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arxiv: 2512.06324 · v3 · submitted 2025-12-06 · 🧮 math.ST · stat.TH

Subsampling Confidence Bound for Persistent Diagram via Time-delay Embedding

Donghyun Park , Junhyun An , Taehyoung Kim , Jisu Kim This is my paper

Pith reviewed 2026-05-17 01:37 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords time-delay embeddingpersistent homologypersistence diagramsubsamplingconfidence boundsperiodicity testtopological data analysisstatistical inference
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The pith

Subsampling produces asymptotic confidence bounds for persistence diagrams of time-delay embeddings and supports a formal test for periodicity.

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

The paper seeks to add statistical rigor to topological analysis of time series by attaching uncertainty measures to the loops that persistent homology detects after time-delay embedding. It first proves that periodic signals yield embeddings homotopy equivalent to a circle while non-periodic signals yield contractible spaces, with the reach of the embedding bounded below to keep features stable. A subsampling procedure then constructs confidence bounds around the persistence diagram under standard manifold regularity on the support. These pieces combine into a hypothesis test for periodicity whose type I and type II error rates are controlled asymptotically. A reader would care because the approach turns an exploratory topological tool into a decision procedure that can be applied to real signals with quantifiable reliability.

Core claim

The central claim is that subsampling yields confidence bounds with asymptotic guarantees for the persistence diagram obtained from the sliding-window time-delay embedding of a time series. The topological characterization establishes that the embedded trajectory is homotopy equivalent to the circle for periodic sampling functions and contractible for non-periodic ones. A lower bound on the reach of the embedding ensures that the persistence features remain stable. The resulting test decides whether the underlying function is periodic while controlling both false-positive and false-negative rates at any prescribed level in the large-sample regime.

What carries the argument

Subsampling construction of confidence bounds around the persistence diagram of the time-delay embedding.

Load-bearing premise

The support of the embedded point cloud must satisfy standard manifold regularity conditions so the subsampling bounds achieve their stated asymptotic coverage.

What would settle it

A sequence of Monte Carlo trials on samples of increasing size drawn from a non-manifold distribution in which the empirical coverage of the proposed intervals falls materially below the nominal level.

Figures

Figures reproduced from arXiv: 2512.06324 by Donghyun Park, Jisu Kim, Junhyun An, Taehyoung Kim.

Figure 1
Figure 1. Figure 1: Example of (Ξ, ϵ)-non-periodic function Assumption 3.3. The true sampling function f : R → R is either (Ξ, ϵ)-periodic for some Ξ, ϵ > 0, or ϵ-non-periodic for some ϵ > 0. We also have the regularity condition tailored for periodicity detection. Assumption 3.4. The true sampling function f : R → R is C 2 , and there exists some δ > 0 such that for all t ∈ R, either |f ′ (t)| ≥ δ or |f ′′(t)| ≥ δ, and for a… view at source ↗
Figure 2
Figure 2. Figure 2: Diagram of Hypothesis Testing We define a decision function ϕ(X) as whether there is any element of persistence diagram P(X) in the region R. ϕ(Xm) = ( 0 if |P(Xm) ∩ R| = 0 (Accept H0) 1 if |P(Xm) ∩ R| ≥ 1 (Reject H0) The following theorems state that our proposed hypothesis test is significant under condition of a ≥ 4cα in terms of type I and type II error. The cα decreases as the sample size and subsampl… view at source ↗
read the original abstract

Time-delay embedding is a fundamental technique in Topological Data Analysis (TDA) for reconstructing the phase space dynamics of time-series data. Persistent homology effectively identifies global topological features, such as loops associated with periodicity. Nevertheless, a statistically rigorous way to quantify uncertainty in the resulting topological features has remained underdeveloped -- a problem that we aim to challenge. First, we analyze the topological characterization of time-delay embeddings under both periodic and non-periodic conditions. Precisely, the embedded trajectory is homotopy equivalent to a circle ($S^1$) for periodic signals and is contractible for non-periodic ones. We also prove that the reach of the sliding window embedding is lower-bounded, ensuring stable persistence features. Next, we propose a subsampling-based method to construct confidence bounds for persistence diagrams derived from time-delay embeddings. Specifically, we derive confidence bounds with asymptotic guarantees, under the assumption that the support satisfies standard manifold regularity. Integrating the results, we propose a statistical testing framework to determine the periodicity of the underlying sampling function. This framework provides a principled statistical test for periodicity with asymptotically controlled type I and type II error rates. Simulation studies demonstrate that our method achieves detection performance comparable to the Generalized Lomb-Scargle Periodogram on periodic data while exhibiting superior robustness in distinguishing non-periodic signals with time-varying frequencies, such as chirp signals. Finally, it successfully captured the periodicity when applied to the BIDMC dataset.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a subsampling-based approach to obtain confidence bounds for persistence diagrams of time-delay embeddings. It shows homotopy equivalence to S^1 for periodic signals and contractibility for non-periodic signals, proves a lower bound on the reach of the embedding, and derives asymptotic guarantees for the bounds assuming standard manifold regularity on the support. These are combined into a periodicity test with asymptotically controlled type I and type II errors. Simulations compare favorably to the Generalized Lomb-Scargle Periodogram, especially for chirp signals, and the method is applied to the BIDMC dataset.

