Identifiable Decomposition of Submovements in Human Hand Trajectories

When is a submovement decomposition of human kinematics mathematically identifiable, and can a scalable algorithm use that criterion to recover the true primitive parameters in long, overlapping movements?

Existing submovement decomposition methods either scale poorly or impose unmeasured assumptions, and none provides a principled way to tell whether a recovered decomposition is informative or merely a spurious fit. Temporal overlap makes basis functions nearly collinear, so the inverse problem loses uniqueness and false decompositions can match the data with very low residual error. Velocity-only methods also accumulate integration drift in position over long horizons.

Prior work falls into three families: exact global search methods such as branch-and-bound and Scattershot, which become intractable beyond roughly ten primitives; search-space reduction heuristics like Gowda and fixed primitive libraries, which scale but introduce unmeasured bias; and learned methods such as SSSUMO, which are trained on labels from earlier algorithms and inherit their biases while being fixed to a sampling rate and spatial dimensionality. None of these provides an identifiability diagnostic, and they tend to drift or over-segment on long-horizon data.

Sub-ID combines three stages: heuristic peak detection on tangential speed to initialize primitives, greedy residual refinement to add basis functions until about 2% velocity error, and adaptive-ridge optimization in position space that penalizes high spatiotemporal kernel correlation. The correlation criterion combines temporal overlap with the cosine of spatial direction and is derived from the Cramér–Rao bound, so it quantifies when primitive amplitudes are distinguishable. Temporal extraction uses coordinate-invariant speed, while spatial amplitudes are recovered with regularized regression in position space to avoid integration drift.

On 100 synthetic trials per condition with K=3, 8, and 15 unimodal primitives, Sub-ID maintains velocity RMSE of at most 2% and position RMSE of 0.012 m at K=15. Gowda position error starts at 0.015 m for K=3 and rises to about 0.06 m at K=15; SSSUMO reaches 0.120 m position error at K=15; Scattershot converges at K=3, degrades at K=8, and fails to converge at K=15. For identifiability, Sub-ID keeps critical overlaps above 0.8 to 3.00% and maximum pairwise correlation to 0.45 at K=15, versus Gowda with 12.30% critical overlaps and ρ≈0.68, and SSSUMO with 51.83% critical overlaps and ρ≈0.96. On bimodal synthetic data with K=15, Sub-ID recovers the two modes but sometimes estimates 14–16 primitives instead of 15. In an extreme-overlap synthetic regime the method reports the ambiguity rather than returning a misleading decomposition. Real-data experiments include UCI Character Trajectories, 3D Object Moving, and custom 2D and 3D PushT tasks with 50 trials per user; detailed quantitative real-data metrics are not reported in the paper excerpt.

The authors state that the identifiability diagnostic depends on recovered direction vectors, so a spurious solution with cancelling bases could in principle report low correlation and provide false assurance; the metric should be treated as necessary but not sufficient without further validation. SSSUMO was evaluated only as released, so retraining on Sub-ID labels was not tested. The final position-domain fit uses Cartesian coordinates, which may not match human motor control, and the least-squares noise model assumes white position noise rather than Brownian motor noise, so the objective is not maximum-likelihood under that model. The paper is a preprint, the real-data portion is not quantified in the provided section, and the evaluation is offline; no robotic hardware experiments are reported.