✓✓ Beats tuned baseline 2026

Continued-Fraction Lacunary Features

Implementation & benchmark of arXiv:2608.25821 — Regularity, quantitative deviation, and non-rigidity of a lacunary skew product

Usefulness6/10

Difficulty3/10

Novelty5/10

Architecture Generalization Sample efficiency Scalability Embedding Mlp Transformer Harmonic analysis Number theory Approximation theory

Source paper: Regularity, quantitative deviation, and non-rigidity of a lacunary skew product arXiv:2608.25821 ⓘ · analyzed Aug 29, 2026

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 Not peer-reviewed.

Idea description

Replace random Fourier features or a dense sinusoidal positional encoding with a compact bank whose frequencies are the continued-fraction denominators of an irrational number. Inverse-frequency amplitudes provide multiscale structure with a controlled sub-Lipschitz regularity profile, while lacunarity reduces the number of frequencies needed to represent oscillatory structure.

Formulas

$$\alpha=[a_0;a_1,a_2,\ldots],\qquad q_{j+1}=a_{j+1}q_j+q_{j-1}$$

$$h(x)=\sum_{j=1}^{\infty}\frac{\cos(2\pi q_jx)}{q_j}$$

$$h_J(x)=\sum_{j=1}^{J}\frac{\cos(2\pi q_jx)}{q_j},\qquad \phi_J(x)=\left[\frac{\cos(2\pi q_jx)}{\sqrt{q_j}},\frac{\sin(2\pi q_jx)}{\sqrt{q_j}}\right]_{j=1}^{J}$$

$$\forall\,0<\beta<1:\quad |h(x)-h(y)|\le C_\beta |x-y|^\beta,\qquad h\ \text{is not Lipschitz}$$

Mathematical statement

For an irrational rotation number alpha with continued-fraction convergents p_j/q_j, the denominators obey q_{j+1}=a_{j+1}q_j+q_{j-1}. The paper constructs h(x)=sum_{j>=1} cos(2 pi q_j x)/q_j. Continued-fraction denominators grow at least exponentially, so the frequencies are lacunary and the coefficient series is absolutely summable. The paper states that h is Holder continuous for every exponent beta with 0<beta<1, while a Fourier argument shows that h is not Lipschitz. For neural networks, truncate the series at J terms and use both sine and cosine channels. The square-root scaling in phi_J keeps each frequency pair's squared feature energy proportional to 1/q_j; direct 1/q_j scaling reproduces the scalar series exactly.

Implementation notes

Integrate this at the input embedding of a coordinate MLP, neural field, transformer position encoder, or world-model state encoder. The simplest MVP is coordinate regression: concatenate the normalized coordinate x with phi_J(x), then pass the result through the first linear layer. Choose an irrational alpha, preferably the golden-ratio conjugate (sqrt(5)-1)/2, compute its denominators using q_minus1=0, q_0=1, and q_{j+1}=a_{j+1}q_j+q_{j-1}, and retain terms with q_j less than or equal to a chosen frequency cutoff. Pseudocode is: initialize q_prev=0 and q_curr=1; for j in range(J), append q_curr, then set q_next=a[j+1]*q_curr+q_prev and shift the two variables; for each q_j append cos(2*pi*q_j*x)/sqrt(q_j) and sin(2*pi*q_j*x)/sqrt(q_j). Concatenate these channels with x and train normally. The denominator recursion and inverse-frequency weighting come from the paper; the cutoff, channel normalization, and number of coordinates are empirical choices. Estimate effective regularity by fitting the slope of log absolute feature differences versus log coordinate separation over several scales. First test on a 1D or 2D neural-field task such as fitting a band-limited image or a smooth-plus-sawtooth signal. Compare raw coordinates, standard NeRF Fourier features, random Fourier features, and this encoding at equal input dimension and FLOPs. Success is lower test error or better extrapolation from a training subinterval without increased training cost.

Verification

Beats tuned baseline

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Confirmed ✓

Stage 2 · Practical benchmark result: Significant win ✓

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 8/10

Built continued-fraction lacunary features using golden-ratio denominators and inverse-square-root scaling, plus a reproducible coordinate-regression benchmark. The recursion matched Fibonacci denominators, the finite-series modulus had a log-log slope of 0.841 consistent with sub-Lipschitz behavior, and the finite derivative bound grew linearly with truncation size. On extrapolation to an unseen coordinate interval, lacunary features achieved RMSE 0.0734 versus 0.9800 for NeRF features and 2.3991 for raw coordinates, showing a clear promising effect in this toy setting.

Agent confidence: 8/10

Baseline: Raw coordinates: train RMSE 0.0666, extrapolation RMSE 2.3991; NeRF Fourier features: train RMSE 0.00382, extrapolation RMSE 0.9800.
Idea: Continued-fraction lacunary features: train RMSE 0.01399, extrapolation RMSE 0.07340; input dimension 17 and 5,377 parameters versus NeRF's 16 dimensions and 5,313 parameters.

Limitations: Only a 1D synthetic continued-fraction-series target and one train/test interval split were tested. The benchmark did not include random Fourier features, multiple seeds, 2D/image fields, wall-clock comparisons, or a strictly parameter-matched architecture; the regularity estimate uses a finite truncation and finite scale range.

How to run: python3 experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.25821) Sep 3, 2026

Beats the tuned baseline by 98.2% (p=0.0081, wins 8 / 8 paired seeds; metric: lower is better).

Benchmark: Oscillatory_poisson_dirichlet
Model: mlp_tiny
Paired seeds: 8
Baseline mean: 0.0004
Idea mean: 0
Effect (Δ): -0.0004 (−98.2%; negative = idea better)
Wins: 8 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: n/a
Mechanism: Confirmed ✓
Practical effect: Helps
Baseline tuning: swept over 3 configs

Limitations:

Only the registered oscillatory PDE track was tested; CIFAR vision, sequence, dynamics, wall-clock cost, and broader cross-domain transfer were not evaluated.

How to run: python3 bench_run.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Sep 3, 2026 latest Worked ✓

Agent confidence: 8/10

Implemented continued-fraction lacunary input features and evaluated them against a raw-coordinate MLP on the registered oscillatory elliptic PDE track with identical architecture, budget, and 8 paired seeds. The idea won all 8 seeds, improving mean test MSE from 0.0004323764 to 0.0000079388; the paired permutation test gave p=0.0081, so the effect is significant on this structurally matched track.

Baseline: Tuned raw-coordinate MLP, lr=0.003, epochs=60: mean MSE 0.0004323764.
Idea: Lacunary-feature MLP, lr=0.003, epochs=60: mean MSE 0.0000079388; 8/8 wins.

Benchmark result