Reduction-Robust Pole Regularization
Implementation & benchmark of arXiv:2609.01329 — Pole-Zero Geometry, Model Reduction, and Identifiability in Sensory Adaptation
Source paper: Pole-Zero Geometry, Model Reduction, and Identifiability in Sensory Adaptation arXiv:2609.01329 ⓘ · analyzed Sep 2, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Train a latent state-space neural network so that its effective pole geometry remains consistent when identified by low-frequency moments and finite-window trajectories. Penalize disagreement between the two reductions, and penalize proximity to the oscillatory/non-oscillatory boundary, to reduce spurious ringing after distillation or context truncation.
Formulas
Mathematical statement
For a two-state reduced drift with characteristic polynomial p(s) = s^2 + a1 s + a0, the pole discriminant is Delta = a1^2 - 4 a0. Delta less than zero gives a complex-conjugate pair, while Delta greater than or equal to zero gives real poles. The paper reports rho_moment = 4.50 and rho_window = 3.31 for the same higher-order real-relaxation system, with a pole-class boundary at rho = 4. For a neural transition z_(t+1) = F_theta(z_t,u_t), estimate a local Jacobian J_t = partial F_theta / partial z_t, obtain local poles from s_i = log(lambda_i(J_t))/dt, and fit the reduced pole coordinate rho using both reduction protocols.
Implementation notes
Integrate this into a small linear or locally linear state-space network, gated RNN, or latent world model. After each validation rollout, collect the latent trajectory and fit the same two-state reduced model in two ways. First, match low-frequency response moments, such as DC gain and the first derivative of the transfer function at zero frequency. Second, fit the reduced model by least squares over a finite trajectory window of length W. Extract both pole coordinates and add their absolute disagreement to the task loss. Pseudocode is: traj = rollout(model, inputs); model_m = fit_moment_reduction(traj); model_w = fit_window_reduction(traj,W); rho_m = pole_coordinate(model_m); rho_w = pole_coordinate(model_w); loss = task_loss + lam*abs(rho_m-rho_w). The reduction procedures and boundary rho = 4 come from the paper; trajectories, Jacobians, fit uncertainty, and local poles are estimated empirically. The first cheap experiment should use a diagonal-plus-low-rank SSM on Copy-Memory and damped-sine sequence prediction, comparing ordinary training with this penalty. The falsifiable prediction is a sharp increase in ringing and long-horizon error when either fitted rho crosses 4. The regularizer should reduce absolute protocol disagreement by at least 50 percent, while the measured oscillation transition should occur within 20 percent of rho = 4.
Verification
Stage 1 · Toy mechanism gate: Failed ✗
Stage 2 · Mechanism transferred to benchmark: Not tested
Stage 2 · Practical benchmark result: Not run
Methodology: Toy-system gate first; the benchmark stage runs only after a pass. How verification works
Stage 1 — Mechanism check agent confidence 8/10
Built a reproducible toy reduction benchmark implementing low-frequency moment rho, finite-window two-state fitting, the discriminant check, and the proposed disagreement/boundary penalty. The discriminant behaved exactly as claimed: rho<4 produced Delta<0 and complex poles, while rho>=4 produced real poles. However, the sampled stable three-pole system did not reproduce the paper's opposite-side reduction (rho_moment=4.468, rho_window=4.008), and regularization reduced disagreement only from 0.460 to 0.418 (~9%) while adding task error; therefore the proposed effect was not observed.
- Agent confidence
- 8/10
- Baseline
- task MSE 0.000000; rho_moment 4.4680; rho_window 4.0080; disagreement 0.4600
- Idea
- task MSE 0.0000212; rho_moment 4.4256; rho_window 4.0080; disagreement 0.4176 (~9.2% lower)
Limitations: This is a small deterministic transfer-function toy, not a neural SSM, gated RNN, Copy-Memory task, or damped-sine prediction experiment. The exact higher-order cascade-parallel parameters and original bacterial sampling schedule were not available in the paper excerpt, so the paper's rho=4.50 versus 3.31 pair was not exactly reproduced. The exploratory search was limited by nonlinear-fit runtime and encountered numerical overflow for some unstable optimizer candidates.
How to run: python3 experiment.py
Verdict computed by deterministic test code from paired-seed statistics — not by the language model.