The RG-Flow Transformer: Encoding Scale-Free Dynamics in Scarce EEG

Abstract

Brain field potentials are scale-free: their power spectra follow a $1/f^2$ law whose aperiodic exponent ${\hat{I}}^2$ tracks cortical state, and sleep depth in particular is a shift in ${\hat{I}}^2$. We ask whether a transformer endowed with an explicit renormalization-group (RG) inductive bias the RG-Flow Transformer, which couples ordinary self-attention to a scale-aware stream with a learnable anomalous dimension ${\hat{I}}^3$, block-spin coarse-graining, and an entropy-gated synchronization bridge has an advantage over a parameter-matched vanilla transformer on \emph{real, scarce} EEG. Using the PhysioNet Sleep-EDF corpus with a strict leakage-free by-subject hold-out, we (i) benchmark RG-Flow against a param-matched vanilla transformer and a hierarchy-only ablation on 5-class AASM sleep staging, (ii) sweep the per-subject data budget to look for the inductive-bias crossover predicted when data are scarce, and (iii) test whether RG-Flow's learned ${\hat{I}}^3$ tracks the measured spectral exponent ${\hat{I}}^2$ out-of-sample a quantity the vanilla model does not possess. Across $5$ subjects and $5$ seeds under leave-one-subject-out cross-validation, RG-Flow and the vanilla transformer are statistically indistinguishable on 5-class staging (77.3\% vs 77.0\% accuracy; paired $p=0.294$), and the predicted scarce-data crossover does not appear: vanilla is numerically ahead at every data-limited budget. What does separate the models is interpretability RG-Flow recovers the continuous spectral exponent out-of-sample (${\hat{I}}^2$-recovery $R^2=0.416$), a capability the vanilla architecture has no analogue for.

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