Stable Distance in Mathematical Analysis

• Stable distance is a term describing metrics and deficit functionals that remain controlled under perturbations, crucial for ensuring robustness across various mathematical frameworks.
• It facilitates rigorous comparisons in persistence diagrams, edit distances, and alignment problems by providing quantitative bounds to capture meaningful feature variations.
• Applications range from materials science classification and time series analysis to SDE convergence and adversarial matching, emphasizing stable, computable, and discriminative measures.

Across the cited works, “stable distance” does not denote a single universally standardized object. It is used for distances, metrics, and deficit functionals whose values are controlled under perturbations of the underlying data, function, model, or coefficient field. In topological data analysis, this control is expressed by inequalities against $\|f-g\|_\infty$, interleaving distances, or convergence of persistence diagrams; in analysis it appears as a remainder term that measures deviation from a family of formal extremals; in stochastic analysis it becomes a quantitative bound on the distance between solution laws or sample paths; and in matching or alignment problems it refers either to the distance of a stable pair or to an optimization objective designed to converge stably under iteration (Maroulas et al., 2018, Tralie et al., 2022, Bauer et al., 2013, Memoli, 2017, Leng, 13 Jun 2026).

1. Stability as a mathematical pattern

A useful editorial taxonomy is that the literature employs “stable distance” in several distinct but structurally related ways. One family consists of bona fide metrics on combinatorial or topological summaries, such as persistence diagrams, merge trees, Reeb graphs, and filtered spaces. A second family consists of finitely stable edit distances, where the Lipschitz constant depends on the size or rank of the objects being compared. A third family consists of deficit identities and remainder estimates, where the “distance” is not a metric on pairs of objects but a quantitative measure of deviation from an optimizer or “virtual extremal.” A fourth family concerns random or operational distances arising from stable matchings or adversarial alignment procedures.

Setting	Object	Stated control
Persistence diagrams (Maroulas et al., 2018)	$d_p^c$	If $d_p^c(A,A_i)\to0$, then $d_p^c(X^k,X_i^k)\to0$
Ordered time series (Tralie et al., 2022)	DOPE	$\mathrm{dope}(x,y)\le\|x^c-y^c\|_1$
Reeb graphs (Bauer et al., 2013)	Functional distortion distance	$d_{FD}(G_f,G_g)\le\|f-g\|_\infty$
Filtered spaces (Memoli, 2017)	Tripod distance $d_{\mathcal F}$	$d_B(D_k(X,f_X),D_k(Y,f_Y))\le d_{\mathcal F}$
Merge trees (Pegoraro, 2021)	Finitely stable edit distance $d_E$	$d_E(T,T')\le 2(\dim(T)+\dim(T'))\varepsilon$ for $d_p^c$0-interleaved RAMTs
Carleman inequality (Leng, 13 Jun 2026)	Deficit $d_p^c$1	$d_p^c$2

This comparison shows that the common content of stability is controlled variation rather than a fixed formalism. It also shows that several works explicitly pair stability with additional desiderata—discriminativity, informativity, computability, or monotonic optimization—rather than treating stability as sufficient on its own.

2. Persistence-diagram distances and cardinality sensitivity

A particularly explicit instance is the stable cardinality distance $d_p^c$3 on persistence diagrams. For persistence diagrams $d_p^c$4 and $d_p^c$5 in the plane, with $d_p^c$6, exponent $d_p^c$7, and penalty $d_p^c$8, it is defined by

$d_p^c$9

The term $d_p^c(A,A_i)\to0$0 truncates the matching cost, while the term $d_p^c(A,A_i)\to0$1 is an explicit penalty for unmatched points in the larger diagram (Maroulas et al., 2018).

The main theorem is a continuity-type stability result. If $d_p^c(A,A_i)\to0$2 is a finite point cloud, $d_p^c(A,A_i)\to0$3 is a sequence of finite point clouds with $d_p^c(A,A_i)\to0$4, and $d_p^c(A,A_i)\to0$5, $d_p^c(A,A_i)\to0$6 are the $d_p^c(A,A_i)\to0$7th-homology persistence diagrams of $d_p^c(A,A_i)\to0$8 and $d_p^c(A,A_i)\to0$9, then

$d_p^c(X^k,X_i^k)\to0$0

The proof outline proceeds by showing that eventually $d_p^c(X^k,X_i^k)\to0$1, then matching points of $d_p^c(X^k,X_i^k)\to0$2 and $d_p^c(X^k,X_i^k)\to0$3 under the optimizing permutation so that pairwise distances converge, and then concluding that births and deaths of homology classes can be matched with small cost in the $d_p^c(X^k,X_i^k)\to0$4 metric (Maroulas et al., 2018).

