Unverified 2026

Parity-Copula High-Order Interaction Benchmark

Implementation & benchmark of arXiv:2608.18286 — How far are $d$-dimensional copulas with uniform $(d-1)$-marginals from (total) independence?

Usefulness6/10

Difficulty3/10

Novelty6/10

Training Accuracy Generalization Sample efficiency Transformer Mlp Data augmentation Attention Probability Statistics Information theory

Source paper: How far are $d$-dimensional copulas with uniform $(d-1)$-marginals from (total) independence? arXiv:2608.18286 ⓘ · analyzed Sep 1, 2026

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 Not peer-reviewed.

Idea description

Construct training and evaluation examples whose every (d-1)-variable marginal is exactly independent, but whose full d-variable distribution contains a parity interaction. This isolates genuine high-order reasoning from shortcuts based on pairwise or lower-order statistics and can expose whether attention or MLP architectures learn the intended interaction.

Formulas

$$\sup\{d_{\infty}(A,\Pi_d):A\in\mathcal{C}_{d}^{\Pi_{d-1}}\}=\frac{1}{2^d},\qquad \sup\{D_1(A,\Pi_d):A\in\mathcal{C}_{d}^{\Pi_{d-1}}\}=\frac{1}{3}.$$

$$\Pi_d(u_1,\ldots,u_d)=\prod_{i=1}^{d}u_i,\qquad d_{\infty}(A,\Pi_d)=\sup_{u\in[0,1]^d}|A(u)-\Pi_d(u)|.$$

$$\Pr(Z=z)=2^{-(d-1)}\mathbf{1}\!\left\{\prod_{i=1}^{d}z_i=+1\right\},\qquad U_i\mid Z_i\sim\mathrm{Uniform}\left(\begin{cases}[0,1/2],&Z_i=-1,\\(1/2,1],&Z_i=+1.\end{cases}\right).$$

$$\Pr\left(\bigwedge_{i=1}^{d}Z_i=z_i\right)-\prod_{i=1}^{d}\Pr(Z_i=z_i)=2^{-d}\quad\text{for every parity-allowed }z.$$

Mathematical statement

The paper's main theorem states that for the family $\mathcal{C}_{d}^{\Pi_{d-1}}$ of d-dimensional copulas $A$ whose every (d-1)-dimensional marginal equals the product copula $\Pi_{d-1}$, $\sup_{A\in\mathcal{C}_{d}^{\Pi_{d-1}}} d_{\infty}(A,\Pi_d)=2^{-d}$, and $\sup_{A\in\mathcal{C}_{d}^{\Pi_{d-1}}}D_1(A,\Pi_d)=1/3$. Here $\Pi_d(u_1,\ldots,u_d)=\prod_i u_i$, $d_{\infty}(A,B)=\sup_{u\in[0,1]^d}|A(u)-B(u)|$, and $D_1$ is the paper's conditioning-based dependence metric. A directly implementable construction samples signs $Z_i\in\{-1,+1\}$ uniformly subject to $\prod_{i=1}^d Z_i=+1$, then samples $U_i$ uniformly from $[0,1/2]$ when $Z_i=-1$ and from $(1/2,1]$ when $Z_i=+1$. Any subset of at most $d-1$ signs is iid uniform, so every lower-order marginal of $U$ is the product copula, while the full parity event is deterministic. At a sign-cell corner, the probability differs from independence by $2^{-d}$.

Implementation notes

Integrate this at the data-generation and evaluation layer, not inside the model. Choose d input channels or token features and generate a latent sign vector Z with one parity constraint: sample d-1 independent Rademacher signs and set Z_d=\prod_{i<d}Z_i. For each coordinate, draw U_i uniformly from the lower half of [0,1] if Z_i=-1 and from the upper half if Z_i=+1; optionally transform U_i through a random monotone map so the marginal is nontrivial while remaining uniform in copula space. Create labels from the parity bit, or embed the parity-conditioned features into real task examples. Pseudocode is: sample z[0:d-1] independently; set z[d-1]=prod(z[0:d-1]); sample u[i]=Uniform(0,.5) if z[i]<0 else Uniform(.5,1); x=feature_map(u); y=prod(z). Verify empirically that every subset of d-1 coordinates has near-zero lower-order mutual information, while the full parity statistic is deterministic. The theorem supplies the target dependence scale d_infinity=2^{-d}; empirical checks estimate deviations with histograms or empirical CDFs. First experiment: compare a 2-layer MLP, a small Transformer, and a model with only pairwise feature products on d=4,6,8 parity data at matched parameter count and sample size. Success is high accuracy for the full model but chance accuracy for pairwise-only baselines, followed by improved performance on a mixed dataset where 30% of examples contain this hidden high-order interaction. Measure accuracy versus d, sample efficiency, and robustness after randomizing all lower-order marginals.

Verification

This idea has not been verified yet.

Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.

Artifacts

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