Interleaving Distance: Metrics & Applications

• Interleaving distance is a metric that quantifies the minimal shift required to compare persistence modules and categorical objects.
• It generalizes classical persistence by applying lax monoidal actions to yield Lawvere metric spaces and relates to standard metrics like the Hausdorff distance.
• It underpins stability in topological data analysis while presenting computational challenges in multi-parameter settings and merge tree comparisons.

Searching arXiv for the foundational paper and closely related work on interleaving distance. Interleaving distance is a family of comparison distances that originated in Topological Data Analysis as a metric on persistence modules and was later generalized to functor categories, categories with a flow, sheaf-theoretic settings, merge trees, mapper graphs, and related categorical structures. In its classical form, an $\varepsilon$-interleaving compares two objects after shifting one by $\varepsilon$ and requiring coherence of the resulting comparison maps; the interleaving distance is the infimum of such $\varepsilon$ (Silva et al., 2017). In the framework of categories with a flow, interleaving distance is defined on arbitrary categories equipped with a lax monoidal action of $([0,\infty),+,0)$, and the resulting object class carries the structure of a Lawvere metric space (Silva et al., 2017). This perspective unifies classical persistence-theoretic interleavings, generalized persistence on posets, and several familiar metrics, including the Hausdorff distance and the $L^\infty$-norm, as instances of interleaving distances (Silva et al., 2017).

1. Origins and classical formulation

The interleaving distance was originally defined in the field of Topological Data Analysis by Chazal et al. as a metric on the class of persistence modules parametrized over the real line (Silva et al., 2017). A persistence module is a functor

$F:(\mathbb{R},\le)\to \mathrm{Vect}_k.$

For $\varepsilon\ge 0$, the translation functor $T_\varepsilon:(\mathbb{R},\le)\to(\mathbb{R},\le)$ is defined by $T_\varepsilon(t)=t+\varepsilon$, together with the unit natural transformation $$\eta_\varepsilon:\mathrm{Id}\Rightarrow T_\varepsilon$$ (Silva et al., 2017).

An $\varepsilon$0-interleaving between persistence modules $\varepsilon$1 and $\varepsilon$2 is a pair of natural transformations

$\varepsilon$3

whose composites agree with the canonical “$\varepsilon$4-step” maps (Silva et al., 2017). The interleaving distance is then

$\varepsilon$5

with the convention $\varepsilon$6 if no $\varepsilon$7-interleaving exists (Silva et al., 2017).

In the one-parameter setting, the interleaving distance is an extended pseudometric, and in standard finiteness regimes it coincides with the bottleneck distance on barcodes (Bjerkevik et al., 2018). The same basic pattern extends to modules indexed by $\varepsilon$8 with product order, where an $\varepsilon$9-interleaving is given by morphisms

$\varepsilon$0

such that

$\varepsilon$1

and

$\varepsilon$2

(Bjerkevik et al., 2018).

This classical formulation underlies “soft stability” proofs: constructions from data to persistence modules are $\varepsilon$3-Lipschitz for this distance (Silva et al., 2017). A plausible implication is that interleaving distance is best understood not merely as a metric on outputs, but as a structural device for proving non-expansiveness of entire TDA pipelines.

2. Categories with a flow and Lawvere metric structure

A major abstraction replaces translations on a poset by a flow on an arbitrary category. Let $\varepsilon$4 be viewed as a strict monoidal category. A flow on a category $\varepsilon$5 is a lax monoidal functor

$\varepsilon$6

consisting of endofunctors $\varepsilon$7, multiplication natural transformations

$\varepsilon$8

and a unit natural transformation

$\varepsilon$9

satisfying the usual lax monoidal coherence axioms (Silva et al., 2017).

Given a category with a flow $([0,\infty),+,0)$0, an $([0,\infty),+,0)$1-interleaving between $([0,\infty),+,0)$2 is a pair of morphisms

$([0,\infty),+,0)$3

such that the composites

$([0,\infty),+,0)$4

and

$([0,\infty),+,0)$5

agree with the canonical maps $([0,\infty),+,0)$6 and $([0,\infty),+,0)$7, where

$([0,\infty),+,0)$8

(Silva et al., 2017).

