# Parity-Copula High-Order Interaction Benchmark

- ID: 2509
- Canonical URL: https://synthcore.org/idea/2509/parity-copula-high-order-interaction-benchmark
- API JSON: https://synthcore.org/api/idea/2509.json
- API Markdown: https://synthcore.org/api/idea/2509.md
- Verification status: unverified
- Source: [arXiv:2608.18286](https://arxiv.org/abs/2608.18286)
- Category: training
- Solves: accuracy, generalization, sample-efficiency
- ML areas: transformer, mlp, data-augmentation, attention
- Math tags: probability, statistics, information-theory
- Ratings: usefulness 6/10; difficulty 3/10; novelty 6/10

## 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.

## 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 \(2^{-d}\).

## Key formulas

- $$\sup\{d_{\infty}(A,\Pi_d):A\in\mathcal{C}_{d}^{\Pi_{d-1}}\}=\frac{1}{2^d},\qquad \sup\{D_1(A,\Pi_d):A\in\mathcal{C}_{d}^{\Pi_{d-1}}\}=\frac{1}{3}.$$
- $$\Pi_d(u_1,\ldots,u_d)=\prod_{i=1}^{d}u_i,\qquad d_{\infty}(A,\Pi_d)=\sup_{u\in[0,1]^d}|A(u)-\Pi_d(u)|.$$
- $$\Pr(Z=z)=2^{-(d-1)}\mathbf{1}\!\left\{\prod_{i=1}^{d}z_i=+1\right\},\qquad U_i\mid Z_i\sim\mathrm{Uniform}\left(\begin{cases}[0,1/2],&Z_i=-1,\\(1/2,1],&Z_i=+1.\end{cases}\right).$$
- $$\Pr\left(\bigwedge_{i=1}^{d}Z_i=z_i\right)-\prod_{i=1}^{d}\Pr(Z_i=z_i)=2^{-d}\quad\text{for every parity-allowed }z.$$

## 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.

## Disclaimer

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 Not peer-reviewed.