Significance. If the central claims hold, particularly the transfer of asymptotic guarantees to the embedded trajectories, the paper would provide a valuable statistical framework for topological periodicity detection in time series. This could bridge TDA and classical time series analysis, offering robustness advantages for non-stationary periodic signals.

major comments (2)
  1. §4 (Subsampling confidence bounds): The asymptotic guarantees for the subsampling-based confidence bounds are stated under the assumption that the support satisfies standard manifold regularity. However, while §3 establishes homotopy equivalence and a reach lower bound for the sliding-window embedding, these do not automatically ensure the additional conditions (e.g., bounded curvature, sufficient sampling density) typically required for subsampling asymptotics on persistence diagrams. This is load-bearing for the claimed type I and type II error control in the periodicity test.
  2. §5 (Statistical testing framework): The periodicity test relies on the confidence bounds to control errors, but the manuscript does not provide explicit verification or discussion of whether the manifold regularity holds for non-periodic signals or chirp signals in the simulations. Without this, the asymptotic control may not be justified in all regimes considered.
minor comments (2)
  1. Abstract: The phrase 'standard manifold regularity' is used without definition or reference; a brief specification or citation to the precise conditions assumed would improve clarity.
  2. Simulation studies: The description lacks details on how error bars are computed and any data exclusion criteria, which would help in reproducing and assessing the reported performance comparisons.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and indicate the revisions we plan to incorporate.

read point-by-point responses

  1. Referee: §4 (Subsampling confidence bounds): The asymptotic guarantees for the subsampling-based confidence bounds are stated under the assumption that the support satisfies standard manifold regularity. However, while §3 establishes homotopy equivalence and a reach lower bound for the sliding-window embedding, these do not automatically ensure the additional conditions (e.g., bounded curvature, sufficient sampling density) typically required for subsampling asymptotics on persistence diagrams. This is load-bearing for the claimed type I and type II error control in the periodicity test.

    Authors: We thank the referee for highlighting the need to connect the topological results more explicitly to the statistical assumptions. For periodic signals, §3 shows the sliding-window embedding is homotopy equivalent to S¹, a compact smooth manifold with positive reach and bounded curvature; these properties directly satisfy the standard manifold regularity conditions (including bounded curvature and the feasibility of sufficient sampling density) required for the subsampling asymptotics on persistence diagrams. The lower bound on reach established in §3 further supports the stability of the diagrams. For non-periodic signals the embedding is contractible and we retain the stated assumption on the support. In the revised manuscript we will add a remark in §4 that explicitly links the homotopy equivalence and reach bound from §3 to the manifold regularity conditions needed for the asymptotic guarantees, with emphasis on the periodic case that drives the periodicity test. revision: yes

  2. Referee: §5 (Statistical testing framework): The periodicity test relies on the confidence bounds to control errors, but the manuscript does not provide explicit verification or discussion of whether the manifold regularity holds for non-periodic signals or chirp signals in the simulations. Without this, the asymptotic control may not be justified in all regimes considered.

    Authors: We agree that an explicit discussion of the regularity assumptions in the simulation regimes would strengthen the justification. For periodic signals the embedding is diffeomorphic to S¹ and therefore satisfies the conditions. For the chirp signals used in the simulations, the time-delay embedding yields contractible trajectories that remain locally manifold-like and obey the reach lower bound; the observed empirical performance is consistent with the claimed error control. We acknowledge that a complete theoretical verification for arbitrary non-periodic signals lies outside the present scope. In the revision we will add a short paragraph in §5 that discusses the applicability of the manifold regularity assumption to the non-periodic and chirp examples, reiterating that the test is derived under the assumptions stated in the manuscript and that the simulations provide supporting evidence. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on independent topological proofs plus external regularity assumption

full rationale

The paper first proves homotopy equivalence of the time-delay embedding to S^1 (periodic) or contractible (non-periodic) and a lower bound on reach; these are self-contained geometric arguments. It then invokes the standard manifold regularity assumption to import asymptotic guarantees for the subsampling confidence bounds on persistence diagrams. The periodicity test is obtained by integrating these two pieces. No equation reduces a claimed prediction to a fitted parameter by construction, no self-citation chain carries the central claim, and the regularity assumption is stated explicitly rather than smuggled in. The derivation therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on one explicit regularity assumption for the data support and on standard results from manifold learning and persistent homology; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption the support satisfies standard manifold regularity
    Invoked to obtain asymptotic guarantees for the subsampling confidence bounds on persistence diagrams

pith-pipeline@v0.9.0 · 5555 in / 1203 out tokens · 30360 ms · 2026-05-17T01:37:09.694768+00:00 · methodology

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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/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    the embedded trajectory is homotopy equivalent to a circle (S¹) for periodic signals and is contractible for non-periodic ones. We also prove that the reach of the sliding window embedding is lower-bounded

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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