The resulting interleaving distance

$([0,\infty),+,0)$9

satisfies reflexivity, symmetry, and the triangle inequality (Silva et al., 2017). Thus $L^\infty$0 is a symmetric extended pseudometric; equivalently, if one ignores symmetry, it is a Lawvere metric space (Silva et al., 2017).

The same paper proves a functoriality statement: the construction

$L^\infty$1

is functorial from the metacategory of flows and colax equivariant functors to the metacategory of Lawvere metric spaces and $L^\infty$2-Lipschitz maps (Silva et al., 2017). Later work further generalizes this perspective using monoidal actions and $L^\infty$3-categories, defining three increasingly general interleaving distances and proving that the resulting infima define extended pseudometrics (McFaddin et al., 2023).

3. Generalizations beyond classical persistence

Bubenik et al. extended the definition from $L^\infty$4-indexed persistence modules to categories of functors on a poset, whose objects may be regarded as generalized persistence modules (Silva et al., 2017). For a poset $L^\infty$5 and category $L^\infty$6, generalized persistence modules are functors $L^\infty$7. A chosen “superlinear family of translations” $L^\infty$8 yields an interleaving distance on $L^\infty$9 (Silva et al., 2017). In the flow formalism, a superlinear family of translations on a poset is precisely a flow $F:(\mathbb{R},\le)\to \mathrm{Vect}_k.$0 on $F:(\mathbb{R},\le)\to \mathrm{Vect}_k.$1, and the induced flow on $F:(\mathbb{R},\le)\to \mathrm{Vect}_k.$2 is given by precomposition $F:(\mathbb{R},\le)\to \mathrm{Vect}_k.$3 (Silva et al., 2017).

This subsumes many TDA settings, including merge trees and Reeb graphs, beyond the original $F:(\mathbb{R},\le)\to \mathrm{Vect}_k.$4 (Silva et al., 2017). For strict flows on $F:(\mathbb{R},\le)\to \mathrm{Vect}_k.$5, the categorical-flow interleaving distance agrees with the generalized interleaving distance of Bubenik–Scott–de Silva–Wagner; for lax flows, the flow-based distance is no larger:

$F:(\mathbb{R},\le)\to \mathrm{Vect}_k.$6

(Silva et al., 2017).

A separate line of work addresses posets that do not admit interesting translations. The relative theory of interleavings defines shifts on a poset $F:(\mathbb{R},\le)\to \mathrm{Vect}_k.$7 relative to a map $F:(\mathbb{R},\le)\to \mathrm{Vect}_k.$8, where $F:(\mathbb{R},\le)\to \mathrm{Vect}_k.$9 carries a superlinear family of translations (Botnan et al., 2020). Using Kan extensions, the relative $\varepsilon\ge 0$0-shift is

$\varepsilon\ge 0$1

and the relative weak interleaving distance satisfies the extend–restrict isometry

$\varepsilon\ge 0$2

(Botnan et al., 2020). This framework applies to zig-zag posets, face relation posets of cell complexes, down-set lattices, and grid structures for cosheaves over Euclidean space (Botnan et al., 2020).

A more recent generalization replaces translation structures by a height-difference function $\varepsilon\ge 0$3 on an arbitrary poset. Given $\varepsilon\ge 0$4, one defines $\varepsilon\ge 0$5-latching and $\varepsilon\ge 0$6-matching endofunctors

$\varepsilon\ge 0$7

with adjunction $\varepsilon\ge 0$8, and then defines the height-interleaving distance $\varepsilon\ge 0$9 (Aoki, 14 Mar 2026). When $T_\varepsilon:(\mathbb{R},\le)\to(\mathbb{R},\le)$0 and $T_\varepsilon:(\mathbb{R},\le)\to(\mathbb{R},\le)$1, this recovers the classical multiparameter interleaving distance (Aoki, 14 Mar 2026). However, in general $T_\varepsilon:(\mathbb{R},\le)\to(\mathbb{R},\le)$2 need not satisfy the triangle inequality; under the connected intersections property and with additive defect $T_\varepsilon:(\mathbb{R},\le)\to(\mathbb{R},\le)$3, one has

$T_\varepsilon:(\mathbb{R},\le)\to(\mathbb{R},\le)$4

(Aoki, 14 Mar 2026). This suggests that the classical translation-based theory is a particularly rigid case of a broader interleaving-type paradigm.

