Supermixing Latent Recurrence
Implementation & benchmark of arXiv:2608.13876 — Mixing for Free Semigroup Actions of Blaschke Products on the Circle
Source paper: Mixing for Free Semigroup Actions of Blaschke Products on the Circle arXiv:2608.13876 ⓘ · analyzed Sep 1, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Replace or augment the transition map of a recurrent state-space model with bounded analytic maps of a latent complex coordinate, using several finite Blaschke generators that share a fixed point. Enforcing a superattracting fixed point of local degree p creates a tunable hierarchy of memory erasure: the theory predicts double-exponential decorrelation with exponent log p, while a merely attracting fixed point gives ordinary exponential decay.
Formulas
Mathematical statement
Let B_j be a finite Blaschke product mapping the unit disk D to itself, with common fixed point a in D. In the local coordinate w=z-a, assume B_j(a)=a and B_j(a+w)=c_j w^{p_j}+O(w^{p_j+1}), where c_j is nonzero and p_j is the first nonzero local power. Define p=min_j p_j. For a composition W_t=B_{i_t} composed with ... composed with B_{i_1}, the local degree is at least p^t. The paper's mechanism predicts analytic-observable correlations bounded by C exp(-c p^t), where C and c are positive constants. Consequently, the asymptotic slope of -log(-log(|Corr_t|)) versus t is log p. If instead B_j'(a)=lambda_j with nonzero multiplier of magnitude less than one, local contraction is linear and the expected decay is exponential in t rather than double exponential.
Implementation notes
1. Exact integration point: implement this as the transition of a small recurrent state-space model. Represent each complex latent coordinate z by two real channels, constrain it to the unit disk with z <- z / max(1, |z|), and use K trainable Blaschke generators B_j. Parameterize a shared fixed point a and impose B_j(a)=a by constructing each map in local coordinates around a. Initialize a repeated local factor so the first nonzero local degree is p, with experiments for p=2 and p=3. Add a residual gate h_next=(1-g)h+g decode(B_j(z)) so the mixer can retain task-relevant information. 2. Pseudocode: sample j from categorical probabilities; compute z_next=B_j(z); compute the task output; backpropagate task loss plus an optional weak correlation-shaping penalty. Estimate Corr_t between hidden states or analytic probe functions separated by t steps. 3. Compute B_j and its Jacobian exactly; estimate p by fitting log|B_j(a+epsilon u)-a| against log|epsilon| for small epsilon. Estimate c and C empirically rather than treating them as known. 4. First experiment: use a one-layer latent SSM on sequential MNIST or permuted sequential MNIST. Compare tanh recurrence, contractive linear recurrence, and Blaschke recurrences with p=2 and p=3. Record accuracy, gradient norms, hidden-state correlations, and rollout stability for t from 1 to 20. The quantitative prediction is that plotting -log(-log(|Corr_t|)) against t yields an approximately linear curve with slope log(2)=0.693 for p=2 and log(3)=1.099 for p=3, within 20 percent. A nonzero fitted slope that is independent of p, or failure of the double-log plot to become linear, falsifies transfer of the claimed mechanism.
Verification
This idea has not been verified yet.
Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.
Artifacts
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