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Topological Periodicity Test (TopPT) via Confidence Bound of Time-Delay Embeddings

Conference: NeurIPS2026
arXiv: 2512.06324
Area: Time Series
Keywords: periodicity testing, time-delay embedding, persistent homology, subsampling confidence bounds, bounded interpolation error

TL;DR

TopPT turns an early-born, long-lived loop in a delay point cloud into a statistical test with a subsampling confidence radius, providing asymptotic error control over restricted, quantitatively separated signal classes; in synthetic experiments, the raw rule detects both periodic signals without rejecting four structured non-periodic signals, but interpolation-error correction substantially reduces detection, and real respiratory data only assess local loop detection.

Background & Motivation

Periodicity detection usually starts with power spectra or autocorrelation. The Generalized Lomb–Scargle periodogram (GLS) and Fisher's \(g\)-test identify frequency components in scalar sequences, but a strong frequency is not the same as a system repeatedly following a coherent trajectory in phase space. Trends, low-frequency variation, and nonstationary oscillations can all produce significant spectral peaks. Time-delay embedding offers a different perspective: combine values of the same signal at several delayed times into a vector, then examine whether these vectors trace a closed loop. SW1Pers established connections between sliding-window geometry and periodic structure, but a long bar in persistent homology alone does not establish whether a feature exceeds finite-sampling uncertainty.

The missing component is not a more elaborate periodicity score, but a complete inferential chain from the continuous signal through its embedded support to a finite-sample test. An arbitrary signal that is not strictly periodic need not lack loop-like geometry; even a circular theoretical support can have its birth and death scales altered by finite sampling and linear interpolation. The authors therefore first restrict the testable signal classes, requiring quantitative phase separation, smoothness, and nondegenerate trajectories. They then prove a lower bound on reach and convert point-cloud coverage error into a persistence-diagram confidence radius. These assumptions are part of the result, not details that can be discarded while retaining the slogan that periodic signals form circles and non-periodic signals form intervals.

Core Idea: establish small-scale topological separation between two restricted signal classes, estimate persistence-diagram error by subsampling, and reject the topological non-periodicity null only when a loop is born sufficiently early and lives beyond the confidence threshold.

Method

Overall Architecture

The input consists of scalar observations and timestamps; the output is a rejection or non-rejection decision at a specified window, delay, and embedding dimension. The procedure runs through phase-separated embedding, subsampling confidence radius, interpolation-error correction, and early-birth long-persistence testing. The first stage identifies the theoretical loop to seek, the middle stages quantify observation uncertainty, and the final stage compares persistence-diagram points with the resulting threshold.

There is no supervised learning, training label, or neural network. The theoretical branch uses exact delay observations; discrete scalar observations are first linearly interpolated to construct the observed delay cloud. The raw rule uses only the subsampling radius, whereas the robust rule additionally accounts for bounded measurement and interpolation errors. Experiments use Vietoris–Rips (VR) persistent homology as a computational surrogate, while support-level birth-scale results concern distance persistence, equivalently the Čech filtration.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Scalar observations<br/>and timestamps"] --> B["Phase-separated<br/>embedding"]
    B --> C["Subsampling<br/>confidence radius"]
    C --> D["Interpolation-error<br/>correction"]
    E["Known error and<br/>smoothness bounds"] -.->|Required for robust| D
    D -->|Raw omits correction| F["Early-birth<br/>long-persistence testing"]
    F --> G["Reject or do not reject"]

Key Designs

1. Phase-separated embedding: reduce restricted periodicity to the topology of circles versus intervals

At each anchor, the embedding collects the scalar value at that time and at \(m\) subsequent delayed times. The authors use \(m\) for the number of delay steps, so the vector has \(m+1\) dimensions. The total window span is \(m\tau\): neither \(m\) nor \(\tau\) alone determines the geometry.

\[ SW_{m,\tau}f(t)=[f(t),f(t+\tau),\ldots,f(t+m\tau)]^{\top}. \]

The decisive restriction is the definition of the signal classes. In Definition 3.2, non-periodicity requires the phase trajectory \(\Phi_f(t)=(f(t),f'(t))\) to be injective on the relevant interval: two distinct times cannot share both the function value and its first derivative. This is substantially stronger than simply lacking a global period. Its quantitative version additionally requires times separated by at least \(d\) to differ by at least \(\epsilon\) in either the function value or first derivative. For the periodic class, separation is measured using time distance modulo the period \(\Xi\). The two hypothesis classes are disjoint but are not complements: near-constant behavior, local recurrence, mixed rhythms, and some nonstationary oscillations are not all placed in the null.

