# Spectral-gap adaptive polynomial filtering

- ID: 1370
- Canonical URL: https://synthcore.org/idea/1370/spectral-gap-adaptive-polynomial-filtering
- API JSON: https://synthcore.org/api/idea/1370.json
- API Markdown: https://synthcore.org/api/idea/1370.md
- Verification status: failed_benchmark
- Source: [arXiv:2607.24643](https://arxiv.org/abs/2607.24643)
- Category: dynamics
- Solves: speedup, stability, accuracy
- ML areas: optimizer, training-dynamics, ssm, diffusion-sampling
- Math tags: spectral-theory, polynomial-approximation, dynamical-systems, numerical-analysis
- Ratings: usefulness 7/10; difficulty 7/10; novelty 7/10

## Idea description

Use the paper's sharp sK approximately equal to 1 phase transition to choose between conservative Fejer averaging and higher-order polynomial filtering. When the local fixed-point spectrum is separated from eigenvalue 1, use a Jackson-type filter; near the critical regime, use the safe Fejer filter instead of unrestricted Anderson extrapolation.

## Mathematical statement

Let L be the local linearized resolvent or fixed-point operator, sigma(L) its spectrum, and s=dist(1,sigma(L)) the spectral distance from the slow mode at 1. The paper states that a Jackson-kernel polynomial achieves residual O(d_0/(K^2 s)) when sK tends to infinity. At the critical scale s_crit=sin(pi/(2K+2)), which is asymptotically proportional to 1/K, the extremal instance still has exact minimax residual d_0/(K+1). Thus high-order acceleration is justified only when the estimated gap satisfies sK substantially greater than one. The safe baseline is the Fejer estimator y_hat_K=(K+1)^(-1) sum from j=0 to K of (2J-I)^j y_0.

## Key formulas

- $$s=\operatorname{dist}(1,\sigma(L)).$$
- $$\|r(y_K)\|=O\!\left(\frac{d_0}{K^2s}\right)\quad\text{when }sK\to\infty.$$
- $$s_{\mathrm{crit}}=\sin\!\left(\frac{\pi}{2K+2}\right)\asymp\frac{1}{K},\qquad \|r(y_K)\|_{\mathrm{minimax}}=\frac{d_0}{K+1}.$$
- $$\widehat y_K^{\mathrm{safe}}=\frac{1}{K+1}\sum_{j=0}^{K}(2J-I)^j y_0.$$

## Implementation notes

Apply this to the solver inside a DEQ, recurrent state-space block, proximal transformer layer, or iterative diffusion sampler. At state y, define F(y)=J(y) and estimate the local Jacobian L=DF(y) with matrix-free automatic differentiation. Draw m=8 random Rademacher vectors v_i and compute L v_i using Jacobian-vector products. For a cheap gap proxy, estimate the largest eigenvalue of L near 1 by three to five power iterations, or form the small Krylov matrix from vectors v, Lv, L^2v and estimate the smallest singular value of I-L on that subspace. Set the conservative gap estimate s_hat to the clipped value in [0,1]. Select K from {3,7,15}. If s_hat*K<2, use the safe Fejer block: z_0=y, z_(j+1)=2J(z_j)-z_j, and y_out=(K+1)^(-1) sum_j z_j. If s_hat*K>=2, apply a precomputed Jackson polynomial p(t)=sum from j=0 to K of c_j t^j with p(1)=1 to the same iterates; validate the coefficients first on diagonal matrices with known spectra. Reject any filtered output whose normalized residual increases by more than 10 percent, and fall back to Fejer averaging. The first cheap experiment should use a 1M-parameter DEQ MLP on CIFAR-10, with plain iteration and always-on Anderson acceleration as baselines. Measure residual reduction per J evaluation, solver failures, wall-clock cost, and task accuracy separately for gapped and near-critical synthetic Jacobian spectra. The target is at least 1.5x fewer evaluations in the gapped regime with no worse stability near s_hat*K approximately 1.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: inconclusive
- Verdict: Built a readable toy MVP for Fejer and Jackson-type polynomial filtering on exact rotation spectra, including coefficient validation, explicit iterate equivalence, adaptive selection, and a 10% residual safeguard. The mechanism manifested: the critical residual matched 1/(K+1) to approximately 1e-6 or better, while fixed sK=8 produced nearly constant K^2*s*residual around 3.4, consistent with O(1/(K^2 s)). The nominal sK=2 switch was too aggressive on this spectrum, but the safeguard rejected the approximately 1.45x degradation; Jackson became beneficial around sK=3.5 and was substantially better by sK=8.

### Mechanism check

- Verdict: Built a readable toy MVP for Fejer and Jackson-type polynomial filtering on exact rotation spectra, including coefficient validation, explicit iterate equivalence, adaptive selection, and a 10% residual safeguard. The mechanism manifested: the critical residual matched 1/(K+1) to approximately 1e-6 or better, while fixed sK=8 produced nearly constant K^2*s*residual around 3.4, consistent with O(1/(K^2 s)). The nominal sK=2 switch was too aggressive on this spectrum, but the safeguard rejected the approximately 1.45x degradation; Jackson became beneficial around sK=3.5 and was substantially better by sK=8.
- Confidence: 8/10
- Limitations: Only synthetic diagonal rotation spectra were tested; no matrix-free Jacobian estimation, DEQ/recurrent model, CIFAR-10 task accuracy, wall-clock evaluation budget, Anderson baseline, noisy gap estimation, or nonlinear operator behavior was tested. The Jackson construction is a simple squared-Fejer polynomial approximation rather than a full paper-derived coefficient implementation.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.05675
- Paired wins: 1/8
- Benchmark verdict: no significant win

## Artifacts

- [REPORT.md](https://synthcore.org/code/535/REPORT.md)
- [analyze.py](https://synthcore.org/code/535/analyze.py)
- [bench_report.json](https://synthcore.org/code/535/bench_report.json)
- [report.json](https://synthcore.org/code/535/report.json)
- [report.md](https://synthcore.org/code/535/report.md)
- [report_bench_2026-08-31T170820.md](https://synthcore.org/code/535/report_bench_2026-08-31T170820.md)
- [results.json](https://synthcore.org/code/535/results.json)
- [spectral_gap_filter.py](https://synthcore.org/code/535/spectral_gap_filter.py)
- [stage2_bench.py](https://synthcore.org/code/535/stage2_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/535)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.