PROJECTION ORDER MATTERS: LEARNING FROM UNREACHABLE TARGETS

Robot faces learning from human mouth motion, prostheses from able-bodied motion capture and impaired musculoskeletal models from a healthy expert learn from targets they cannot reach. Some existing methods filter such demonstrations or retarget each onto the feasible set before training. When every target is unreachable, nothing survives strict feasibility filtering, and it is unclear whether to project targets before or after learning. This order matters through two competing effects; we make three contributions. (1) An exact finite-rank theorem: when the reachable set is a linear subspace, projecting first keeps unreachable directions from consuming a rank-limited linear student's capacity. (2) On curved reachable sets, projection and conditional averaging need not commute, so projecting noisy targets can bias the learning target; constrained-Bayes distillation (CBD) estimates the conditional mean with an offline teacher, projects it and distils the feasible target into a small deployment model. In controlled sweeps, CBD improves on project-then-fit at every tested curvature, while raw fitting is the best of the three orders in most ample-capacity cells. (3) The inverse that projects a target also returns commands, useful supervision even where the projected outputs are worse labels. On a saturating trajectory plant, pretraining a feasible-output student on them turns the worst learner into the best at smaller widths, a gain compute-matched alternatives do not reproduce; its advantage over them replicates on a torque-driven two-link arm in a predeclared design. Projection order is thus a design choice, not a convention.