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Shifting the Paradigm: A Diffeomorphism Between Time Series Data Manifolds for Achieving Shift-Invariancy in Deep Learning
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Proposes a differentiable bijective phase-based diffeomorphism that maps time-shifted time series to a canonical shift via a learned guidance network, giving shift-invariance for time series without shift-range limits.
Problem
Deep networks for time series are highly sensitive to circular time shifts, and existing shift-invariance methods designed for images (low-pass filtering, adaptive subsampling) fail to give full shift-invariance on time series and often hurt accuracy.
Approach
Use the Fourier phase of a harmonic whose period is at least the signal length to uniquely index all circular shifts; apply a linear phase shift to all harmonics so that every shifted variant maps to the same point on a manifold, with a guidance network predicting the target angle and a variance-based loss.
Key results
- Ours achieves 100% shift consistency on all nine datasets while the baseline ResNet reaches as low as 32% on DaLiA heart rate and prior methods stay in the 50-98% range.
- On IEEE SPC22 heart rate, RMSE drops from 18.39 (baseline) to 16.25 bpm, and on Sleep-EDF accuracy rises from 75.41 to 77.90.
- Previous shift-invariance methods (LPF, APS) often degrade heart rate performance, with error increasing 7-10% on DaLiA.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add learned guidance to the transformation rather than mapping every sample to a single fixed manifold (ablation) | Proposed diffeomorphism · IEEE SPC22 | RMSE: 19.18 -> 16.25 | Helped |
| Add learned guidance to the transformation rather than mapping every sample to a single fixed manifold (ablation) | Proposed diffeomorphism · UCIHAR | Acc: 84.67 -> 85.81 | Helped |
| Add learned guidance to the transformation rather than mapping every sample to a single fixed manifold (ablation) | Proposed diffeomorphism · HHAR | Acc: 92.33 -> 91.83 | Hurt |
| Add learned guidance to the transformation rather than mapping every sample to a single fixed manifold (ablation) | Proposed diffeomorphism · Sleep-EDF | Acc: 75.54 ±2.39 -> 77.80 ±1.95 | Helped |
| Add learned guidance to the transformation rather than mapping every sample to a single fixed manifold (ablation) | Proposed diffeomorphism · Chapman | Acc: 91.82 -> 92.10 | Helped |
| Add learned guidance to the transformation rather than mapping every sample to a single fixed manifold (ablation) | Proposed diffeomorphism · Respiratory | Acc: 21.28 ±7.43 -> 33.10 ±5.12 | Helped |
| Train the guidance network with classification cross-entropy only, omitting the angle-variance term (ablation) | Proposed diffeomorphism · IEEE SPC22 | RMSE: 17.16 -> 16.25 | Hurt |
| Train the guidance network with classification cross-entropy only, omitting the angle-variance term (ablation) | Proposed diffeomorphism · UCIHAR | Acc: 84.30 -> 85.81 | Helped |
| Train the guidance network with classification cross-entropy only, omitting the angle-variance term (ablation) | Proposed diffeomorphism · Sleep-EDF | Acc: 77.21 ±1.51 -> 77.80 ±1.95 | Helped |
| Train the guidance network with classification cross-entropy only, omitting the angle-variance term (ablation) | Proposed diffeomorphism · Respiratory | Acc: 27.17 ±6.71 -> 33.10 ±5.12 | Helped |
| Add an angle-variance term that encourages greater diversity of mapped manifolds (ablation) | Proposed diffeomorphism · IEEE SPC22 | RMSE: 17.00 -> 16.25 | Mixed |
| Add an angle-variance term that encourages greater diversity of mapped manifolds (ablation) | Proposed diffeomorphism · UCIHAR | F1: 81.99 -> 83.81 | Hurt |
Surprising findings
- Existing shift-invariance techniques (LPF, adaptive subsampling) decrease performance on heart rate estimation tasks.
- Short shifts of 10-100 ms can change heart rate predictions by over 80 bpm in trained ResNets.
Limitations
- Relies on a harmonic with period at least the signal length and the Fourier transform length equal to the signal length.
- Gains in some tasks are marginal (e.g., HHAR accuracy drops 0.5% with the guidance network).
Experimental setup
Datasets: IEEE SPC 2015 DaLiA UCIHAR HHAR Chapman-Shaoxing ECG PhysioNet 2017 Clemson step counting Sleep-EDF Respiratory@TR
Benchmarks: IEEE SPC 2015 DaLiA UCIHAR HHAR Chapman-Shaoxing ECG PhysioNet 2017 Clemson Sleep-EDF Respiratory@TR
Metrics: shift consistency accuracy F1 RMSE MAE MAPE AUC Cohen's kappa
Baselines: Baseline ResNet Augmentation LPF APS WaveletNet Canonicalize
Builds on: low-pass filtering anti-aliasing adaptive polyphase sampling wavelet networks time-delay neural networks
Closest papers by meaning
- Representing Closed Transformation Paths in Encoded Network Latent Space (AAAI 2020 · similarity 0.85)
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