A Shared Observation Shields Collective Fluctuations while Preserving Local Independence

What the math gives to ML

The paper identifies a transferable mechanism for shared-observation conditioning: independent hidden trajectories acquire weak pairwise correlations concentrated in the collective direction visible to an observation. Girsanov conditioning produces a centered-square penalty in that direction, with pair covariance of order z^-1 but a finite aggregate effect because many weak correlations add coherently. In neural networks, this suggests a low-rank regularizer for branches, experts, or particles that share a prediction or routing signal. The regularizer should suppress observation-visible collective fluctuations while preserving diversity in directions not seen by the shared output.