Structured Trajectory Supervision for Reliable Physiological Time-Series Regression

Many regression targets are summary statistics of an unobserved structured trajectory: the scalar label is a reduction of a higher-resolution signal the model never sees. We ask whether estimating the trajectory as the model's representation—and reducing it to the scalar afterwards—is a beneficial form of dense structured supervision, and what can be learned about a model's behaviour from that representation. Using wrist photoplethysmography (PPG) heart-rate (HR) estimation as the testbed, our estimator, PPI-Net, predicts a 128-point pulse-to-pulse-interval trajectory with per-point uncertainty and derives HR by inverse-variance-weighted aggregation. On an identical backbone, replacing a scalar head by the trajectory head lowers participant-level MAE from 5.11 to 4.04 BPM under leave-one-subject-out evaluation (paired $-1.07$ BPM, better on 13/15 subjects, $P=.010$)—three times a re-run noise floor of $\pm$0.34 BPM that we measure and treat as the threshold of interpretability. The representation also supports mechanistic analysis that a scalar cannot: the signed error decomposes into shrinkage toward the cohort mean under subject-level shift (unbiased point $\approx$92 BPM, slope $-0.18$), and the delta-method uncertainty inherits a ${\hat{h}}^2$ factor that inverts its ranking in the high-HR stratum, which we derive and then verify. Under one corrected protocol, PPI-Net outperforms a standard 1D-CNN and a classical adaptive filter and is on par with the strongest estimator we could retrain from public code, BeliefPPG (3.64 vs. 3.83$\pm$0.20; $P=.17$), with a complementary error profile; the shared failure reproduces in three held-out cohorts, and a calibration–test-separated confidence score turns it into selective regression (MAE $4.04\to 1.43$ at 50% coverage). We state explicitly what the evidence does not support, including a sensor ablation that reverses our own earlier conclusion.