△ Mechanism confirmed, baseline not beaten 2026

Lunar Color-Connectivity Regularizer

Implementation & benchmark of arXiv:2608.27118 — Lunar Generalizations of the Euclidean Minimum Spanning Tree in the Plane and their Expected Costs

Usefulness6/10

Difficulty6/10

Novelty6/10

Regularization Accuracy Generalization Stability Embedding Regularization Graph nn Metric geometry Topology Combinatorics Optimization

Source paper: Lunar Generalizations of the Euclidean Minimum Spanning Tree in the Plane and their Expected Costs arXiv:2608.27118 ⓘ · analyzed Aug 29, 2026

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 Not peer-reviewed.

Idea description

Add a regularizer that penalizes expensive component births and merges in an embedding when points are partitioned into multiple colors, such as classes, modalities, or augmentation identities. Unlike ordinary contrastive learning, it encourages local regions to contain all required colors and uses the full merge hierarchy rather than only selected positive and negative pairs.

Formulas

$$f_{\max}(x)=\max_{0\leq i\leq s}\min_{a\in A_i}\|x-a\|_2$$

$${\rm Cost}(A_0,\ldots,A_s)=2\sum_{i=1}^{m}d_i-2\sum_{i=1}^{m}b_i$$

$$f_{\max}^{-1}[0,r]=\{x:f_{\max}(x)\le r\}=\bigcup_{a_i\in A_i}\bigcap_{i=0}^{s}B(a_i,r)$$

$$\rho(H)=\min_x\max_{0\le i\le s}\|z_{a_i}-x\|_2,\qquad L_{\rm lunar}=2\sum_i\widetilde d_i-2\sum_{i\ge1}\widetilde b_i$$

Mathematical statement

For color sets $A_0,\ldots,A_s\subset\mathbb R^2$, define the max-nearest-color field $f_{\max}(x)=\max_{0\le i\le s}\min_{a\in A_i}\|x-a\|_2$. Its sublevel set is $f_{\max}^{-1}[0,r]=\{x:f_{\max}(x)\le r\}$, equivalently the union of radius-$r$ intersections of one disk around a point from each color, called colorful lunes. Let $b_0\le\cdots\le b_m$ be radii at which connected components are born and $d_1\le\cdots\le d_m$ be radii at which components merge. The lunar cost is ${\rm Cost}=2\sum_{i=1}^m d_i-2\sum_{i=1}^m b_i$, with $b_0$ deliberately omitted. For a neural implementation, approximate the continuous plane by a graph or hypergraph of embedding samples. A colorful hyperedge $H=(a_0,\ldots,a_s)$ has radius $\rho(H)=\min_x\max_i\|z_{a_i}-x\|_2$, the smallest radius at which all corresponding disks intersect. Activate hyperedges in increasing $\rho(H)$, record component births and merges, and use the same birth-minus-death expression as the loss. The exact filtration cost comes from the paper; graph restriction, soft extrema, and differentiable sorting are adaptations.

Implementation notes

(1) Integrate the regularizer after a projection head of a metric-learning or classification network. For each minibatch, compute embeddings $z_j\in\mathbb R^2$ or a learned two-dimensional projection of a higher-dimensional embedding, and assign colors $c_j\in\{0,\ldots,s\}$. Colors can represent semantic classes, augmentation identities, or modalities. (2) Construct candidate colorful hyperedges using at most $K$ nearest neighbors from every color for each anchor. For $s=1$, use cross-color pairs; for $s=2$, enumerate restricted triples. Estimate each hyperedge radius by initializing $x$ at the mean of its vertices and running $T$ gradient steps on a smooth maximum: $x\leftarrow x-\eta\nabla_x[\beta^{-1}\log\sum_i\exp(\beta\|x-z_{a_i}\|_2)]$. Use the final smooth maximum as $\rho(H)$. (3) Sort candidate radii, activate hyperedges in ascending order, and maintain connected components with union-find. Record the radius when a new vertex or hyperedge creates a component and the radius when an activated hyperedge joins two existing components. Compute $L_{\rm lunar}=2\sum_i\widetilde d_i-2\sum_{i\ge1}\widetilde b_i$, normalize by the minibatch diameter, and add $\lambda L_{\rm lunar}$ to cross-entropy or supervised contrastive loss. Union-find decisions may be stop-gradient routing decisions while gradients flow through the selected radii; a softer alternative uses edge probabilities $p_H(r)=\sigma((r-\rho(H))/\tau)$ and a differentiable connectivity surrogate. The mathematical quantities are the colorful lune radii and birth/death cost; $K,T,\beta,\tau$ are engineering approximations. First test CIFAR-10 with ResNet-18, a two-dimensional projection head, and $s=1$ using two augmentation colors. Compare cross-entropy, supervised contrastive learning, and the combined objective at equal FLOPs. Measure test accuracy, embedding k-NN accuracy, normalized lunar cost, missing-color component count, and collapse indicators. Success means better k-NN or few-shot accuracy at equal classification accuracy, reduced cross-color connection radii, and stable optimization.