4. Realizations as classical metrics and stability mechanisms

One of the most consequential observations in the flow framework is that several standard metrics arise as interleaving distances. If $T_\varepsilon:(\mathbb{R},\le)\to(\mathbb{R},\le)$5 is a metric space and $T_\varepsilon:(\mathbb{R},\le)\to(\mathbb{R},\le)$6 denotes the poset of nonempty subsets ordered by inclusion, the flow

$T_\varepsilon:(\mathbb{R},\le)\to(\mathbb{R},\le)$7

induces an interleaving distance equal to the Hausdorff distance:

$T_\varepsilon:(\mathbb{R},\le)\to(\mathbb{R},\le)$8

(Silva et al., 2017).

Likewise, if $T_\varepsilon:(\mathbb{R},\le)\to(\mathbb{R},\le)$9 is viewed as the poset $T_\varepsilon(t)=t+\varepsilon$0 with coordinatewise order and one defines the strict flow $T_\varepsilon(t)=t+\varepsilon$1, then

$T_\varepsilon(t)=t+\varepsilon$2

(Silva et al., 2017).

For functions, one may work in the slice category $T_\varepsilon(t)=t+\varepsilon$3. The flow

$T_\varepsilon(t)=t+\varepsilon$4

recovers the $T_\varepsilon(t)=t+\varepsilon$5 distance on functions up to homeomorphism:

$T_\varepsilon(t)=t+\varepsilon$6

(Silva et al., 2017). The same construction extends to $T_\varepsilon(t)=t+\varepsilon$7 for any metric space $T_\varepsilon(t)=t+\varepsilon$8 using graph thickening (Silva et al., 2017).

These identifications clarify why interleavings are so effective in stability theory. If $T_\varepsilon(t)=t+\varepsilon$9 is colax $$\eta_\varepsilon:\mathrm{Id}\Rightarrow T_\varepsilon$$0-equivariant, with natural transformations

$$\eta_\varepsilon:\mathrm{Id}\Rightarrow T_\varepsilon$$1

then $$\eta_\varepsilon:\mathrm{Id}\Rightarrow T_\varepsilon$$2 is $$\eta_\varepsilon:\mathrm{Id}\Rightarrow T_\varepsilon$$3-Lipschitz:

$$\eta_\varepsilon:\mathrm{Id}\Rightarrow T_\varepsilon$$4

(Silva et al., 2017). This yields concise “soft stability” proofs by verifying colax equivariance of each pipeline component (Silva et al., 2017). Later formulations in weighted $$\eta_\varepsilon:\mathrm{Id}\Rightarrow T_\varepsilon$$5-categories prove analogous stability theorems for Lipschitz $$\eta_\varepsilon:\mathrm{Id}\Rightarrow T_\varepsilon$$6-functors (McFaddin et al., 2023).

In sheaf theory, the interleaving/convolution pseudo-distance is defined using kernels $$\eta_\varepsilon:\mathrm{Id}\Rightarrow T_\varepsilon$$7 and thickening functors

$$\eta_\varepsilon:\mathrm{Id}\Rightarrow T_\varepsilon$$8

on $$\eta_\varepsilon:\mathrm{Id}\Rightarrow T_\varepsilon$$9 (Petit et al., 2021). Under good geometric hypotheses and constructibility assumptions, distance zero implies isomorphism: if $\varepsilon$00 and $\varepsilon$01, then $\varepsilon$02 (Petit et al., 2021). This answers, in that setting, the separation question for the sheaf-theoretic interleaving distance (Petit et al., 2021).

