Is spectral predictability enough? Testing {\Omega}-matching for synthetic pretraining under distribution shift
The paper tests whether matching spectral predictability $\Omega$ between synthetic pretraining data and target time series is sufficient for effective transfer learning under distribution shift. They evaluate this "predictability-aware" hypothesis using LSTM, PatchTST, and TCN across controlled synthetic conditions and the real-world ETTm1 benchmark. The core finding is a negative result: while $\Omega$-matching works under ideal conditions, it fails across different synthetic source families on real data, suggesting practitioners need more nuanced source-selection criteria.
The work presents a carefully scoped negative result that is persuasive for the tested setting but limited in external validity. The authors structure the paper as a hypothesis test, first verifying that $\Omega$-matching succeeds in a fully controlled synthetic-to-synthetic regime before showing it breaks down when transferring to real data. The central empirical observation—that two $\Omega$-matched synthetic sources transfer to ETTm1 in near-uncorrelated ways ($r=-0.02$)—is striking, but the evidence base is narrow: one benchmark, two source families, and three random seeds.
The synthetic-to-synthetic verification is the strongest element of the paper, cleanly establishing that $\Omega$-matching can in principle confer large benefits. Under the ideal down-gap condition ($\omega_{pre}=0.7 \to \omega_{tgt}=0.3$), the predictability-aware strategy exceeds even the oracle for LSTM (+46.8% MSE reduction). The cross-source comparison on real data is also well designed: by holding $\Omega$ fixed and swapping sine-plus-noise for chaotic ODE trajectories, the authors isolate whether $\Omega$ alone determines transfer.
The most significant limitation is restricted coverage: ETTm1 is a single domain (grid load electricity data) and the two synthetic sources are structurally very different, with the dysts ODE family seemingly ill-suited a priori to grid forecasting. The authors concede that the candidate factors they discuss (smoothness, correlation structure, phase continuity) are motivated by the dysts failure rather than isolated in a controlled ablation. Additionally, the random_synthetic baseline draws $\Omega$ uniformly from $[0,1]$, which may effectively average to $\Omega \approx 0.5$, close to many target values; a more adversarial random baseline could strengthen the claim that matching matters. Only three seeds provide a coarse variance estimate, and the $\omega=0.9$ regime was not fully achieved ($\Omega \approx 0.80$–$0.89$), introducing a confound in that condition.
The evidence strongly supports the central negative claim for the tested configurations: sine-plus-noise pretraining at matched $\Omega$ yields only a marginal mean improvement (+2.3% averaged over all three architectures), whereas $\Omega$-matched chaotic ODE trajectories systematically degrade performance ($-13.2\%$). The comparison with related work is appropriately modest—the paper frames itself as testing a hypothesis rather than proposing a new method, and it correctly cites foundational work on transfer learning and recent time-series foundation models. However, the paper does not compare against existing domain-adaptation or source-selection baselines, so readers cannot gauge whether any alternative criterion currently performs better than $\Omega$-matching.
The experimental protocol is described in sufficient detail to allow independent replication in principle, with explicit model architectures, TCN hyperparameters (kernel_size=2, dilations=[1,2,4,8,16]), dataset sizes $T \in \{2000, 10000\}$, and a two-stage pretrain-then-finetune procedure. However, no code repository, data generation scripts, or model weights are released; the paper contains no reproducibility statement or artifact links. With only three seeds (123, 456, 789) and $\omega=0.9$ reflecting achieved rather than target values, run-to-run variance is only coarsely estimated. The dysts system selection procedure—selecting the ODE whose empirical $\Omega$ most closely matches the target—also lacks sufficient detail to guarantee exact reproduction without the selection code.
Synthetic pretraining under distribution shift requires deciding what synthetic data to use. A natural criterion is to match the spectral predictability $\Omega$ of the source to the target, but is $\Omega$ sufficient? We test this hypothesis, which we call predictability-aware pretraining, across three architectures (LSTM, PatchTST, TCN) and three seeds. Under ideal synthetic-to-synthetic conditions, $\Omega$-matched pretraining can outperform even an oracle baseline (+46.8% MSE reduction, LSTM), though the benefit is already architecture-dependent. On the real-world ETTm1 benchmark, however, this relationship does not hold: sine-plus-noise pretraining at matched $\Omega$ yields only a marginal mean improvement +2.3% averaged over all three architectures, up to +65.5% on individual channels), whereas $\Omega$ matched chaotic ODE trajectories systematically degrade performance -13.2%). The two sources agree on the direction of transfer for only 53% of configurations across all three architectures (Pearson r=-0.02), indicating that $\Omega$ carries little information about which source will transfer beneficially. We further observe an architecture-dependent asymmetry in the response to the direction of the $\Omega$ gap. We conclude that $\Omega$ matching can help in controlled settings but is not sufficient to predict transfer across sources, and we discuss additional signal properties that a source-selection criterion may need to consider.
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