Dataset Condensation for Time Series Classification via Dual Domain Matching

What came before

Dataset condensation for time series had not been systematically studied; the few applicable methods were image/graph condensation techniques (DD, DC, DSA, DM, MTT, IDC, HaBa) applied unmodified to raw time-domain sequences, using image-style random-pixel initialization and image-style augmentations (jittering, cropping).

The problem

Directly porting image/graph condensation to time series performs poorly because it only uses the time domain, ignoring that time series have periodicity/seasonality structure that is far more separable in the frequency domain (confirmed by the paper's own t-SNE visualization, Fig. 3, showing better class separation in frequency space); it also inherits image-style random-pixel initialization, which the paper finds empirically performs poorly for time series.

The idea

Match the same surrogate objectives (multi-step gradient matching, parameter-space embedding matching) used by prior image methods, but compute them in both the time domain and the Fourier-transformed frequency domain, and enrich the synthetic set at each step with additional frequency-based augmented views (low-pass filtering, phase perturbation, magnitude perturbation) so the matching signal draws on frequency structure the time domain alone does not expose.

How it works

Initialize the synthetic set $\mathcal{S}$ via K-means cluster centroids per class (not random samples, per the empirical finding above). At each training epoch, for each of four augmented views (raw, LPF, phase-perturbed, magnitude-perturbed — Eq. 5-8, sequentially applied via FFT/inverse-FFT), compute both the time-domain and frequency-domain (via FFT) versions of the view; train a network on $\mathcal{S}$ for $N{=}10$ steps and on real data $\mathcal{T}$ for $M{=}1000$ steps from the same sampled initialization $\theta_0\sim p(\theta)$ (approximated by real-data training trajectories, as in MTT), separately in each domain, to get $\hat\theta_N$ and $\bar\theta_M$; compute a multi-step gradient- matching loss $\mathcal{L}^{grad}=\|\hat\theta_N-\bar\theta_M\|^2/\|\theta_0-\bar\theta_M\|^2$ (Eq. 9-11, an MTT-style trajectory-matching term) and a parameter-space embedding-matching loss $\mathcal{L}^{emb}=\|\text{Mean}(\hat{H}_N)-\text{Mean}(\bar{H}_M)\|^2$ (Eq. 12-15, a DM-style distribution-matching term, but computed on embeddings from the trained trajectory parameters rather than random-network embeddings) in each domain; sum all four views' losses ($\mathcal{L}=\mathcal{L}_{raw}+\mathcal{L}_{LPF}+\mathcal{L}_{PP}+\mathcal{L}_{MP}$, each itself $\mathcal{L}_t^{grad}+\mathcal{L}_f^{grad}+\lambda(\mathcal{L}_t^{emb}+\mathcal{L}_f^{emb})$) and backpropagate to update $\mathcal{S}$. Stored artifact: only the time-domain synthetic sequences and hard labels (frequency-domain versions and augmented views exist only during synthesis).

Evidence

Table 2 (5 datasets — HAR, Electric, Insect, FD, Sleep — CNN+BN evaluator, hard labels, 10 baselines including DD/DC/DSA/DM/MTT/IDC/HaBa and coreset methods): CondTSC beats every baseline at every spc/ratio with statistical significance (t-test p<0.01), e.g. HAR spc=1 (0.1%): 61.38% vs. best baseline (DC) 58.30% (full data 93.14%); Insect spc=1 (0.05%): 45.15% vs. best baseline (K-means) 16.70% (full 70.78%) — the largest margin in the paper, on the dataset where frequency structure matters most; FD spc=20 (0.72%): 90.78% vs. best baseline (MTT) 74.63% (full 98.51%). NAS downstream task (Table 3, HAR, 324-ConvNet search space, 1% ratio): CondTSC's condensed-set accuracy ranking correlates best with full-data ranking (Spearman 0.665 vs. next-best MTT 0.582) at comparable time cost (~24.5 min for all methods). Ablation (Table 4, Base=MTT-only, +A=augmentation, +T=dual-domain training, +M=dual-objective matching): each module adds accuracy monotonically; the Dual Domain Training module (+T) gives the largest jump on frequency-friendly datasets (Insect spc=1: 18.58%→44.64%, FD spc=1: 51.41%→69.62%), directly isolating the frequency domain as the dominant source of CondTSC's advantage over MTT. Initialization ablation (Table 5): K-means initialization outperforms random, but CondTSC still substantially improves every initialization scheme tested (11-26% relative gains), showing the method is not solely reliant on its K-means initialization choice.

Limitations

Evaluation network is limited to a single CNN+BN architecture in the main results (cross-architecture results are relegated to an appendix); only classification is addressed, not forecasting (the complementary time-series problem condtsf-2024 targets); the frequency-domain augmentations (LPF, phase/ magnitude perturbation) are hand-designed and dataset-agnostic hyperparameters (e.g. 50% low-pass cutoff) rather than learned or tuned per dataset; cost is reported only as aggregate wall-clock in the NAS experiment (~24.5 min, comparable to baselines) without a breakdown of the added frequency-domain computation's overhead.