To prevent local collapse of the embedded trajectory, Assumption A.1 requires \(f\) to be \(C^{2,1}\), meaning twice continuously differentiable with a Lipschitz second derivative. The absolute first and second derivatives and the Lipschitz constant of the second derivative are controlled by \(L\), and at every relevant time at least one of \(|f'|\) and \(|f''|\) is no smaller than \(\delta\). These conditions must cover the extended interval \(J_{m,\tau}=[T_{\min},T_{\max}+m\tau]\), not merely the anchor interval. In the periodic case, the anchor observation interval must also contain at least one full period; otherwise an observed segment of a periodic orbit may only be an open arc.

Local curvature control and distant phase separation jointly prevent intersections and near-intersections. Specifically, write \(\nu_{\delta,L}=\frac{\delta}{16}\min\{1,\frac{\delta}{8L}\}\). Scale compatibility requires \(d<\frac{\pi\nu_{\delta,L}^{2}}{2L^{2}}\), and the delay parameters must satisfy \(m\tau>\max\{\delta/(2L),\epsilon/(4L)\}\) and \(\tau<\min\{\delta/(8L),\epsilon/(16L)\}\). The total window must be sufficiently long and the individual steps sufficiently fine for scalar phase separation to imply separation of complete delay vectors.

Reach is the minimum distance from the support to its medial axis, interpretable as the radius of a tubular neighborhood in which nearest-point projection remains unique. Theorems 4.1 and 4.3 provide the following common positive lower bound. Its two terms account for global separation and local curvature, respectively:

\[ a_{m,\tau}=\sqrt{m+1}\min\left\{\frac{\epsilon}{32}\min\left\{1,\frac{\epsilon}{16L}\right\},\frac{\nu_{\delta,L}^{2}}{2L}\right\}. \]

Under these conditions, the non-periodic support is homeomorphic to an interval, and its neighborhoods below this bound are contractible. The periodic support is homeomorphic to a circle, whose small-scale neighborhoods retain one one-dimensional loop. In the corresponding Čech support diagram, the null has no one-dimensional features born below \(a_{m,\tau}\), whereas the alternative has a point \((0,D)\) with \(D\geq a_{m,\tau}\). The null may still acquire artificially closed loops at larger scales. The final test must therefore check birth location as well as persistence, rather than simply select the longest-lived feature.

2. Subsampling confidence radius: quantify diagram uncertainty through point-cloud coverage error

The theoretical sampling model is not a general stochastic time-series model. It independently samples anchor times from a distribution \(Q\) on an interval for a fixed smooth function, then obtains the corresponding delay vectors. The time density has positive lower and upper bounds and is Lipschitz, preventing parts of the support from remaining inaccessible to sampling. Regular time grids, overlapping windows, and observations with correlated noise are not automatically covered by this i.i.d. theorem. Interpolation-error correction does not itself resolve that dependence issue.

For a size \(n\) point cloud \(X\), draw size \(b\) subsets \(X_b^{(j)}\) without replacement and compute the Hausdorff distance between the complete cloud and each subset. Since a subset is contained in the cloud, this distance is the maximum, over full-cloud points, of the distance to the nearest subset point. It measures the coverage scale lost by a sparse subset, rather than spectral-peak variation after resampling a time series. The theoretical tail function averages over all subsets; Algorithm 1 approximates it with \(B\) Monte Carlo subsamples.

\[ L(t)=\binom{n}{b}^{-1}\sum_j\mathbf{1}\{d_H(X,X_b^{(j)})>t\},\qquad c_\alpha=2\inf\{t\geq0:L(t)\leq\alpha\}. \]