The paper places this distance against the bottleneck and $d_p^c(X^k,X_i^k)\to0$5-Wasserstein distances. The latter allow surplus points to be matched to the diagonal at no explicit cardinality cost, whereas $d_p^c(X^k,X_i^k)\to0$6 imposes a hard penalty $d_p^c(X^k,X_i^k)\to0$7 per unmatched point. The stated consequence is that $d_p^c(X^k,X_i^k)\to0$8 is sensitive to diagram cardinality and never “hides” a large difference in counts of topological features by sending points to the diagonal at arbitrarily small cost (Maroulas et al., 2018).

Computation is framed as a truncated assignment problem of size $d_p^c(X^k,X_i^k)\to0$9. The paper describes a cost matrix

$\mathrm{dope}(x,y)\le\|x^c-y^c\|_1$0

and reports that one may use a standard $\mathrm{dope}(x,y)\le\|x^c-y^c\|_1$1 Hungarian-style algorithm or more recent $\mathrm{dope}(x,y)\le\|x^c-y^c\|_1$2 optimizers. In practice, the authors pre-compute pairwise $\mathrm{dope}(x,y)\le\|x^c-y^c\|_1$3 distances, threshold them by $\mathrm{dope}(x,y)\le\|x^c-y^c\|_1$4, and use a dedicated assignment solver; a simple grid-search on $\mathrm{dope}(x,y)\le\|x^c-y^c\|_1$5 over 10–20 geometrically spaced values is described as sufficient in applications (Maroulas et al., 2018).

The application domain is materials-science classification from synthetic atom probe tomography. For sparse and noisy atomic neighborhoods drawn from BCC or FCC lattices, the method computes $\mathrm{dope}(x,y)\le\|x^c-y^c\|_1$6- and $\mathrm{dope}(x,y)\le\|x^c-y^c\|_1$7-dimensional persistence diagrams and then forms eight statistics: the average and variance of $\mathrm{dope}(x,y)\le\|x^c-y^c\|_1$8 to all training diagrams of type BCC or FCC for $\mathrm{dope}(x,y)\le\|x^c-y^c\|_1$9. A logistic-decision-tree on these eight features attains better than $d_{FD}(G_f,G_g)\le\|f-g\|_\infty$0 cross-validated accuracy uniformly over $d_{FD}(G_f,G_g)\le\|f-g\|_\infty$1 with $d_{FD}(G_f,G_g)\le\|f-g\|_\infty$2 of atoms missing, outperforming both a classifier built on Wasserstein distances alone and a purely counting classifier (Maroulas et al., 2018).

3. Reeb graphs, merge trees, and filtered spaces

For Reeb graphs, Bauer, Ge, and Wang define the functional distortion distance by combining a graph-intrinsic metric with function-value discrepancies. Given continuous maps $d_{FD}(G_f,G_g)\le\|f-g\|_\infty$3 and $d_{FD}(G_f,G_g)\le\|f-g\|_\infty$4, the distance takes the infimum of the maximum of three quantities: metric distortion $d_{FD}(G_f,G_g)\le\|f-g\|_\infty$5, $d_{FD}(G_f,G_g)\le\|f-g\|_\infty$6, and $d_{FD}(G_f,G_g)\le\|f-g\|_\infty$7. The paper proves the stability estimate

$d_{FD}(G_f,G_g)\le\|f-g\|_\infty$8

for tame functions on the same domain under a right-inverse assumption on the quotient maps. It also proves lower bounds from ordinary and extended persistence, namely

$d_{FD}(G_f,G_g)\le\|f-g\|_\infty$9

and

$d_{\mathcal F}$0

so the functional distortion distance is both stable and more discriminative than persistence-diagram bottleneck distance in the stated sense (Bauer et al., 2013).

Mémoli’s tripod distance addresses the case of filtrations on different finite spaces. A tripod is a common parameter set $d_{\mathcal F}$1 with surjections $d_{\mathcal F}$2 and $d_{\mathcal F}$3, and the distance is

$d_{\mathcal F}$4

The pullback construction lifts combinatorial stability to different ground sets: $d_{\mathcal F}$5 The same paper constructs explicit constant-speed geodesics by linear interpolation on a minimizing tripod, and proves a strengthened stability theorem bounding the bottleneck-length of the barcode path along such a geodesic (Memoli, 2017).

