PaperlayerView exact revision on arXiv

Source-linked AI summary

TopTimeNet: Topologically-assisted time-series classification model

Sharareh Sayyad, Sophia Bazzi

arXiv:2609.39792v2cs.LGcs.CGmath.DSnlin.CDphysics.app-ph

TL;DR

Distinguishing periodic from chaotic dynamics is difficult when only finite, noisy time series are available. TopTimeNet extracts fixed geometric and topological descriptors before a lightweight classifier learns from them. On a 49-system benchmark, its compact configuration matches a much larger one’s mean accuracy and performs comparably to CNNs and better than the average of converged Transformer runs, while raw-signal noise remains a distinct robustness challenge.

  • Problem

    Periodic and chaotic dynamics must often be distinguished from finite, noisy time series when governing equations or complete state measurements are unavailable.

    Source: §I · p. 1

  • Method

    TopTimeNet computes 42 fixed geometric and topological descriptors from delay-embedded signals and persistent homology, then uses a lightweight learnable stage for classification.

    Source: §I · p. 2

  • Results

    A 1,638-parameter classifier matches the mean accuracy of a configuration with 33× more trainable parameters and achieves mean accuracy comparable to CNNs and higher than the mean of converged Transformer runs.

    Source: §V · p. 18

  • Takeaways & Limitations

    The benchmark supports separating fixed feature construction from lightweight discrimination as a way to obtain comparable classification accuracy with substantially fewer trainable parameters.

    Source: §V · pp. 17–18

    Robustness to raw-signal acquisition noise cannot be assumed from persistent homology’s stability properties and must be established empirically.

    Source: §V · p. 18

Abstract

from arXiv · show

Distinguishing periodic from chaotic dynamics in a time series is a fundamental challenge in both physics and engineering. Yet, end-to-end learned architectures must discover both a representation and a decision boundary from data, at substantial cost. We introduce TopTimeNet, which decouples these tasks: a fixed, non-learned stage extracts a $42$-dimensional geometric and topological descriptor from Takens delay embeddings and persistent homology, and a lightweight learnable stage performs classification. On a benchmark of $49$ nonlinear dynamical systems, a $1{,}638$-parameter configuration matches the mean accuracy of one with $33\times$ more trainable parameters. Additionally, this approach delivers mean accuracy comparable to convolutional neural networks and surpasses the average performance of converged Transformer models, while requiring three to four orders of magnitude fewer trainable parameters. Robustness also depends sharply on where noise is introduced: TopTimeNet degrades gracefully under perturbations to its precomputed features, but degrades sharply when noise is introduced into the raw signal and the full feature-extraction pipeline is recomputed, showing that robustness to perturbations of the precomputed features does not imply robustness of the complete raw-signal-to-prediction pipeline. These results show that decoupling fixed geometric and topological feature construction from a lightweight discriminative stage can achieve comparable classification accuracy with substantially fewer trainable parameters.

I. INTRODUCTION

Periodic and chaotic regimes must often be inferred from finite, noisy time series when governing equations or full state measurements are unavailable. TopTimeNet makes geometric and topological structure explicit before learning, then tests how much trainable capacity classification requires and how noise location affects accuracy.

I. INTRODUCTION

  • Finite data and measurement noise can obscure distinctions between chaotic and stochastic signals, while common nonlinear measures probe particular signal properties.The introduction frames the problem around limited observations and the need to infer regimes when equations or state variables are unavailable.Source: §I · p. 1
  • TopTimeNet computes geometric and topological features before a learnable stage projects, combines, and classifies them, shifting learning toward interactions among feature groups.The 42 descriptors are arranged in geometric, entropy, lifetime, Betti-curve, and persistence-image groups.Source: §I · p. 2
  • A 33-fold parameter-count difference between searched TopTimeNet configurations yields nearly identical mean classification accuracy.The study also compares TopTimeNet with CNNs and Transformers trained directly on raw time series.Source: §I · p. 2
  • Accuracy degrades more sharply when raw-signal noise triggers feature recomputation than when noise perturbs precomputed features.Robustness to feature-level perturbations therefore does not establish robustness of the full raw-signal-to-prediction pipeline.Source: §I · p. 2

A. Dataset generation

  • The benchmark uses 49 nonlinear systems, segmented into univariate signals and balanced to 22,600 time series across periodic and chaotic classes.The dataset comprises 96 system-state pairs and 239 individual signals before segmentation and balancing.Source: A. Dataset generation · pp. 2–3

B. TopTimeNet model

TopTimeNet transforms each segmented signal into a delay-coordinate point cloud, then computes geometric and persistent-homology descriptors. These fixed feature groups form a representation for the learnable fusion and classification stages.

