  Collaborative filtering is the de facto standard for analyzing users'
activities and building recommendation systems for items. In this work we
develop Sliced Anti-symmetric Decomposition (SAD), a new model for
collaborative filtering based on implicit feedback. In contrast to traditional
techniques where a latent representation of users (user vectors) and items
(item vectors) are estimated, SAD introduces one additional latent vector to
each item, using a novel three-way tensor view of user-item interactions. This
new vector extends user-item preferences calculated by standard dot products to
general inner products, producing interactions between items when evaluating
their relative preferences. SAD reduces to state-of-the-art (SOTA)
collaborative filtering models when the vector collapses to 1, while in this
paper we allow its value to be estimated from data. Allowing the values of the
new item vector to be different from 1 has profound implications. It suggests
users may have nonlinear mental models when evaluating items, allowing the
existence of cycles in pairwise comparisons. We demonstrate the efficiency of
SAD in both simulated and real world datasets containing over 1M user-item
interactions. By comparing with seven SOTA collaborative filtering models with
implicit feedbacks, SAD produces the most consistent personalized preferences,
in the meanwhile maintaining top-level of accuracy in personalized
recommendations. We release the model and inference algorithms in a Python
library https://github.com/apple/ml-sad.