5. Merge trees, ordered variants, and geometric reinterpretations

Merge trees constitute one of the most extensively studied non-linear settings for interleaving distance. A merge tree is a pair $\varepsilon$03, where $\varepsilon$04 is a rooted tree and $\varepsilon$05 is a continuous height function strictly increasing toward the root, with $\varepsilon$06 (Beurskens et al., 2023). For merge trees $\varepsilon$07 and $\varepsilon$08, a pair of maps $\varepsilon$09 and $\varepsilon$10 is a $\varepsilon$11-interleaving if for all $\varepsilon$12 and $\varepsilon$13,

$\varepsilon$14

$\varepsilon$15

(Beurskens et al., 2023). The induced distance is

$\varepsilon$16

(Beurskens et al., 2023).

Equivalent formulations are available through $\varepsilon$17-good maps and label-based constructions. For labelled merge trees $\varepsilon$18, the induced matrix

$\varepsilon$19

yields a label distance

$\varepsilon$20

and one has

$\varepsilon$21

(Beurskens et al., 2023). The intrinsic property of interleaving distance on spaces of labelled and unlabelled merge trees was established later: in the labelled setting, geodesics arise by linear interpolation of cophenetic matrices, and the distance is strictly intrinsic; in the unlabelled setting, interleaving distance is intrinsic as well (Gasparovic et al., 2019).

Ordered merge trees strengthen the model by equipping each level set with a consistent total order (Beurskens et al., 2023). The monotone interleaving distance $\varepsilon$22 requires interleaving maps to be order-preserving, and satisfies

$\varepsilon$23

(Beurskens et al., 2023). The paper "Relating Interleaving and Fréchet Distances via Ordered Merge Trees" shows that for ordered merge trees, if $\varepsilon$24 and $\varepsilon$25 are the induced $\varepsilon$26D curves obtained from in-order traversals, then

$\varepsilon$27

(Beurskens et al., 2023). As a consequence, the monotone interleaving distance can be computed exactly in near-quadratic time in the complexity of the trees (Beurskens et al., 2023). By contrast, the classical interleaving distance for merge trees is NP-hard to compute (Beurskens et al., 2023).

Recent work extends merge-tree interleaving in two directions. One line defines average merge trees: if $\varepsilon$28, a constructed representative average merge tree $\varepsilon$29 satisfies

$\varepsilon$30

(Touli et al., 28 Feb 2026). Another line develops heuristic polynomial-time algorithms for labelled and partially labelled merge trees, using leaf-based assignments and induced matrices to approximate interleaving distance in difficult regimes (Touli et al., 19 Sep 2025). Exact computation remains NP-hard, but improved fixed-parameter algorithms based on $\varepsilon$31-good maps and path-preserving maps yield substantially better parameter dependence than earlier approaches (P et al., 12 Feb 2026).

6. Computation, hardness, and approximation

The computational status of interleaving distance is sharply stratified. In one parameter, interleaving distance is computable in polynomial time because it coincides with bottleneck distance on barcodes (Bjerkevik et al., 2018). In multiple parameters, computing the interleaving distance is NP-hard (Bjerkevik et al., 2018). More precisely, deciding whether two bigraded persistence modules are $\varepsilon$32-interleaved is NP-complete, already for bigraded, interval-decomposable modules, and approximation within any factor smaller than $\varepsilon$33 is NP-hard (Bjerkevik et al., 2018).

The underlying reduction uses constrained matrix invertibility: given zero patterns $\varepsilon$34, one constructs bigraded modules $\varepsilon$35 such that

$\varepsilon$36

$\varepsilon$37

(Bjerkevik et al., 2018). The same hardness persists for indecomposable modules and for one-sided stability problems such as deciding existence of injections or surjections between persistence modules (Bjerkevik et al., 2018). Earlier work had already shown CI-hardness for $\varepsilon$38-indexed vector-space-valued modules and GI-completeness for isomorphism of $\varepsilon$39-indexed set-valued modules, including Reeb-graph isomorphism (Bjerkevik et al., 2017).

