# Cardinality-Stratified Interaction Decomposition for Interpretable Pairwise and Higher-Order Structure in Transactional Basket Data

arXiv: https://arxiv.org/abs/2609.34191 · HTML: https://mlsanity.com/paper/2609.34191/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 39 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 7 of 7)

- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [The physics of higher-order interactions in complex systems](https://mlsanity.com/paper/2110.06023/) (2021)
- [Inverse statistical problems: from the inverse Ising problem to data science](https://mlsanity.com/paper/1702.01522/) (2017)
- [Inferring Higher-Order Couplings with Neural Networks](https://mlsanity.com/paper/2501.06108/) (2025)
- [Estimation in Tensor Ising Models](https://mlsanity.com/paper/2008.12882/) (2020)
- [Bayesian Inference of Minimally Complex Models with Interactions of Arbitrary Order](https://mlsanity.com/paper/2008.00520/) (2020)
- [Modeling Discrete Data with High-Order Vector Potts Models](https://mlsanity.com/paper/2606.03429/) (2026)

## Abstract (arXiv)

Transactional basket data can reveal associations among items, but observed co-occurrence conflates item-specific relations with basket-size structure and unmodeled higher-order dependence. We introduce Cardinality-Stratified Interaction Decomposition (CSID), an interpretable framework that decomposes log-odds contrasts stratified by the number of remaining items into item-set-specific and cardinality-common components, without fitting a global joint distribution. CSID uses an information-weighted, gauge-constrained ridge projection to estimate pair and triple components and to diagnose higher-order contributions to pairwise structure. CSID is designed primarily for interpretable decomposition of association structure rather than for full-distribution prediction. In a simulation with zero pair effects, increasingly strong small-basket cardinality potentials drive ordinary Ising couplings spuriously negative, whereas CSID pair estimates remain centered near zero. Detection power rises with the magnitude of planted triple effects, and local deprojection reduces pair-coefficient RMSE from 0.244 to 0.073. Across three grocery datasets, high-information triple components are reproducible over time. In the matched cross-period partial-transfer evaluation, transferred CSID triple components show closer agreement with later-period stratified contrasts than the nodewise-symmetrized cardinality-aware higher-order pseudolikelihood comparator, with gains in weighted Lin's concordance correlation of 0.038--0.122. These results support CSID as an exploratory and interpretable decomposition framework for pairwise and higher-order association structure in transactional data.