle inequality follow from the corresponding conditions on the metric dM.
- Metric equivalence: The sublinear projections derived from asymmetric and symmetric Hausdorff distances are equal.This removes the need to choose between the two constructions for the resulting metric.
- Induced metric: The induced metric is an extended pseudometric defined by the sublinear projection associated with the metric on the underlying space.It is denoted by ωℓ=ωd in the paper.
- Offsets: The ε-offset Aε contains points whose distance from A is at most ε, and the operation A ↦ Aε is a translation.The associated translations form a superlinear family, and the two interleaving distances agree.
- Inverse images: A function f:X→M induces an order-preserving inverse-image map from subsets of M to subsets of X.This inverse-image map is therefore a functor and can be used to build persistence modules.
F : P SubsetsM SubsetsX Top
The paper models inverse-image persistence by sending functions into generalized persistence modules over subsets of a metric space. Under suitable translation conditions, this construction is non-expanding for the induced metrics.
- The inverse-image construction sends a function f:X→M to a generalized persistence module in Top over a poset P of subsets of M.
- The induced function metric d∞ and persistence-module metric dω motivate asking how Inv behaves between these metric spaces.
- A family P has enough translations when each weak ε-offset is included in a translation Γ(A) whose weight is at most η for every 0≤ε<η.
- Offset and weak-offset operations provide translations for subsets, closed subsets, compact subsets in proper metric spaces, and open subsets with an η-dependent construction.
- The inverse-image stability theorem states that Inv is 1-Lipschitz, so nearby functions yield persistence modules no farther apart under the induced metrics.
- The proof uses translations associated with η>ε and obtains the bound by letting η approach the function distance ε.
P SubsetsM SubsetsX
Functoriality transfers inverse-image stability through arbitrary target functors, yielding a common framework for standard persistence theories. The framework distinguishes soft interleaving bounds from hard stability for persistence diagrams.
- Applying any functor H:Top→E to inverse-image modules preserves the stability conclusion.
- Sublevelset persistence: For sublevelset persistence, ||f−g||∞≤ε implies that the corresponding persistent homology modules are ε-interleaved.
- Multidimensional persistence: The same ε-interleaving implication extends to multidimensional sublevelset persistent homology using lower quadrants in R^n with the ℓ∞ metric.
- Other theories: Levelset, circular, and copresheaf persistence arise as inverse-image theories with corresponding stability bounds.
- Soft and hard stability: These interleaving-distance bounds are called soft stability theorems, whereas hard stability additionally relates interleaving distance to bottleneck distance between persistence diagrams.
4. Monoidal structures
The paper interprets interleaving metrics through monoidal structures and adjunctions, and extends the framework to vector-valued persistence. It also emphasizes broad categorical applicability while separating general soft stability from problem-specific hard invariants.
- Vector persistence: Comparing translations with [0,∞]^n or [0,∞)^n yields stable vector persistence, while multidimensional persistence admits multiple natural metrics because its module theory is wild.
- An adjunction relation between ω and Ω makes both maps monotone and equates the sublinear-projection and superlinear-family conditions.
- Translation composition makes TransP a strict monoidal category, while nonnegative real values form monoidal categories under addition.
- Superlinear families correspond to lax monoidal functors, and monotone sublinear projections correspond to oplax monoidal functors.
- Closing remarks: The closing framework applies to arbitrary indexing preorders, target categories, metric-valued functions, and alternative algebraic-topology functors.
- Closing remarks: The authors distinguish constructing stable generalized modules from designing discrete invariants that are computationally useful and sufficiently discriminating.