Unbiased Asymmetric Co-purchase Relationship (UACR)

• UACR is a formalism that measures the directional likelihood of one item's purchase causing another, eliminating confounding factors like popularity and user preference.
• It employs a dual-embedding framework and probabilistic models to derive unbiased, asymmetric co-purchase scores for constructing interpretable item graphs.
• Empirical evaluations on large datasets show that UACR improves recommendation diversity and accuracy compared to traditional co-occurrence models.

An Unbiased Asymmetric Co-purchase Relationship (UACR) is a formalism for quantifying the directional, causally-meaningful likelihood that the purchase of one item leads to the purchase of another, with explicit removal of confounding effects from item popularity and individual user preferences. UACR is foundational for constructing directed item-item graphs that model genuine complementarity rather than mere co-occurrence, enabling more robust and diverse recommendation systems and interpretable discovery of product relationships (Zhang et al., 19 Dec 2025, Xu et al., 2019).

1. Mathematical Definition and Formal Properties

The UACR formalism is built on the premise that “$i$ complements $j$” denotes that item $j$ is disproportionately likely to be purchased after $i$, with asymmetry ($i\to j \not\Rightarrow j\to i$) and sporadic non-transitivity as essential features.

Let $\mathcal I$ be the set of items and $\mathcal U$ the set of users. The strength of the directed complementarity $(i \to j)$ is encoded as a real-valued score

$s(i, j) = \langle \tilde z^I_i, z^I_j \rangle$

where $\tilde z^I_i$ (“item-out” embedding) and $j$0 (“item-in” embedding) are distinct $j$1-dimensional learned representations. This dual-embedding structure ensures the model captures directionality: in general, $j$2 (Xu et al., 2019).

The unbiased property is achieved by separating out user-specific and popularity-driven confounders, so that $j$3 reflects only higher-order, contextually-aware item-item effects—not user affinity or overall frequency.

2. Learning UACR: Dual-Embedding and User Bias Decomposition

The estimation of UACR leverages a multitask neural embedding framework:

• Each item $j$4 is represented by two vectors: $j$5 for outgoing complementarity ($j$6) and $j$7 for incoming complementarity ($j$8).
• Each user $j$9 and any relevant context token $j$0 are also embedded: $j$1, $j$2, and $j$3 for item-/user-contexts.
• The purchase event at time $j$4 is modeled as

$j$5

where $j$6 models user-item bias, and

$j$7

is the user-independent, unbiased item-item complementarity.

A critical fact is that $j$8 remains unbiased by any individual user, thus producing an item-item relation appropriate for generalization and structural inference (Xu et al., 2019).

3. Partial-Observability Formulation and Probabilistic Inference

An alternative UACR estimation method is grounded in the partial-observability bivariate probit model, adapted from network inference (Rozenas et al., 2017). The true but unobserved network of item-to-item “desires” is represented as a directed graph $j$9 with $i$0 expressing a net positive co-purchase propensity.

However, only the symmetrized, undirected co-purchase graph $i$1 is observed, where $i$2 indicates co-purchase above a threshold. The model assumes

$i$3

where $i$4 and $i$5 are latent directional utilities. A bivariate probit model with joint normal errors is fit using MCMC and data augmentation. Explicit estimation of both $i$6 and $i$7 ensures unbiasedness for the latent, directional process even though the observed graph is undirected.

This probabilistic framework allows the recovery of unbiased, asymmetric co-purchase propensities from noisy observed data and supports wide classes of covariate and latent-variable extensions (Rozenas et al., 2017).

4. Objective Function, Training Methodology, and Geometric Interpretation

The empirical training objective for UACR in the deep learning paradigm is a joint negative-log-likelihood, incorporating negative sampling: $i$8 where $i$9. Each observation is updated in an asynchronous SGD (Hogwild!) regime with learning-rate decay, and negatives sampled with power-law smoothing.

Samples involving only the item-item term in the loss isolate $i\to j \not\Rightarrow j\to i$0, revealing a geometric interpretation: for item $i\to j \not\Rightarrow j\to i$1, optimizing

$i\to j \not\Rightarrow j\to i$2

corresponds to fitting a separating hyperplane between positive (“$i\to j \not\Rightarrow j\to i$3 precedes $i\to j \not\Rightarrow j\to i$4”) and negative (“$i\to j \not\Rightarrow j\to i$5 not preceded by $i\to j \not\Rightarrow j\to i$6”) distributions, with the user bias $i\to j \not\Rightarrow j\to i$7 acting as the intercept (Xu et al., 2019).

5. Empirical and Practical Implications

UACR estimation yields directed, unbiased graphs of item complementarity that are structurally interpretable and empirically validated:

• Asymmetry: e.g., TV $i\to j \not\Rightarrow j\to i$8 TV-mount, but not TV-mount $i\to j \not\Rightarrow j\to i$9 TV.
• Non-transitivity: TV $\mathcal I$0 mount, mount $\mathcal I$1 cover, but TV $\mathcal I$2 cover.
• Higher-order recommendations: combining items in the input can reveal composite complements, not just unions of one-step complements.

Empirical evaluation on datasets such as Instacart (50K items, 3M orders) and Walmart.com (2M items, 100M orders) demonstrates superior performance of UACR-based methods compared to item2vec, prod2vec, BPR, and others, both on product classification and within-basket/next-purchase recommendation tasks, using micro/macro F1, AUC, NDCG, and Hit@K metrics (Xu et al., 2019).

Method	Dept. F1 (micro)	Cat. F1 (macro)	AUC	NDCG
item2vec	0.377	0.075	0.941	0.116
triplet2vec	0.382	0.082	0.960	0.127
Ours (UACR)	0.666	0.535	0.965	0.151

Ablation studies confirm the necessity of simultaneously modeling item context and user bias for maximizing unbiasedness.

6. Extensions and Advanced Topics

Several advanced directions are enabled by the UACR formalism:

• Dynamic UACR: Allow $\mathcal I$3 to evolve over time via a dynamic probit or latent-space evolution.
• User segmentation: Model UACR within customer subpopulations, capturing heterogeneity of complementarity relationships.
• Latent-space augmentation: Introduce network-latent-space factors for items (cf. Hoff 2005) to accommodate unmeasured sources of complementarity.
• Zero-inflated modeling: Adapt UACR estimation to extremely sparse co-purchase matrices characteristic of large-scale e-commerce.

These generalizations expand UACR's applicability to dynamic and high-dimensional recommendation environments while preserving the unbiased and directional nature essential for diversity- and causality-oriented tasks (Rozenas et al., 2017, Zhang et al., 19 Dec 2025).

7. Role in Contemporary Recommendation Frameworks

The UACR is deployed as a core component in recent frameworks such as Cadence, which integrate deconfounded, directed item graphs derived from UACR into LightGCN-based recommendation architectures. Here, UACR facilitates the identification and high-exposure simulation of causally-relevant, diverse items that have not yet been presented to the user, driving gains in both diversity and accuracy relative to prior baselines (Zhang et al., 19 Dec 2025).

A plausible implication is that as item co-purchase data continues to increase in scale and complexity, techniques such as UACR that analytically separate user bias and popularity from directional complementarity will become central to explainable, fair, and effective recommendation ecosystems.