# Forward-Backward Representations

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arXiv Mar 21, 2026
Temporal abstraction is shown to align the spectrum of forward-backward representations, stabilizing learning at high discount factors in continuous control
The work analyzes temporal abstraction as a mechanism to mitigate the spectral mismatch in forward-backward (FB) representations, showing by characterizing the spectral properties of the transition operator that temporal abstraction acts analogously to a low-pass filter suppressing high-frequency spectral components, thereby reducing the effective rank of the induced successor representation while preserving a formal bound on the value function error, and empirically showing that this alignment is a key factor for stable FB learning at high discount factors.
The work analyzes temporal abstraction as a mechanism to mitigate the spectral mismatch in forward-backward (FB) representations, showing by characterizing the spectral properties of the transition operator that temporal abstraction acts analogously to a low-pass filter suppressing high-frequency spectral components, thereby reducing the effective rank of the induced successor representation while preserving a formal bound on the value function error, and empirically showing that this alignment is a key factor for stable FB learning at high discount factors.
University of Freiburg
Seyed Mahdi B. Azad（First author）
Forward-Backward Representations Temporal Abstraction Successor Representation
arXiv Sep 30, 2026
CrossBFM distills Unitree G1's latent behavior space onto three humanoids in under one GPU-hour, losing only 0.025 rad in tracking
CrossBFM freezes the backward map of a Behavior Foundation Model pretrained on the Unitree G1 and, using the frame-level cross-embodiment correspondence supplied by retargeted data, distills that latent space onto three humanoids (Inhouse M3, Booster T1, Fourier N1) with a unified encoder that has no robot-specific parameters, training the encoder in under one GPU-hour and each tracker in about 10 more GPU-hours; all three prompting modes transfer (motion tracking, goal reaching between poses, reward optimization), the latent-conditioned policy loses only 0.
CrossBFM freezes the backward map of a Behavior Foundation Model pretrained on the Unitree G1 and, using the frame-level cross-embodiment correspondence supplied by retargeted data, distills that latent space onto three humanoids (Inhouse M3, Booster T1, Fourier N1) with a unified encoder that has no robot-specific parameters, training the encoder in under one GPU-hour and each tracker in about 10 more GPU-hours; all three prompting modes transfer (motion tracking, goal reaching between poses, reward optimization), the latent-conditioned policy loses only 0.
Tan-Dzung Do（First author）
CrossBFM Forward-Backward Representations Knowledge Distillation Motion Retargeting Humanoid Whole-Body Control

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Temporal abstraction is shown to align the spectrum of forward-backward representations, stabilizing learning at high discount factors in continuous control

CrossBFM distills Unitree G1's latent behavior space onto three humanoids in under one GPU-hour, losing only 0.025 rad in tracking