Verification

Mechanism confirmed, baseline not beaten

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Confirmed ✓

Stage 2 · Practical benchmark result: No measurable effect

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 two-color lunar EMST cost verifier using exact brute-force minimum enclosing circles, plus a small PyTorch classification experiment with a differentiable nearest cross-color surrogate. The math check produced nonnegative costs for 20 random configurations, and at identical 100% classification and 3-NN accuracy the regularizer reduced cross-color nearest distance from 0.426 to 0.156 and normalized lunar cost from 0.750 to 0.687. This is a promising connectivity signal, but not an accuracy win; the experiment is too small to establish generalization gains.

Agent confidence: 7/10

Baseline: CE: accuracy=1.000, kNN3=1.000, cross-color nearest=0.4260, lunar cost=4.1656, normalized lunar cost=0.7505
Idea: lambda=0.5: accuracy=1.000, kNN3=1.000, cross-color nearest=0.1560, lunar cost=3.5730, normalized=0.6870; lambda=2: accuracy=1.000, kNN3=1.000, cross-color nearest=0.1159, lunar cost=2.9533, normalized=0.6893

Limitations: No CIFAR-10 or ResNet experiment, no multi-seed statistics, and no full differentiable hierarchy implementation. The training penalty is a nearest cross-color approximation rather than the exact lunar cost; the toy data are linearly structured and classification is already saturated.

How to run: python3 lunar_experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.27118) Sep 3, 2026

No significant difference vs. the tuned baseline (+0.7%, p=0.6628, wins 4 / 8 paired seeds).

Benchmark: Tabular regression (Friedman)
Model: mlp_tiny
Paired seeds: 8
Baseline mean: 16.5426
Idea mean: 16.6558
Effect (Δ): 0.1133 (+0.7%; negative = idea better)
Wins: 4 / 8 paired seeds
p-value: 0.6628 (permutation test, 20 000 shuffles)
Smallest detectable effect: n/a
Mechanism: Confirmed ✓
Practical effect: No effect
Baseline tuning: swept over 3 configs

Limitations:

Only the built-in tabular track was tested. The implementation is a soft nearest-pair surrogate, not the full colorful hyperedge hierarchy; regularization weight and temperature were fixed, and the signature measured idea-side trained embeddings only.

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 Sep 3, 2026 latest Failed ✗

Agent confidence: 9/10

The Lunar Color-Connectivity regularizer used the same 2D-projection MLP as baseline, differing only by the differentiable symmetric soft nearest cross-color penalty. Baseline test MSE was 16.542597 and idea MSE was 16.655847, with paired delta_mean=+0.113250 and p_value=0.6628, so the idea did not win. The mechanism signature was confirmed for the trained idea models because cross-color nearest distance was measured as 0.072500 on average, but this did not translate to better task performance.

Baseline: MSE mean 16.542597; best lr=0.01; per_seed=[16.479937,13.553504,18.048687,14.999381,14.820083,16.000380,16.129124,22.309685]
Idea: MSE mean 16.655847; best lr=0.01; per_seed=[17.079338,14.434269,16.626240,15.863789,14.660068,16.299858,16.102301,22.180918]

Benchmark result