For merge trees, the classical interleaving distance is NP-hard (Beurskens et al., 2023). For mapper graphs, the interleaving distance arising from discretizations of graph-with-function data is also NP-hard to compute (Chambers et al., 4 Apr 2025, Chambers et al., 2023). Recent work therefore emphasizes computable upper bounds rather than exact values. A general loss-function framework on concrete categories defines “assignments” that need not commute and measures their failure via a loss $\varepsilon$40. For a $\varepsilon$41-assignment, one obtains the upper bound

$\varepsilon$42

(Olave et al., 13 Jan 2026). In finite and vector-space-valued settings, the loss is computable in polynomial time, including for certain $\varepsilon$43-parameter persistence modules (Olave et al., 13 Jan 2026).

For mapper graphs specifically, the analogous loss on basis assignments gives

$\varepsilon$44

and the loss can be optimized as an integer linear program (Chambers et al., 4 Apr 2025). On small examples where the true interleaving distance is known, the optimized upper bound matches the interleaving distance (Chambers et al., 4 Apr 2025). This suggests that, in some structured settings, exact computation may be effectively replaced by optimization over assignment spaces, even though the general decision problem remains intractable.

A recurring misconception is that barcode-style combinatorial summaries should suffice for all interleaving computations. The multi-parameter hardness results, the dependence on field characteristic for interval-decomposable modules, and the divergence between interleaving and bottleneck-type distances in higher-dimensional settings all indicate otherwise (Bjerkevik et al., 2018, Bjerkevik et al., 2017). A plausible implication is that interleaving distance should be regarded less as a uniformly computable invariant and more as a structurally canonical metric whose exact algorithmics depend heavily on the indexing category and the chosen representation class.

7. Broader interpretations and current directions

Interleaving distance has gradually shifted from a persistence-specific construction to a general mechanism for comparing functorial objects under controlled transformations. One categorical route interprets interleavings of functors with common codomain as solutions to an extension problem, leading to categorical analogues of Hausdorff distance and Gromov–Hausdorff distance (Bubenik et al., 2017). In that framework, the interleaving distance between functors is defined by minimizing Hausdorff distance over weighted pairwise embeddings that admit a common extension (Bubenik et al., 2017). This recovers shift equivalence of discrete dynamical systems as a particular interleaving notion (Bubenik et al., 2017).

Another route uses monoidal actions and $\varepsilon$45-categories. For a strict monoidal functor

$\varepsilon$46

with monoidal weight $\varepsilon$47, the actegory interleaving distance

$\varepsilon$48

defines an extended pseudometric (McFaddin et al., 2023). The resulting theory recovers classical categories-with-a-flow, locally persistent categories, and generalized persistence modules, while connecting interleavings to group-action distances from statistical shape analysis and diffeomorphism-invariant constructions (McFaddin et al., 2023).

A further recent development reformulates the interleaving distance itself as an edit distance on finitely presented single- and multi-parameter persistence modules. Using graded free presentations, Galois connections, and a path metric built from poset morphisms with adjoints, one proves

$\varepsilon$49

for suitable edit categories $\varepsilon$50 (Kim et al., 29 Sep 2025). This recasts interleavings as controlled edits of generators and relations and suggests new routes for proving stability of multiparameter invariants (Kim et al., 29 Sep 2025).

Across these variants, several themes recur. Interleaving distances quantify approximate equivalence after a prescribed family of shifts, thickenings, or actions; they tend to induce extended pseudometrics or Lawvere metrics; and stability is usually obtained by functoriality or equivariance rather than by direct estimation. At the same time, separation, triangle inequalities, and algorithmic tractability become subtle outside the classical one-parameter setting. The sheaf-theoretic separation theorem (Petit et al., 2021), the relaxed triangle inequalities for height-interleavings (Aoki, 14 Mar 2026), and the widespread NP-hardness phenomena (Bjerkevik et al., 2018, Bjerkevik et al., 2017) show that “interleaving distance” is not a single metric formula but a broad family of categorical constructions whose exact properties depend on the ambient indexing and action structure.

In that sense, the modern theory of interleaving distance is both unifying and diagnostic. It unifies persistent homology, merge trees, mapper graphs, sheaves, and acted-on categories within a common language of controlled comparison; and it diagnoses which aspects of those objects are stable, computable, or intrinsically resistant to simplification.