1. Feature extraction

  • A segmented signal is mapped to a delay-coordinate point cloud using embedding dimension 2 and time delay 20, fixed throughout evaluation.The embedding creates T − (d_emb − 1)τ points per cloud.Source: §1 · p. 3
  • The geometric branch summarizes the cloud with its diameter, nearest-neighbor distances, and a correlation-dimension proxy based on pairwise distances.The proxy uses two distance percentiles rather than a fitted scaling regime.Source: §1 · p. 3
  • Persistent homology tracks connected components and higher-dimensional holes as the Vietoris–Rips filtration grows, recording each feature’s birth and death thresholds.The persistence diagrams provide the input to four descriptor branches.Source: §1 · p. 4
  • The four topological branches summarize persistence through entropy, lifetime statistics, sampled Betti curves, and persistence images.Betti curves count live features across filtration thresholds; persistence images smooth birth–persistence pairs into a fixed grid.Source: §1 · pp. 4–6
  • Empty diagrams receive defined zero-valued summaries in the entropy, lifetime, Betti-curve, and persistence-image branches.For an empty Betti curve, the bimodality coefficient instead remains a large finite value.Source: §1 · p. 6

2. Learnable fusion and classification

TopTimeNet normalizes and projects five deterministic feature groups into tokens, combines information across groups with a selectable fusion strategy, and classifies the fused representation.

2. Learnable fusion and classification

  • Each deterministic feature group has its own batch-normalization and projection module, producing five embeddings in a shared space.The five tokens are concatenated or otherwise combined by the selected fusion layer before classification.Source: §2 · pp. 6–7
  • Fusion strategies range from pairwise group interactions and gated residual updates to linear attention, low-rank cross-group MLPs, and token attention.The low-rank MLP mixes information across concatenated tokens while limiting parameters through its bottleneck dimension.Source: §2 · pp. 7–8
  • Linear attention applies query–key–value interactions across the five tokens using a positive feature map instead of softmax.The feature map is ELU(x) + 1.Source: §2 · p. 7
  • The classification head batch-normalizes the fused representation and maps it to class logits through an MLP or a single linear layer.The MLP can use configurable hidden layers, activations, and dropout.Source: §2 · p. 8

1. Training protocol

TopTimeNet’s learnable components are trained jointly with cross-entropy, with optimization settings selected by configuration. Evaluation uses segment-level splits, which constrain claims about generalization to unseen trajectories or systems.

1. Training protocol

  • Training uses cross-entropy, validation-based early stopping with patience 50, best-checkpoint restoration, and learning-rate reduction on validation-loss plateaus.Runs last up to 500 epochs; the optimizer, label smoothing, gradient clipping, and input normalization are selected separately for each configuration.Source: §1 · p. 8
  • Segments from the same trajectory may cross the stratified 80/10/10 partitions, potentially inflating test performance and leaving unseen-trajectory or unseen-system generalization unmeasured.All benchmarked architectures use the same segment-level splits, enabling controlled comparisons under that protocol.Source: §1 · p. 8
  • The small configuration standardizes the 42-dimensional feature vector using training-set statistics, while the large configuration does not.Raw signals retain their native amplitudes when geometric and topological features are computed.Source: §1 · p. 8
  • A validation-set temperature is fit after training, constrained to [0.5, 5.0], to improve alignment between predicted confidence and empirical accuracy without changing trained weights..Source: §1 · p. 8

2. Hyperparameter search

The search samples architectural, regularization, and optimization settings, while keeping TDA feature parameters fixed. It selects both a highest-scoring configuration and a parameter-efficient alternative, then retrains each 30 times.

