When Can A Trajectory Be Aligned? An Information Bound for Amplitude-Phase Separation

Elastic functional data analysis separates trajectories into amplitude and phase by warping each one onto a common template and taking the warps as the phase, and recent work makes the warp a neural network that integrates a positive velocity field. Developed for densely sampled curves, it is now applied to trajectories with very few samples: contrast-enhanced MRI with three post-contrast frames, or ICU chemistry panels drawn three times in two days. We compute the Fisher information a set of measurement times carries about a warp parameter. Under the within-exam baseline normalisation these pipelines apply, it depends on the centred derivative of the response and the tangent direction of the warp family, neither of which involves the estimator, so the bound follows from acquisition metadata alone and holds for a network of arbitrary capacity. It implies that a warp pinning both ends of the time axis, the construction these differentiable pipelines are built from, cannot represent a rigid arrival delay at all; that a response with constant derivative carries no timing information once its onset precedes the window, at any sample size or SNR; and that fitting an unknown amplitude costs $3.7\times$ on a near-monotone response. Across six warp families and three encoders, in simulation and on real acquisitions, no configuration exceeds the bound, and the endpoint-pinned family recovers $0.595$ of the arrival variation against $1.03$ for free-endpoint ones: impaired on real scans rather than disabled as it is in simulation. On $3,303$ prostate DCE-MRI examinations across two protocols, thinning a dense acquisition degrades a measured arrival estimate from $0.25$ s at forty samples to $20.1$ s at three, and the bound tracks that degradation across a $119$-fold range ($r=0.96$). Sample count alone orders those schedules equally well; holding it at six and moving only where the six fall, the bound still orders the measured error. Applied to PhysioNet 2012 timestamps alone, the same calculation spans $10.6\times$ across $25$ ICU variables.