PRISM: Polynomial Representations for Interaction-Structured Motor Control

01 / Method

PRISM learns polynomial interactions.

PRISM is learned end-to-end, uses no added sensors, and remains backbone-compatible with existing policies.

PRISM integration in reinforcement learning and imitation learning. A learned latent polynomial interaction transforms proprioception before the existing policy network. — **One representation across RL and imitation.** PRISM transforms proprioception while preserving the policy and action interface.

02 / Results

Polynomial interactions improve control.

PRISM outperforms standard and larger controls across controlled and stronger policy backbones.

Reinforcement learning

Humanoid-Gym

PRISM nearly doubles survival; the wider MLP stays at baseline.

MLP51.0

Larger MLP52.25

PRISM92.5

Linear error 0.2099 Episode length 2233.4

Imitation learning

Task-specific LIBERO

PRISM reaches 91% success without force as a policy input.

Diffusion63.8

MCC-Sensorless47.8

MCC-Oracle64.5

PRISM91.0

Same RGB + state input MCC-Oracle alone uses force

BFM-Zero / humanoid RL

Tracking EMD ↓

Polynomial interactions reduce tracking error under nominal and shifted dynamics.

Nominal

1.104 · 1.090 · 1.050

Low friction

1.582 · 1.589 · 1.548

Payload

1.121 · 1.114 · 1.073

Shorter bars indicate lower error

SmolVLA / multi-task VLA

LIBERO success ↑

PRISM improves average success, with its largest gain on long-horizon tasks.

Average
63.50 · 64.90 · 66.55

Spatial

69.8 · 64.6 · 70.0

Object

57.2 · 65.6 · 57.4

Goal

81.2 · 85.0 · 85.4

Long

45.8 · 44.4 · 53.4

Official multi-task eval50 at 80K

03 / BFM-Zero robustness video

Low-friction motion tracking.

Video order: GT|BFM-Zero|Larger|PRISM

Low friction Low-friction tracking

Motion 30 across a longer rollout, selected by aligned per-motion EMD.

The selected rollout illustrates the condition-level result; EMD above averages all 40 motions.

SmolVLA / manipulation videos

Contact-rich manipulation rollouts.

LIBERO Long Object transfer

LIBERO Long Cup interaction

LIBERO Goal Goal-conditioned placement

Matched control cases: SmolVLA and Larger fail while PRISM succeeds from the same evaluation resets.

Post-hoc contact diagnostic Motion changes through contact

Force is logged only after rollout.

Selected rolloutContact force norm

SmolVLA Larger PRISM

Loading trace...

The shaded interval marks initial contact; the cursor follows video time.

04 / Representation analysis

Learned features reveal physical structure.

Probes and ablations show what the learned products encode and use.

Locomotion−14.0%

joint-power probe MSE

Locomotion+9.6%

slip-velocity probe PCC

Manipulation−19.9%

contact-impulse probe MSE

Manipulation−11.9%

contact-work probe MSE

Emergent factors

Velocity products drive locomotion actions.

Velocity-memory coupling0.057

Cross-joint velocity0.025

Delayed velocity memory0.009

State-velocity coupling0.008

Post-hoc names, not predefined variables.

Qualitative feature view

Dynamics shifts separate in learned state space.

Nominal Low friction Payload mass

Loading representation...

Qualitative feature view.

Citation

BibTeX

@article{lee2026prism,
  title   = {PRISM: Polynomial Representations for Interaction-Structured Motor Control},
  author  = {Lee, Seung Hyun and Yu, Stella X.},
  journal = {arXiv preprint arXiv:2607.23473},
  year    = {2026},
  doi     = {10.48550/arXiv.2607.23473}
}