Koenigs-Linearized Disk RNN
Implementation & benchmark of arXiv:2608.18781 — Characterizations of extremal hyperbolic rates via Herglotz measures and Koenigs linearization
Source paper: Characterizations of extremal hyperbolic rates via Herglotz measures and Koenigs linearization arXiv:2608.18781 ⓘ · analyzed Sep 1, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Constrain a recurrent latent state to the unit disk and learn an auxiliary Koenigs coordinate in which the recurrent transition is a scalar dilation. The nonlinear transition is trained to satisfy the conjugacy equation, so repeated application has a prescribed asymptotic rate instead of accumulating uncontrolled Jacobian errors.
Formulas
Mathematical statement
The paper studies a holomorphic disk self-map \(g:\mathbb{D}\to\mathbb{D}\) with Denjoy–Wolff point \(\tau\in\partial\mathbb{D}\) and multiplier \(\alpha\in(0,1)\). Its extremal-rate theorem gives \(\rho_{\mathbb{D}}(g^{\circ n}(z),w)=n\log(1/\alpha)+O(1)\), where \(\rho_{\mathbb{D}}\) is the Poincare distance and \(O(1)\) is bounded independently of \(n\). The equivalent boundary condition is \(\int_{\partial\mathbb{D}\setminus\{\tau\}}\log(1/|\zeta-\tau|)\,d\sigma(\zeta)<\infty\), where \(\sigma\) is the boundary measure in the Herglotz–Nevanlinna representation associated with \(g\). The second characterization is \(0<|\angle\lim_{z\to\tau}h(z)/\phi_\tau(z)|<\infty\), where \(h\) is a Koenigs function and \(\phi_\tau:\mathbb{D}\to\mathbb{H}\) maps \(\tau\) to infinity. The operational identity behind this characterization is the Koenigs conjugacy \(h(g(z))=h(z)/\alpha\). Adapt this identity as a learned latent-coordinate constraint, with \(\phi_1(z)=(1+z)/(1-z)\) for a fixed boundary target \(\tau=1\).
Implementation notes
(1) Integrate this into a complex-valued RNN or latent world model. Represent each scalar latent coordinate as \(z_t\in\mathbb{D}\), normalize unconstrained complex outputs by \(z\leftarrow (1-\epsilon)z/(1+|z|)\), and use a learned transition \(z_{t+1}=g_\theta(z_t,x_t)\); initially fix the target boundary point to \(\tau=1\) and learn \(\alpha=\sigma(a)\) with a small positive floor. Add a separate complex MLP \(h_\psi\) that maps disk states to complex Koenigs coordinates. (2) For each minibatch sequence, compute \(z_{t+1}=g_\theta(z_t,x_t)\) and the conjugacy residual \(r_t=h_\psi(z_{t+1})-h_\psi(z_t)/\alpha\). Optimize the task loss plus \(\lambda_c\operatorname{mean}_t|r_t|^2\). Sample extra states near the boundary, \(z=(1-\varepsilon)e^{i\theta}\), and add \(\lambda_b[\log(|h_\psi(z)/\phi_1(z)|+\delta)]^2\) after centering the log-ratio over sampled angles; this discourages the Koenigs coordinate from vanishing or exploding relative to the half-plane coordinate. (3) The paper supplies the mathematical target identity, extremal rate, and logarithmic boundary integrability criterion; estimate \(\alpha\) by gradient descent and evaluate the empirical rate from rollouts using the Poincare distance \(\rho_{\mathbb{D}}(u,v)=2\operatorname{artanh}|(u-v)/(1-\bar v u)|\). (4) First test on a synthetic delayed-copy or sinusoidal-memory task with a 2-layer complex RNN, comparing an unconstrained RNN, disk-only RNN, and disk-plus-Koenigs model at equal hidden width. Measure prediction error versus rollout length, the variance of \(\rho(z_{t+n},z_t)/n\), and the fitted slope against the target \(\log(1/\alpha)\). Success means substantially slower error growth and a slope that remains stable when extrapolating beyond training sequence lengths.
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.
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