Paired translation between quasiperiodic physiological waveforms (i.e., recovering a target oscillatory signal from a source) is central to interpreting cardiovascular signals from wearables placed at different body locations. This mapping carries an inherent geometric structure: phase wraps around a cycle and must be treated as a circular variable; amplitude stays strictly positive; beat‑to‑beat alignment can drift unpredictably across cycles and subjects. Although deep neural networks have been applied to phase estimation and complex‑valued signal modeling, prior work does not explicitly learn phase transport between paired signals. Consequently, neither endpoint‑supervised regression nor the standard affine path used in flow matching captures this phase‑amplitude structure.
We introduce cylindrical geodesic flow matching for paired cardiovascular waveform translation. We show that the affine path commonly used in flow matching distorts intermediate amplitude and instantaneous frequency when interpolating quasiperiodic signals; replacing it with a closed‑form geodesic on the phase‑amplitude cylinder eliminates these artifacts and converts each training pair into dense, geometry‑consistent velocity supervision.
On zero‑shot photoplethysmography (PPG) and limited‑support seismocardiography (SCG) adaptation benchmarks, our method consistently outperforms interpolation baselines and matches or exceeds direct supervised prediction. Compared with the strongest competing baseline, Hilbert Transform, $L_2$, and Dynamic Time Warping ($DTW$) distances are reduced by up to $15\%$. These results suggest that bridge geometry is a critical inductive bias for flow matching on oscillatory signal translation.
Review: This approach leverages a closed‑form geodesic on the phase‑amplitude cylinder to eliminate amplitude and instantaneous‑frequency distortion during interpolation, offering a powerful geometric inductive bias for cross‑site cardiovascular waveform translation.