2. Hyperparameter search

  • Random search samples 400 configurations across architectural, regularization, and optimization choices, while batch size 128 and patience 50 are effectively fixed.Attention configurations with embedding dimensions not divisible by the number of heads are rejected and resampled.Source: §2 · pp. 8–9
  • Takens embedding, delay, Betti-bin, and persistence-image settings are fixed rather than searched, and their features are precomputed and cached across trials.The fixed feature set is valid for the search because those TDA-specific settings do not vary between trials.Source: §2 · p. 9
  • Trials are scored by 0.6·F1 + 0.4·G-mean, while CNN and Transformer baselines use the same scoring formula with α = 0.7 and separate architecture-specific searches.The baseline objective weight therefore differs from TopTimeNet’s.Source: §2 · p. 9
  • The highest-scoring configuration and a Pareto-selected parameter-efficient alternative are each retrained 30 times with independent seeds, and evaluation metrics are reported as mean ± standard deviation.The small alternative is chosen among configurations within a 1% score tolerance while using fewer parameters.Source: §2 · p. 9

III. ILLUSTRATIVE EXAMPLE

A double-pendulum example illustrates how TopTimeNet’s five feature branches offer complementary descriptions of periodic and chaotic dynamics. Across geometric and topological features, the chaotic trajectory generally appears more spatially diffuse and less dominated by a single persistent loop.

Example setup

  • The illustrative example uses a double pendulum with distinct initial conditions for the periodic and chaotic trajectories; their estimated largest Lyapunov exponents are approximately 0.015 and 0.106, respectively.The periodic estimate is close to zero, while the chaotic estimate is positive.Source: §III · p. 9

Geometric branch

  • The chaotic point cloud has a diameter of 3.2792 versus 1.1552 and a correlation-dimension proxy of 1.6585 versus 1.0672, while its four geometric features separate the regimes in this example.Mean nearest-neighbor distance and its standard deviation also increase by approximately 29-fold and 59-fold, respectively.Source: §III · p. 10

Entropy branch

  • H1 entropy is 0.038 for the periodic signal and 5.01 for the chaotic signal, whose persistence diagram contains hundreds of short-lived loops rather than one dominant loop.The H0 entropy values are less strongly separated in this example.Source: §III · pp. 10–11, Figure 3 · p. 10

Lifetime branch

  • For H1, the periodic signal has one significant lifetime and a 0.9959 dominance ratio, whereas the chaotic signal has 169 significant lifetimes and a 0.0254 dominance ratio.The total lifetime rises from 0.9386 to 6.0242 even as the maximum lifetime falls from 0.9348 to 0.1533; these statistics decompose the contrast into scale, concentration, spread, and count.Source: §III · p. 11

Betti-curve branch

  • The H1 Betti curve shows one periodic loop across most of the filtration range, versus a chaotic peak of nearly 50 coexisting loops that decays to near zero by normalized threshold 0.15.For H0, the chaotic curve decays more gradually, with a plateau around normalized thresholds 0.05–0.1.Source: §III · p. 11, Figure 4 · p. 11

Persistence-image branch

  • The H1 persistence-image centroid is 0.8928 for the periodic case and 0.4860 for the chaotic case, whose intensity is more diffuse.The H0 birth centroid of 0.5 in both regimes is an implementation consequence of padding a degenerate birth axis, not a property of the dynamics.Source: §III · pp. 12–13, Figure 5 · p. 12

Overall feature complementarity

  • The five branches describe complementary aspects of the periodic–chaotic contrast, from point-cloud geometry and persistence concentration to loop counts across filtration scales and birth–persistence locations.The example motivates combining all 42 features rather than relying on any single representation.Source: §III · p. 13

IV. RESULTS

On the 49-system extendedteaspoon benchmark, the 1,638-parameter TopTimeNet configuration matched the larger configuration’s mean test accuracy. The results motivate using the compact configuration in subsequent analyses but do not establish a general capacity threshold.

