✓✓ Beats tuned baseline 2026

Differentiable R-Function Geometry Gate

Implementation & benchmark of arXiv:2607.01449 — Geometry-Aware R-Structured Kolmogorov-Arnold Networks

Usefulness7/10

Difficulty4/10

Novelty6/10

Architecture Accuracy Generalization Stability Mlp Loss Regularization Geometry Geometry Approximation theory

Source paper: Geometry-Aware R-Structured Kolmogorov-Arnold Networks arXiv:2607.01449 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

Attach an analytic geometry gate to a KAN or MLP so that known feasible regions, exclusions, and unions are represented by differentiable implicit functions instead of being learned only from samples. Use R-conjunctions for intersections and R-disjunctions for unions, then convert the signed support score into a soft gate that modulates the prediction.

Formulas

$$f(\mathbf{x})=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^{n}\psi_{q,p}(x_p)\right)$$

$$a\wedge_R b=a+b-\sqrt{a^2+b^2},\qquad a\vee_R b=a+b+\sqrt{a^2+b^2}$$

$$a\wedge_{R,\varepsilon} b=a+b-\sqrt{a^2+b^2+\varepsilon^2},\qquad a\vee_{R,\varepsilon} b=a+b+\sqrt{a^2+b^2+\varepsilon^2}$$

$$m(\mathbf{x})=\sigma(\kappa G(\mathbf{x})),\qquad \hat y(\mathbf{x})=m(\mathbf{x})h_{\mathrm{in}}(\mathbf{x})+[1-m(\mathbf{x})]h_{\mathrm{out}}(\mathbf{x})$$

Mathematical statement

The paper states the Kolmogorov-Arnold representation f(x)=sum over q of Phi_q(sum over p of psi_qp(x_p)), where x_p is input coordinate p, psi_qp and Phi_q are continuous univariate functions, and n is input dimension. The transferable geometric construction uses R-functions for signed logical operations. For signed constraints a(x)>=0 and b(x)>=0, the R-conjunction is a AND_R b = a+b-sqrt(a^2+b^2). It is positive when both inputs are positive, zero on the logical boundary, and nonpositive otherwise. The R-disjunction is a OR_R b = a+b+sqrt(a^2+b^2), which is positive when at least one input is positive. For numerical smoothness, replace the square root by sqrt(a^2+b^2+epsilon^2), where epsilon>0. For a combined support score G(x), use m(x)=sigmoid(kappa G(x)), where kappa controls boundary sharpness.

Implementation notes

Integrate the method at the prediction head of a small KAN or MLP. Define each known primitive region with a signed implicit function whose positive side is feasible. For a disk, use a(x,y)=r^2-(x-c_x)^2-(y-c_y)^2. For a rectangle, use a=x-x_min, b=x_max-x, c=y-y_min, and d=y_max-y, then combine all four with the smoothed R-conjunction. For unions of regions, combine their scores with the smoothed R-disjunction. In pseudocode: G = primitive[0]; for each later primitive p, set G = G + p - sqrt(G*G + p*p + eps*eps) for an intersection, or use the plus sign for a union; then compute m = sigmoid(kappa*G) and yhat = m*h_in(x) + (1-m)*h_out(x). The R-function equations and their sign semantics are fixed analytically. Estimate only scale normalization, kappa, epsilon, and optionally the primitive parameters from data. Initialize kappa moderately and anneal it upward. Run a first experiment on a two-dimensional discontinuous regression task with a disk-union-rectangle support. Compare a width- and parameter-matched MLP, a KAN, and the gated model. Measure global RMSE, RMSE within a narrow band around G=0, and boundary intersection-over-union. Success is lower boundary error and faster loss reduction at equal parameter count; robustness should also be tested by shifting one supplied geometric boundary.

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 7/10

Built a differentiable smoothed R-function gate for a disk-union-rectangle regression task and compared it with a small MLP. The predicted R-conjunction/disjunction zero crossings and sigmoid transition widths were confirmed to within about 1e-4 relative/error scale. The idea produced a small global RMSE gain (0.13086 vs 0.13250), but the claimed boundary-error improvement was absent (0.21157 vs 0.20384), and supplying a shifted boundary substantially worsened performance, so this is a mechanism-level signal rather than a robust empirical win.

Agent confidence: 7/10

Baseline: Test RMSE 0.13250; boundary-band RMSE 0.20384; shifted-geometry RMSE 0.13250; parameters 1185.
Idea: Test RMSE 0.13086; boundary-band RMSE 0.21157; shifted-geometry RMSE 0.17568; parameters 194. Math: AND zero crossing observed 0.05656 vs predicted 0.0565685; OR -0.05656 vs -0.0565685; sigmoid width relative errors 1.1e-5 to 1.0e-4.

Limitations: Only one fixed random seed and one small 2D regression task were tested. The comparison used an MLP, not a KAN; no FLOP-normalized or multi-seed study was performed; geometry parameters were supplied rather than learned; and support IoU is not a calibrated classification metric for the sign-changing regression target.

How to run: python3 run_experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.01449) Sep 2, 2026

Beats the tuned baseline by 65.1% (p=0.0081, wins 8 / 8 paired seeds; metric: lower is better).

Benchmark: Geometry support regression
Model: shared_dual_mlp
Paired seeds: 8
Baseline mean: 0.0762
Idea mean: 0.0266
Effect (Δ): -0.0496 (−65.1%; negative = idea better)
Wins: 8 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±7.2%
Mechanism: Confirmed ✓
Practical effect: Helps
Baseline tuning: swept over 3 configs

Limitations:

Only the registered custom 2D geometry-support regression track was tested because no built-in track contains the required known-region geometry structure. Geometry parameters were supplied rather than learned; no shifted or incorrect-boundary robustness test, KAN backbone, FLOP-normalized comparison, or larger real-world geometry dataset was run.

How to run: python3 stage2_bench.py

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

Stage 2 — Benchmark Aug 30, 2026 latest Worked ✓

Agent confidence: 9/10

The registered custom geometry track was evaluated with end-to-end matched baseline and R-function-gated dual-expert MLP systems. The idea achieved lower MSE on all 8 paired seeds: 0.02664203 versus 0.07623758 for the tuned baseline, paired delta -0.04959555 and permutation p=0.0081, yielding the required verdict idea better (significant). The trained-model mechanism signature also confirmed the predicted sigmoid gate width within 7.9% relative error.

Baseline: Best lr=0.003; full 8-seed MSE mean=0.07623758167028427, std=0.00677848986822672; sweep means lr 0.001=0.07988427579402924, lr 0.002=0.07893520779907703, lr 0.003=0.07778701558709145.
Idea: Best lr=0.003, kappa=10.0, eps=0.02; full 8-seed MSE mean=0.026642029639333487, std=0.0023230215134395164; paired delta=-0.049595552030950785; 8/8 wins; permutation p=0.0081.
Math check: R-AND and R-OR zero-crossing absolute errors were 1.4575e-6; analytic sigmoid-width relative errors were approximately 1.12e-5.
Mechanism signature: Registered trained-model signature: predicted gate width in G=0.4394449154672439, observed width=0.4047520160675049, relative error=0.07894709479765279, confirmed=true.

Benchmark result