The threshold has upper-tail probability \(\alpha\), conventionally corresponding to the \(1-\alpha\) quantile of the distance distribution, and it must be multiplied by \(2\). It is not an ordinary lower-tail \(\alpha\) quantile. Diagram stability converts Hausdorff error into bottleneck error. Under \(b\to\infty\), \(b=o(n/\log n)\), and the remaining conditions, Theorem 5.1 bounds coverage failure by \(\alpha+O((b/n)^{1/4})\). Lemma 5.2 establishes that the radius converges to zero in probability for fixed embedding parameters.

Appendices D and F address the endpoints of the non-periodic support by supplying an interval version of the subsampling argument. Small-ball probability mass is one-sided at an endpoint, changing its leading constant relative to the interior; the authors use this to extend the original result for manifolds without boundary. This note retains the paper's stated local-mass regularity conditions rather than applying a general manifold confidence bound to intervals without explanation. The theoretical all-subsets tail and its finite-\(B\) estimate must also be distinguished: the displayed remainder is not an additional exact calibration guarantee for arbitrary Monte Carlo sample counts.

3. Interpolation-error correction: explicitly propagate scalar error into the confidence radius

Practical inputs often provide only discrete scalar observations, and delayed positions need not coincide with the observation grid. Piecewise-linear interpolation is therefore applied first. Assumption 3.5 requires the grid to cover the full extended interval, each scalar measurement error to be bounded by \(\sigma_N\), and a bound \(L_{\mathrm{int}}\) on the true signal's second derivative. If the maximum grid spacing is \(\Delta_N\), scalar error is the sum of measurement error and a second-order interpolation term. A delay vector has \(m+1\) components, amplifying the Euclidean error by \(\sqrt{m+1}\).

\[ \zeta_{m,N}=\sqrt{m+1}\left(\sigma_N+\frac{\Delta_N^2 L_{\mathrm{int}}}{8}\right),\qquad \hat c_\alpha^{\mathrm{err}}=\hat c_\alpha+5\zeta_{m,N}. \]

Here \(\hat c_\alpha\) is the subsampling radius computed on the interpolated observed cloud. Appendix G explains the factor \(5\): perturbations of at most \(\zeta_{m,N}\) to the full cloud and each subset alter their Hausdorff distance by at most \(2\zeta_{m,N}\). Multiplication by the quantile's factor \(2\) yields a radius change of at most \(4\zeta_{m,N}\). Adding the observed diagram's displacement of at most \(\zeta_{m,N}\) from the exact diagram gives \(5\zeta_{m,N}\). This is not an empirically tuned noise coefficient.

Theorem 5.3 extends the same asymptotic coverage form to the corrected radius. If the measurement-error bound holds only on a high-probability event, the final failure probability must additionally include the probability of that event's complement. Unbounded Gaussian or Laplace noise cannot use the deterministic conclusion without further treatment. Convergence of the robust radius to zero also requires \(\zeta_{m,N}\to0\), so increasing anchor count while maintaining a fixed measurement-error bound does not suffice. Synthetic experiments estimate \(L_{\mathrm{int}}\) numerically on a dense grid rather than use a separately certified analytic upper bound; the paper explicitly preserves this experimental qualification.

4. Early-birth long-persistence testing: exclude both diagonal noise and large-scale artificial loops

For each one-dimensional persistence point, let \(b_\gamma\) and \(d_\gamma\) denote its birth and death scales; their difference is its persistence. The confidence radius \(r_\alpha\) is \(c_\alpha\) for the exact rule and \(\hat c_\alpha^{\mathrm{err}}\) for the bounded-interpolation rule. Raw experiments use the observed cloud's \(\hat c_\alpha\) without correction. The null is rejected if at least one point satisfies both conditions below:

\[ \exists\gamma:\quad b_\gamma\leq r_\alpha,\qquad d_\gamma-b_\gamma>2r_\alpha, \qquad A_\alpha^r=\{a_{m,\tau}>4r_\alpha\}. \]

The birth condition makes the feature compatible with the theoretical genuine loop born at zero. The persistence condition places it farther from the diagonal than the confidence radius, so matching it to the diagonal cannot explain it as noise. Boundary conventions matter: birth equal to the radius remains eligible for rejection, but persistence equal to twice the radius does not.

