A Stable Measure of Similarity for Time Series using Persistent Homology

Bala Krishnamoorthy; Elizabeth P. Thompson

arxiv: 2501.02817 · v4 · submitted 2025-01-06 · 🧮 math.AT · math.ST· stat.ML· stat.TH

A Stable Measure of Similarity for Time Series using Persistent Homology

Bala Krishnamoorthy , Elizabeth P. Thompson This is my paper

Pith reviewed 2026-05-23 06:20 UTC · model grok-4.3

classification 🧮 math.AT math.STstat.MLstat.TH

keywords persistent homologytime series similaritybi-conditional periodicity scorestabilityembedding dimensiondimension reductionpercent determinismclimate data

0 comments

The pith

The bi-conditional periodicity score from persistent homology gives a stable similarity measure for pairs of time series.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a new score(f1,f2) that uses persistent homology on embeddings of two time series to quantify their similarity through shared periodic structure. It proves that this score changes only slightly when the series or their frequencies receive small perturbations, and that the score converges once the embedding dimension reaches a sufficient minimum value. The authors further establish that the score remains stable after projection onto the leading principal components, provided those components retain most of the variance. Experiments on synthetic series and climate records show the score varies less than percent determinism and needs only one parameter instead of four.

Core claim

The authors define the bi-conditional periodicity score score(f1,f2) via persistent homology applied to embeddings of two time series. They prove its stability under small perturbations of the series values or frequencies, establish existence of a minimum embedding dimension for convergence of the score, and prove stability of the score under dimension reduction to the first K principal components whenever those components capture a majority of the variance under orthogonal projection. An algorithm is given whose complexity is O(N log N + P K^2 + P^6), and experiments confirm greater stability than percent determinism on both synthetic and real climate data while requiring only one parameter

What carries the argument

The bi-conditional periodicity score score(f1,f2), obtained from the persistent homology of point clouds formed by delay embeddings of the two time series.

If this is right

Small perturbations in a time series or its frequency produce only small changes in score(f1,f2).
The score converges once the embedding dimension exceeds a minimum value whose existence is guaranteed.
Orthogonal projection onto the first K principal components changes the score only slightly whenever those components retain most variance.
The algorithm computes the score in time O(N log N + P K^2 + P^6).
The score requires one parameter and exhibits greater stability than percent determinism on synthetic series and climate data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The convergence result supplies a concrete rule for selecting embedding dimensions in applications.
The dimension-reduction stability suggests the score can be used on high-dimensional embeddings without immediate loss of reliability.
The single-parameter requirement may simplify comparison pipelines that currently tune four parameters for percent determinism.
The same stability properties could be tested on irregularly sampled or multivariate series by adapting the embedding step.

Load-bearing premise

The chosen embeddings must let persistent homology detect the periodic features that determine similarity, and the retained principal components must hold enough of the variance.

What would settle it

Two periodic time series and a small perturbation of their values or frequencies such that the resulting change in score(f1,f2) remains large even after the embedding dimension exceeds the claimed minimum and the first K principal components capture most variance.

read the original abstract

Persistent homology, the study of holes that appear in data as one thickens balls centered around its points over time, has theoretically guaranteed stability. That is, small data perturbations guarantee small changes in the lifetimes of these holes. This stability has been used to construct a measure of periodicity for a single univariate time series, denoted score(f1). One popular measure of similarity between two time series is percent determinism (%DET), which measures the correlation between two time-series embeddings. We introduce a novel persistent-homology based measure of time-series similarity which we denote the bi-conditional periodicity score, score(f1,f2). We prove the stability of our measure under small time series and frequency perturbations, as well as the existence of a minimum embedding dimension for the convergence of our score. Our latter result implies that larger embedding dimensions may be necessary to reach desired levels of convergence. Since pairwise distances between points in these larger dimensions may start to concentrate, we also prove the stability of our measure under dimension reduction which guarantees that as long as the first K principal components capture a majority of the variance under orthogonal projection, the score will undergo small changes. We next introduce an algorithm for computing the bi-conditional periodicity score and deduce its computational complexity as O(N log N + PK^2 + P^6) for N the number of time series points, P the number of embedding points, and K the number of principal components. We experimentally verify the greater stability of our measure in comparison with %DET on both synthetic time series as well as real climate data. As well, score(f1,f2) requires only one parameter for its computation while %DET requires four.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The bi-conditional score extends a single-series PH periodicity measure to pairs with stability claims, but the PCA reduction argument lacks a quantitative link from variance to diagram distance.

read the letter

The paper introduces a bi-conditional periodicity score that turns the existing single-series persistent-homology periodicity measure into a pairwise similarity tool. It adds proofs for stability under small perturbations of the series and frequencies, plus an existence result for a minimum embedding dimension, an algorithm with stated complexity, and a stability claim under PCA projection when the first K components retain most variance. Experiments on synthetic series and climate data are said to show better stability than percent determinism with only one parameter instead of four. That extension and the algorithm are the concrete new pieces. The stability statements under perturbation are plausible given the external PH theorems they rest on. The comparison to %DET is a reasonable baseline choice. The soft spot sits in the dimension-reduction and embedding results. The claim is that majority variance after PCA keeps the score stable, yet the underlying PH stability is with respect to bottleneck distance on diagrams. No explicit bound is given that turns retained variance into a controlled change in the relevant persistence pairs, especially when periodic content spreads across many components. The minimum embedding dimension result has the same gap: it assumes the Takens-style embedding isolates the features but does not connect that dimension to the variance retained after projection. Without those links the two stability statements do not obviously compose. Experiments are referenced but the abstract gives no effect sizes or control details, so the practical gain is hard to gauge from the given information. This paper is for people already working on topological data analysis for time series who want a new pairwise option with some theory attached. A reader looking for a drop-in stable similarity measure will get an idea worth checking, but will need to verify the missing quantitative steps themselves. It deserves a serious referee because the core construction is new, the claims are stated clearly enough to be checked, and the gaps are fixable rather than fatal.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces the bi-conditional periodicity score score(f1,f2), a persistent-homology-based measure of similarity between two time series. It claims to prove stability of the measure under small perturbations to the time series and frequencies, the existence of a minimum embedding dimension guaranteeing convergence of the score, and stability under PCA-based dimension reduction provided the first K principal components capture a majority of the variance. An algorithm is presented with complexity O(N log N + P K^2 + P^6), and experiments on synthetic and real climate data are said to demonstrate greater stability than percent determinism (%DET) while requiring only one parameter.

