Interaction as Interference: A Quantum-Inspired Aggregation Approach

TL;DR

This work redefines causal interaction as a property of the aggregation rule rather than a separate feature, using a quantum-inspired coherent (phase-sensitive) aggregation under the Born rule. It proves in a minimal 2×2 design that the interference cross-term equals the potential-outcome interaction contrast Δ_INT, making the relative phase a direct lever to control synergy versus antagonism. The Interference Kernel Classifier (IKC) operationalizes this with two complex-valued amplitude channels, and two diagnostics, Coherent Gain and Interference Information, to quantify the value of coherence over an incoherent proxy. Across phase-sweep validation, XOR, robustness tests, ablations, and real-world Adult/Bank datasets, coherent aggregation improves likelihood and calibration where interaction is pivotal, while performance on capacity-rich real-world data highlights the need for richer representations to fully harness the proposed inductive bias.

Abstract

Classical approaches often treat interaction as engineered product terms or as emergent patterns in flexible models, offering little control over how synergy or antagonism arises. We take a quantum-inspired view: following the Born rule (probability as squared amplitude), \emph{coherent} aggregation sums complex amplitudes before squaring, creating an interference cross-term, whereas an \emph{incoherent} proxy sums squared magnitudes and removes it. In a minimal linear-amplitude model, this cross-term equals the standard potential-outcome interaction contrast \(Δ_{\mathrm{INT}}\) in a \(2\times 2\) factorial design, giving relative phase a direct, mechanism-level control over synergy versus antagonism. We instantiate this idea in a lightweight \emph{Interference Kernel Classifier} (IKC) and introduce two diagnostics: \emph{Coherent Gain} (log-likelihood gain of coherent over the incoherent proxy) and \emph{Interference Information} (the induced Kullback-Leibler gap). A controlled phase sweep recovers the identity. On a high-interaction synthetic task (XOR), IKC outperforms strong baselines under paired, budget-matched comparisons; on real tabular data (\emph{Adult} and \emph{Bank Marketing}) it is competitive overall but typically trails the most capacity-rich baseline in paired differences. Holding learned parameters fixed, toggling aggregation from incoherent to coherent consistently improves negative log-likelihood, Brier score, and expected calibration error, with positive Coherent Gain on both datasets.

Figures (3)

• Figure 1: E1: Phase sweep validates the interference–interaction identity. Interaction contrast $\Delta_{\mathrm{INT}}$ as a function of the relative phase $\Delta\phi$. The plot overlays the theoretical law $2\,r_A r_B \cos(\Delta\phi)$ from the amplitude–linear model with $(r_0,r_A,r_B)=(0.20,0.35,0.35)$ and empirical plug-in estimates $\widehat{\Delta}_{\mathrm{INT}}$ with two-sided 95% bootstrap confidence bands (balanced $2\times 2$ design; $n{=}2000$ per cell; 121 phase points; 5,000 bootstrap replicates; a fixed random seed).
• Figure 2: E3: Robustness to training label noise. Paired deltas ($\Delta=$ IKC $-$ Best-Baseline; lower is better) versus label-noise level (%) with two-sided 95% bootstrap CIs, using a fixed test split per seed. (a)$\Delta$NLL; (b)$\Delta$Brier. The best baseline is chosen per seed by validation NLL on the same split and retrained on the training and validation data. TS uses a safety switch (fit on the calibration split; applied to the test split only if it lowers calibration NLL)
• Figure 3: E5: Adult and Bank paired test-set deltas. Paired, budget-matched differences $\Delta=$ IKC $-$ Best-Baseline (lower is better) for NLL, Brier, and ECE; mean $\pm$$95\%$ CI over $n_\text{seeds}{=}20$. Best baseline is chosen per seed; all models use TS with a safety switch.