The theoretical error control additionally relies on the separation event \(A_\alpha^r\). Theorem 6.2 first bounds the joint probability of an error and this event, not the error probability conditioned on \(A_\alpha^r\). The error bound without the separation restriction additionally contains \(P((A_\alpha^r)^c\mid H_i)\). Consequently, this is not a uniform finite-sample guarantee at the nominal significance level for arbitrary non-periodic signals. With fixed \(m,\tau\), shrinking radii, and the theoretical assumptions, both error probabilities receive asymptotic upper bounds of \(\alpha+o(1)\). At fixed \(\alpha\), this statement alone also does not prove that both errors necessarily converge to zero.

For a predetermined finite parameter grid of size \(K\), Corollary 6.4 applies level \(\alpha/K\) to each candidate and rejects globally if any candidate rejects, obtaining aggregate error control with the relevant separation events. This does not automatically resolve selection of candidate periods from the same data that will be tested. Real experiments do apply Bonferroni correction over the realized window–period set, while the authors acknowledge the gap between a fixed-grid theorem and this adaptive procedure.

The scale relationship between Čech and VR must also be retained. Section 2.2.1 uses an edge-length parameter and writes \(\mathcal C(\epsilon)\subseteq\mathcal{VR}(2\epsilon)\subseteq\mathcal C(2\epsilon)\), whereas Appendix B defines VR using pairwise distances below \(2r\) and writes \(\mathcal C(r)\subseteq\mathcal{VR}(r)\subseteq\mathcal C(2r)\). These use different parameter conventions, so thresholds cannot be mixed. Interleaving connects the computational surrogate to the theoretical filtration, but does not make their persistence diagrams identical. A rigorous transfer of the Čech support result to an implemented VR decision must still track scales and the corresponding bounds.

A Worked Example

Consider the paper's \(f_1\), with period \(\Xi=2\pi/5\). The synthetic configuration uses \(m=20\) and \(\tau=\Xi/(m+1)\), so each vector has \(21\) components. Observations must extend beyond the end of the anchor interval by \(m\tau\) to complete the last vector. Linear interpolation constructs the cloud; from \(n=336\) anchors, draw subsets of size \(b=120\) for \(500\) iterations, and the upper-tail threshold determines the confidence radius.

If a loop is born early and persists sufficiently long, the raw rule rejects. The robust rule uses the same observed cloud and diagram, adds \(5\zeta_{m,N}\), and checks both inequalities again. Its lower detection rate does not reflect unsuccessful network training: the larger error budget makes the persistence requirement harder to meet and can remove the separation margin \(a_{m,\tau}>4r_\alpha\). The values \(1.000\) and \(0.699\) below are aggregated rejection rates for these two rules on the same signal, not birth or death values from an individual sample.

Loss & Training

TopPT optimizes no learning loss and trains no classifier. Embedding, subsampling, and persistent homology directly process observations. Error and smoothness bounds are inputs to the robust rule, not quantities inferred from test rejection rates. The statistical distinction between a fixed parameter grid and data-adaptive real-data candidates is explained above.

Key Experimental Results

Main Results

The synthetic anchor interval is \([0,2\pi]\), and both periodic signals have known period \(2\pi/5\). They are \(f_1(t)=3/(2-\cos(5t))\) and \(f_2(t)=5\log(\sin(10t)+5)+\exp(\cos(5t))\). The non-periodic examples are a linear trend, a quadratic trend, a sigmoid trend, and a trend plus a low-frequency sinusoid. Each Table 1 row averages over \(100\) seeds and seven error settings: no noise, and uniform or truncated Gaussian noise with amplitude bounds \(0.02\), \(0.05\), and \(0.10\).