IV. RESULTS

  • The benchmark spans 49 nonlinear dynamical systems simulated in periodic and chaotic regimes.Source: §IV · p. 13
  • 97.58% ± 0.22% mean test accuracy for the 1,638-parameter configuration matched 97.58% ± 0.29% for the 54,886-parameter configuration.Both values are means over 30 independent training runs.Source: §IV · pp. 13–14, Table VI · p. 14
  • The small configuration stopped at an average of epoch 306, while the large configuration used its full 500-epoch budget in every run.The small configuration’s stopping epochs ranged from 140 to 500 under the adopted early-stopping criterion.Source: §IV · p. 14
  • Increasing trainable parameters from 1,638 to 54,886 did not improve mean test metrics under the present evaluation protocol.This comparison does not establish a general capacity threshold beyond which additional parameters cannot help.Source: §IV · p. 14

A. Comparison against convolutional and transformer-based baselines

Under the tested protocol, the small TopTimeNet configuration achieved mean accuracy comparable to the CNN and higher than the converged Transformer runs, with substantially fewer trainable parameters. Three of ten Transformer attempts did not converge.

A. Comparison against convolutional and transformer-based baselines

B. Robustness evaluation

TopTimeNet’s accuracy degrades more sharply when noise enters the raw signal and the full feature pipeline is recomputed than when noise perturbs precomputed features. The sweeps use absolute noise levels, and their comparison does not isolate the cause of that difference.

B. Robustness evaluation

  • At σ=0.1, feature-level accuracy remained 97.02% ± 0.25%; raw-signal accuracy had fallen to 71.27% ± 1.47% already at σ=0.025.The raw-signal curve reached 51.51% ± 0.12% at σ=1.0.Source: Figure 6 · p. 16, B. Robustness evaluation · p. 17
  • Feature-level noise perturbed cached 42-dimensional features, whereas raw-signal noise triggered recomputation of the full feature pipeline, including persistent homology.The CNN and Transformer had only raw-signal curves because they lack an intermediate feature representation to perturb separately.Source: Figure 6 · p. 16, B. Robustness evaluation · p. 16
  • The noise grid uses absolute Gaussian standard deviations, so equal σ values need not represent equal relative perturbations across systems or between TopTimeNet’s two sweeps.In one worked example, σ≥0.5 was comparable to or exceeded the clean signal’s scale.Source: B. Robustness evaluation · p. 16
  • Only TopTimeNet received training-time noise augmentation, and that augmentation was applied to features rather than raw signals.The feature-level model was trained with σaug=0.05; the CNN and Transformer received no noise augmentation.Source: B. Robustness evaluation · p. 17
  • At σ=1.0, TopTimeNet’s feature-level ECE was 10.02% ± 1.54%, while its raw-signal ECE reached 47.54% ± 0.45%.All three raw-signal models combined near-chance accuracy with large ECE values at the highest noise levels.Source: Figure 6 · p. 16, B. Robustness evaluation · p. 17
  • The experiment does not isolate why the two TopTimeNet robustness curves differ, although a single example shows raw-signal noise disrupting the point cloud and persistence diagram before classification.That example does not establish how much this mechanism accounts for benchmark-level degradation.Source: B. Robustness evaluation · p. 17
  • For deployment, feature-perturbation robustness should not be taken as evidence of robustness to acquisition-time noise; evaluate by propagating corrupted signals through the complete pipeline.Source: B. Robustness evaluation · p. 17

V. CONCLUSION

TopTimeNet separates deterministic geometric and topological feature construction from a lightweight learnable classifier. On the nonlinear-system benchmark, it achieves comparable mean accuracy with substantially fewer trainable parameters, while raw-signal noise exposes a distinct robustness limitation.