Several merge-tree distances refine this picture. The merge-tree matching distance $d_{\mathcal F}$6 is built from branch decomposition trees, full matchings, and induced zigzag diagrams. It satisfies

$d_{\mathcal F}$7

and is more discriminative than bottleneck distance in the precise sense that

$d_{\mathcal F}$8

The same work proves a persistence-simplification bound: $d_{\mathcal F}$9 when $d_B(D_k(X,f_X),D_k(Y,f_Y))\le d_{\mathcal F}$0 and $d_B(D_k(X,f_X),D_k(Y,f_Y))\le d_{\mathcal F}$1, and reports applications to shape comparison and periodicity detection in the von Kármán vortex street (Bollen et al., 2022).

The finitely stable edit distance $d_B(D_k(X,f_X),D_k(Y,f_Y))\le d_{\mathcal F}$2 on merge trees is presented as an analog of $d_B(D_k(X,f_X),D_k(Y,f_Y))\le d_{\mathcal F}$3-Wasserstein distance for persistence diagrams. For $d_B(D_k(X,f_X),D_k(Y,f_Y))\le d_{\mathcal F}$4-interleaved regular abstract merge trees,

$d_B(D_k(X,f_X),D_k(Y,f_Y))\le d_{\mathcal F}$5

while

$d_B(D_k(X,f_X),D_k(Y,f_Y))\le d_{\mathcal F}$6

The paper states the parallel

$d_B(D_k(X,f_X),D_k(Y,f_Y))\le d_{\mathcal F}$7

mirroring the classical inequalities between bottleneck and $d_B(D_k(X,f_X),D_k(Y,f_Y))\le d_{\mathcal F}$8 for persistence diagrams (Pegoraro, 2021).

A related metric for functions defined on merge trees extends the edit formalism to edge-valued functions in an editable metric monoid. It proves finite stability with constant $d_B(D_k(X,f_X),D_k(Y,f_Y))\le d_{\mathcal F}$9: $d_E$0 whenever $d_E$1. The same work casts the optimization as a binary linear program over match, delete, and insert variables and reports simulated experiments with trees up to 150 leaves per tree (Pegoraro, 2021).

4. Ordered time series and the DOPE distance

For real-valued time series on the interval or circle, the DOPE distance compares the ordered critical-point subsequences rather than persistence diagrams alone. If $d_E$2 and $d_E$3 are the critical-point sequences, an alignment consists of an order-preserving matching together with removal sets made of adjacent min-max pairs, so that every index appears exactly once in a matching or removal. The cost is

$d_E$4

and

$d_E$5

An equivalent formulation views DOPE as an edit distance with one-point matches, pair-matches, deletions, and insertions (Tralie et al., 2022).

The central theorem is $d_E$6-stability: $d_E$7 The proof uses a candidate alignment that matches critical points in order and deletes any excess tail in adjacent min-max pairs; by Euler-characteristic arguments, these tail deletions are legal, and the resulting cost is exactly $d_E$8 (Tralie et al., 2022).

Computation is by dynamic programming. For interval time series, the recurrence on prefix costs $d_E$9 yields an $d_E(T,T')\le 2(\dim(T)+\dim(T'))\varepsilon$0 algorithm, hence $d_E(T,T')\le 2(\dim(T)+\dim(T'))\varepsilon$1 overall since $d_E(T,T')\le 2(\dim(T)+\dim(T'))\varepsilon$2 and $d_E(T,T')\le 2(\dim(T)+\dim(T'))\varepsilon$3. For circular time series, trying two cuts on one sequence and all cyclic shifts of the other gives

$d_E(T,T')\le 2(\dim(T)+\dim(T'))\varepsilon$4

which is $d_E(T,T')\le 2(\dim(T)+\dim(T'))\varepsilon$5 in the worst case $d_E(T,T')\le 2(\dim(T)+\dim(T'))\varepsilon$6 (Tralie et al., 2022).

The paper explicitly contrasts DOPE with persistence-diagram distances. It proves

$d_E(T,T')\le 2(\dim(T)+\dim(T'))\varepsilon$7

stating that DOPE is at least as informative as the ordinary $d_E(T,T')\le 2(\dim(T)+\dim(T'))\varepsilon$8-Wasserstein distance on persistence diagrams. It also remarks that bottleneck distance is only $d_E(T,T')\le 2(\dim(T)+\dim(T'))\varepsilon$9-stable and can miss changes in birth-death pairings until a large critical-height crossing occurs, whereas DOPE, being $d_p^c$00-stable, records small height moves in total. Empirically, on 128 interval time-series tasks from the UCR repository, DOPE typically outperforms bottleneck and $d_p^c$01-Wasserstein distances on nearest-neighbor classification and also rivals DTW on critical-point sequences, while remaining a true metric with provable stability and informativity (Tralie et al., 2022).