The shared configuration is \(N=400\) scalar observations, \(m=20\), \(n=336\) delay anchors, \(b=120\), \(B=500\), and \(\alpha=0.05\). This preserves the experimental use of \(N\) rather than forcing agreement with Assumption 3.5's grid indexed from \(s_0\) to \(s_N\). Periodic rows report detection rates; non-periodic rows report null rejection frequencies. These finite experimental frequencies do not replace theoretical error guarantees.

Signal Class or structure raw TopPT robust TopPT GLS Fisher \(g\) permutation
\(f_1\) Periodic 1.000 0.699 1.000 1.000 1.000
\(f_2\) Periodic 1.000 0.457 1.000 1.000 1.000
\(h_1\) Linear trend 0.000 0.000 1.000 1.000 1.000
\(h_2\) Quadratic trend 0.000 0.000 1.000 1.000 1.000
\(h_3\) Sigmoid trend 0.000 0.000 1.000 1.000 1.000
\(h_4\) Trend plus low-frequency sinusoid 0.000 0.000 1.000 1.000 1.000

Source: main-text Table 1 and Appendix I.1. All three scalar baselines reject the four structured non-periodic examples, showing that spectral significance and delay-space closure target different properties. This does not establish that spectral methods fail on all periodicity tasks.

Real-data experiments use raw TopPT; the results below come from main-text Table 2. Record-level rejection means at least one record-wise corrected local candidate rejects. Window-level rejection means at least one period candidate rejects within that window.

Dataset Records Windows TopPT record rejections TopPT window rejections GLS: records / windows Fisher \(g\), permutation: records / windows
Fantasia 40 280 35/40 142/280 40/40; 280/280 40/40; 280/280
BIDMC 53 318 51/53 249/318 53/53; 318/318 53/53; 318/318

Fantasia uses the first \(90\) seconds, downsamples to \(25\) Hz, and uses starts at \(0,10,\ldots,60\) seconds for seven overlapping \(20\)-second windows. BIDMC uses the first \(180\) seconds, applies a \(0.05\)–\(0.8\) Hz bandpass filter, downsamples to \(25\) Hz, and uses starts at \(0,30,\ldots,150\) seconds for six \(30\)-second windows. Windows are linearly detrended and standardized; embeddings use \(m=8\), at most \(300\) anchors, and \(300\) subsampling iterations.

Appendix I.3 sets the subset size to \(b=0.99n\), which the authors describe as practical calibration to increase local sensitivity, not the asymptotic configuration \(b=o(n/\log n)\). Candidate periods come from autocorrelation and Welch peaks. The strongest hint is multiplied by \(0.9,1,1.1\), clipped, and deduplicated, producing \(825\) window–period tests for Fantasia and \(954\) for BIDMC. TopPT correction covers window–period pairs, whereas scalar baseline correction covers windows; their search objects differ.

These data have neither topological null labels nor certified measurement-error or interpolation-smoothness bounds, so robust real-data testing is not reported. The values \(35/40\) and \(51/53\) are record rejection counts, not accuracy; they cannot establish correct-classification rates or false-positive rates.

Ablation Study

The paper does not present neural-module removal ablations. The comparison of raw and robust rules in Table 1 analyzes error correction. The following nonstationary stress test from Appendix Table 3 illustrates how both rules behave outside the strict hypothesis classes.

Signal Nonstationary mechanism raw TopPT robust TopPT GLS Fisher \(g\) permutation
\(g_1\) Phase drift 0.791 0.137 1.000 1.000 1.000
\(g_2\) Amplitude modulation 1.000 0.714 1.000 1.000 1.000

Source: Appendix I.2 and Table 3; rates again average over \(100\) seeds and seven bounded-error settings. The signals are \(g_1(t)=\sin(5t+0.18t^2)\) and \(g_2(t)=\{1+0.18\cos(0.5t)+0.08t\}\sin(5t)\). Although not strictly periodic, they may have substantial loop-like recurrence. Their rejections should not be interpreted as type I errors under the topological null in Section 6.