V. CONCLUSION

  • TopTimeNet deterministically computes five geometric and topological feature families from Takens delay embeddings and persistent homology, leaving only classification to a lightweight learnable stage.The five families are geometric, entropy, lifetime, Betti-curve, and persistence-image statistics.Source: §V · p. 17
  • On the nonlinear-system benchmark described in Sec. II A, a 1,638-parameter classifier matches the mean accuracy of a configuration with 33× more trainable parameters, is comparable to the CNN, and exceeds converged Transformer runs with three to four orders of magnitude fewer trainable parameters.Source: §V · p. 18
  • Under the tested protocol, no non-converged runs occurred in 30 TopTimeNet runs or 10 CNN runs, while 3 of 10 Transformer training attempts did not converge.Source: §V · p. 18
  • Feature-vector perturbations degrade accuracy gradually, whereas raw-signal noise followed by full feature recomputation causes sharp degradation; CNN and Transformer pipelines also collapse under raw-signal noise.The raw-signal results are under the tested protocol, and the magnitude of a given noise level relative to the signal varies across systems and variables.Source: §V · p. 18
  • Feature-level robustness does not establish raw-signal robustness, and the experiments do not isolate the cause of the different robustness curves.The two experiments perturb different spaces, and only feature-level noise was used for TopTimeNet training augmentation.Source: §V · p. 18
  • Persistence-diagram stability does not guarantee stability of the complete pipeline; a single example illustrates a candidate noise mechanism, not a general periodic-versus-chaotic fragility rule.The noise is added to observed signals after trajectory generation, not to initial conditions; the illustrated periodic signature is more easily destroyed, consistent with its smaller amplitude and simpler structure.Source: §V · p. 18
  • Future tests include multiclass benchmarks spanning periodic, quasiperiodic, chaotic, and stochastic regimes, and testing the framework on quantum dynamics.The reliability of the present feature set for these settings remains to be established; the quantum application is a proposed direction, not an established outcome.Source: §V · p. 18
  • Applying representation/discrimination decoupling beyond periodic-versus-chaotic time series is an open direction, including testing where its efficiency and robustness behavior carry over.The paper proposes testing this idea in other tasks with informative domain-based feature representations.Source: §V · pp. 18–19

VII. AUTHOR CONTRIBUTIONS

S. S. conceived and implemented the project, analyzed the data, and prepared the initial manuscript draft; S. B. contributed to results analysis, and both finalized the manuscript.

VII. AUTHOR CONTRIBUTIONS

  • S. S. conceived the project, implemented the algorithm, analyzed the data, and prepared the initial draft; S. B. contributed to analysis, and both finalized the manuscript.Source: §VII · p. 19

Appendix A: Illustrative raw-signal noise sweep on the double pendulum

The double-pendulum illustration follows raw-signal Gaussian noise through the recomputed feature pipeline. The periodic example loses its clean loop structure and shows much larger feature-vector drift than the chaotic example.

Appendix A: Illustrative raw-signal noise sweep on the double pendulum

  • The illustration adds zero-mean Gaussian noise at five levels to one periodic and one chaotic double-pendulum segment, then recomputes the full feature pipeline.The tested levels are σ ∈ {0, 0.1, 0.2, 0.5, 1.0}.Source: Appendix A · p. 19
  • At σ=0.1, noise is 25% and 15% of periodic and chaotic signal standard deviation, respectively; by σ=0.5 it exceeds the periodic signal’s scale and approaches the chaotic signal’s.At σ=1.0 it exceeds both signal scales, more so for the periodic example.Source: Appendix A · p. 19
  • By σ=0.2, the periodic embedded ring’s central hole has effectively closed, while the chaotic point cloud remains comparatively similar across the noise range.The periodic point cloud is a thin ring at σ=0; the chaotic cloud is already diffuse at σ=0.Source: Appendix A · pp. 19–20, Figure 8 · p. 20
  • The periodic diagram’s dominant H1 bar shrinks as noise rises, while the chaotic diagram’s scattered H1 bars remain qualitatively similar across the tested range.By σ=0.5, the periodic diagram’s once-dominant bar is only marginally distinguishable from the growing cloud of small bars.Source: Appendix A · p. 20, Figure 9 · p. 21
  • Periodic feature-vector drift rises from below 0.1 to above 0.8 between σ=0.05 and 0.15, while chaotic drift stays below 0.13 across the tested range.At σ=0.1, drift is 0.40 versus 0.03; at σ=0.2, it is 0.86 versus 0.04.Source: Appendix A · p. 20, Figure 10 · p. 21
Loading 2609.39792v2…