5. Deficit-based and stochastic notions of stable distance

In the stability theory of Hardy’s and Carleman’s integral inequalities, distance appears as a remainder term measuring deviation from a family of virtual extremals. For Hardy’s inequality, the deficit is

$d_p^c$02

and the paper proves the exact identity

$d_p^c$03

For $d_p^c$04 this becomes

$d_p^c$05

For Carleman’s inequality, the paper defines the local weighted Hellinger-type distance

$d_p^c$06

and proves

$d_p^c$07

The stated significance is that, although the classical inequalities have no genuine extremizers in their natural spaces, their deficits still measure deviation from the corresponding families of virtual extremals (Leng, 13 Jun 2026).

For one-dimensional SDEs with drift terms driven by a symmetric $d_p^c$08-stable process, the main result is a Hölder-type estimate in $d_p^c$09 between two solution paths. The coefficient discrepancies are not measured by a supremum norm but by weighted integral quantities

$d_p^c$10

and

$d_p^c$11

where $d_p^c$12 is the baseline transition density. Under the stated regularity and smallness assumptions, the paper bounds

$d_p^c$13

by a Hölder-type function of $d_p^c$14, $d_p^c$15, and $d_p^c$16, and then derives corresponding convergence rates in probability for $d_p^c$17 (Nakagawa, 29 Oct 2025).

A different stochastic comparison studies the optimal Wasserstein-$d_p^c$18 distance between SDEs driven respectively by Brownian motion and rotationally symmetric $d_p^c$19-stable Lévy noise. For $d_p^c$20 the paper proves

$d_p^c$21

and hence

$d_p^c$22

For a $d_p^c$23-dimensional Ornstein–Uhlenbeck system it also proves

$d_p^c$24

which the authors state indicates that the convergence rate with respect to $d_p^c$25 is optimal (Deng et al., 2023).

In the density-distance version of the Carlen–Frank–Lieb stability theorem, the distance is the $d_p^c$26 distance between normalized densities

$d_p^c$27

The main theorem states that for the family $d_p^c$28 of optimal potentials,

$d_p^c$29

The paper’s novelty is that the Hölder step is replaced by the Leng–Lu $d_p^c$30-stability theorem for probability densities, thereby converting a spectral deficit into a squared density-distance bound (Leng, 2 Jun 2026).

6. Matching, alignment, and broader interpretations

In stable matching theory, “stable distance” refers not to a metric between objects but to the distance of a stable pair. In the one-dimensional Poisson ride-hailing model, passengers and cabs form independent Poisson processes of rates $d_p^c$31 and $d_p^c$32, and $d_p^c$33 denotes the distance in a typical stable blue–red pair. The constrained half-line matching corresponds to an LCFS-PR queue busy period $d_p^c$34, with

$d_p^c$35

For the full stable matching, the paper derives upper-tail and expectation bounds, including

$d_p^c$36

In the discrete hypercube dating-site model, the same paper studies the matching distance $d_p^c$37 under Hamming and Weighted Hamming metrics, proving uniqueness or near-uniqueness regimes and concentration results for $d_p^c$38 (Abadi et al., 2017).

In adversarial distribution alignment, the relevant object is a stable optimization distance rather than a stable pair distance. For linear discriminators, the adversarial logistic objective is dualized into a smooth constrained minimization over weights $d_p^c$39, yielding a block-form objective denoted $d_p^c$40. The paper interprets this as an iteratively reweighted empirical MMD and states that, because the dual objective is smooth and lower-bounded, alternating descent in $d_p^c$41 gives monotonic decrease for sufficiently small step sizes. Empirically, the dual formulation exhibits more stable and monotonic improvement than the primal min-max GAN-like objective and an MMD objective on synthetic point clouds and digit-domain adaptation (Usman et al., 2017).

A further extension of the vocabulary appears in algebraic geometry of neural layers. The generic Euclidean Distance degree $d_p^c$42 is not a metric on a pair of models, but it is a projective invariant measuring the number of optimal approximations of a general point with respect to a general metric. For a fixed architecture, the paper proves that $d_p^c$43 is stably polynomial in the input and output dimensions and depends only on the degree of the activation function. This usage shows that the language of stable distance can also refer to eventual polynomial behavior of a distance-related invariant under dimensional growth, rather than to perturbation bounds alone (Graziani, 22 Jan 2026).

A plausible implication of these diverse constructions is that “stable distance” functions less as the name of a single theory than as a recurrent design principle. The recurring questions are whether the quantity is sensitive to the information that matters, whether it deforms continuously or quantitatively under perturbation, and whether it remains computable in the regimes of interest. The cited works make these tradeoffs explicit: some emphasize discriminativity in addition to stability, some replace exact stability by finite stability, and some translate a deficit into a distance from a virtual optimizer rather than a metric between two observed objects.