Key Findings

  • Bounded-error correction retains zero rejection for the four structured non-periodic examples, while periodic rejection rates fall from \(1.000\) to \(0.699\) and \(0.457\), reflecting the conservativeness of the larger confidence radius.
  • Raw/robust rejection rates are \(0.791/0.137\) for phase drift and \(1.000/0.714\) for amplitude modulation. Loop-like recurrence and strict periodicity are not interchangeable.
  • Respiratory data demonstrate local selectivity, not labeled classification performance. Figure 4's maximum persistence margin is persistence minus \(2\hat c_\alpha\); a positive margin satisfies only the persistence condition, and birth must still be checked.

Highlights & Insights

  • Include birth scale in the test: long persistence alone does not certify a genuine small-scale loop, because an open curve can acquire a hole when covered at a large scale. The early-birth condition links the decision to the actual small-scale distinction between circles and intervals.
  • Separate geometric identifiability from statistical precision: \(a_{m,\tau}\) measures the signal class's topological margin, whereas \(r_\alpha\) measures observation uncertainty. Comparing them explains detection credibility or lack of separation more directly than reporting a large maximum persistence alone.
  • Make the error budget traceable: \(5\zeta_{m,N}\) accounts for perturbations of both diagrams and subsampling radii. This reasoning can inform other topological tests involving interpolation or approximate geometric reconstruction, but their error bounds must be established separately.

Limitations & Future Work

  • The null is not all non-periodic signals: phase injectivity, quantitative separation, nondegenerate smoothness, and full-period coverage restrict applicability. Constants, partial recurrence, quasi-periodicity, and mixed dynamics require separate models.
  • Theoretical and practical sampling differ: theory assumes i.i.d. anchors, whereas real sequences and windows involve dependence, and the real configuration \(b=0.99n\) is outside the stated subsampling asymptotic regime. Coverage theory for dependent sampling or practical window construction remains necessary.
  • Adaptive search lacks the same guarantee: fixed finite-grid Bonferroni results do not directly certify candidate periods estimated from the same data. Sample splitting or post-selection inference are research directions, not steps completed in this paper.
  • The computational surrogate requires scale alignment: experimental VR diagrams and theoretical Čech diagrams are not identical objects. Rigorous experimental certification should specify filtration conventions and threshold conversions induced by interleaving.
  • Certified inputs for the robust rule are limited: synthetic experiments use numerical second-derivative bounds, and real data lack certified error bounds. Finite Monte Carlo quantile error must also be distinguished from the theoretical all-subsets bound. Strengthening these certification steps may be more valuable than increasing detection alone.
  • Non-rejection does not mean no oscillation: it means no loop satisfies this confidence rule at the tested window, candidate period, and embedding scale. Noise, rhythmic drift, or an overly conservative error radius can all lead to non-rejection.
  • vs SW1Pers / Perea–Harer: both represent periodic structure using persistent loops in sliding windows. This paper emphasizes quantitative reach, finite-cloud confidence radii, and an explicit rejection region rather than only constructing a periodicity score.
  • vs Fasy et al.'s persistence-diagram confidence sets: subsampling and Hausdorff-to-bottleneck stability are existing tools. This paper connects them to signal-specific circle/interval separation and discusses curves with endpoints and scalar interpolation errors.
  • vs GLS, Fisher \(g\), and permutation max-periodogram: spectral tests assess significant frequency structure in a scalar sequence, whereas TopPT assesses an early-born, long-lived loop beyond the confidence scale in a delay cloud. These perspectives can complement each other; differences on the present structured nulls do not establish a universal ranking.

Rating

  • Novelty: 4/5 — Connects restricted signal topology, reach, and subsampling confidence bounds into an interpretable testing chain.
  • Experimental Thoroughness: 3/5 — Includes structured nulls, error correction, nonstationary stress cases, and real data, but practical calibration and theoretical certification remain distinct.
  • Writing Quality: 4/5 — Clearly distinguishes assumptions and raw/robust rules; Čech/VR parameter conventions and some extracted formulas require careful reading.
  • Value: 4/5 — Adds a geometric perspective that combines identifiability and uncertainty, currently most suitable for research with explicit assumptions.