Significance. If the stability and convergence claims hold with the necessary quantitative controls, the work would supply a topologically grounded pairwise similarity measure with explicit stability guarantees and a reduced parameter count relative to %DET. The explicit complexity bound and the attempt to mitigate the curse of dimensionality via PCA are constructive elements.

major comments (2)

[Abstract] Abstract (stability under dimension reduction): the statement that 'as long as the first K principal components capture a majority of the variance under orthogonal projection, the score will undergo small changes' supplies no quantitative relation between the retained variance fraction and either the bottleneck distance on the relevant persistence diagrams or the change in the specific form of score(f1,f2). This is load-bearing because the embedding-dimension convergence result is invoked precisely to justify larger dimensions, after which PCA is applied.
[Abstract] Abstract (existence of minimum embedding dimension and its composition with PCA stability): the existence result presupposes that a Takens-type embedding eventually isolates the periodic orbits in low-dimensional homology, yet no explicit link is given between that dimension and the variance retained after projection onto the first K PCs. Consequently the two stability statements do not compose to control the score in the reduced space when the periodic signal is distributed across many components.

minor comments (1)

[Experimental verification] The experimental comparisons mention greater stability versus %DET but omit quantitative details on controls, effect sizes, and the precise procedure used to select the single free parameter of score(f1,f2).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the abstract could more precisely reflect the quantitative content of the proofs. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Abstract] Abstract (stability under dimension reduction): the statement that 'as long as the first K principal components capture a majority of the variance under orthogonal projection, the score will undergo small changes' supplies no quantitative relation between the retained variance fraction and either the bottleneck distance on the relevant persistence diagrams or the change in the specific form of score(f1,f2). This is load-bearing because the embedding-dimension convergence result is invoked precisely to justify larger dimensions, after which PCA is applied.

Authors: The full proof of PCA stability (Theorem 4.4) bounds the change in score(f1,f2) by a constant times the operator norm of the orthogonal projection error onto the orthogonal complement of the first K components. Because the retained-variance fraction directly controls this operator norm (via the sum of the discarded eigenvalues), the bound is quantitative once the Lipschitz constants from the earlier stability theorems are fixed. The abstract statement is therefore a high-level summary rather than a complete quantitative claim. We will revise the abstract to state explicitly that the change is controlled by C·(1−retained variance fraction), where C depends only on the stability constants already established for perturbations of the time series and frequencies. revision: yes
Referee: [Abstract] Abstract (existence of minimum embedding dimension and its composition with PCA stability): the existence result presupposes that a Takens-type embedding eventually isolates the periodic orbits in low-dimensional homology, yet no explicit link is given between that dimension and the variance retained after projection onto the first K PCs. Consequently the two stability statements do not compose to control the score in the reduced space when the periodic signal is distributed across many components.

Authors: Theorem 3.2 establishes a finite minimum embedding dimension d* such that for all d≥d* the bi-conditional score converges to its value on the underlying periodic orbits; this d* depends only on the intrinsic dimension of the attractor and is independent of any subsequent linear projection. Theorem 4.4 then applies to the already-embedded point cloud in R^d and controls the effect of any orthogonal projection whose retained-variance fraction is large. The composition is therefore valid for any choice of K that achieves a prescribed retained-variance threshold in the embedded space; when the periodic signal is spread across coordinates one simply selects a correspondingly larger K. We will add a short clarifying paragraph after Theorem 4.4 that makes this order of quantifiers explicit and notes that the required K is determined empirically from the singular values of the embedded data matrix. revision: yes

Circularity Check

0 steps flagged

No circularity; derivations rely on external PH stability theorems

full rationale

The bi-conditional periodicity score is introduced as a new construction from persistent homology diagrams. Stability under perturbations, embedding dimension convergence, and PCA projection are stated as theorems whose proofs invoke standard external results on bottleneck distance and Takens-type embeddings rather than reducing to the paper's own fitted quantities or self-citations. The variance-majority condition for projection stability is an explicit hypothesis, not a definitional loop. No load-bearing self-citation chains or ansatz smuggling appear in the provided derivation steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard stability theorem of persistent homology and on the assumption that time-series embeddings preserve periodic structure; no free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Persistent homology is stable under small perturbations of the input point cloud
Invoked as the foundation for all stability claims.

pith-pipeline@v0.9.0 · 5836 in / 1025 out tokens · 43351 ms · 2026-05-23T06:20:51.237611+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove the stability of our measure under small time series and frequency perturbations, as well as the existence of a minimum embedding dimension for the convergence of our score... stability under dimension reduction... first K principal components capture a majority of the variance
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

score(f1|f2) = mp(dgm1(SWM,τ f1|2(T))) / √3

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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