A Ternary Non-Commutative Latent Factor Model for Scalable Three-Way Real Tensor Completion

TL;DR

This paper introduces a novel non-commutative tensor factorization model for scalable three-way tensor completion, leveraging algebraic structures to better capture complex relations, and demonstrates its effectiveness on real datasets.

Contribution

The paper presents a new non-commutative latent factor model using algebraic operations for tensor completion, advancing beyond standard PARAFAC methods.

Findings

Outperforms PARAFAC on MovieLens and Fannie Mae datasets

Utilizes non-commutative algebraic operations for modeling

Provides a scalable approach for large-scale tensor completion

Abstract

Motivated by large-scale Collaborative-Filtering applications, we present a Non-Commuting Latent Factor (NCLF) tensor-completion approach for modeling three-way arrays, which is diagonal like the standard PARAFAC, but wherein different terms distinguish different kinds of three-way relations of co-clusters, as determined by permutations of latent factors. The first key component of the algebraic representation is the usage of two non-commutative real trilinear operations as the building blocks of the approximation. These operations are the standard three dimensional triple-product and a trilinear product on a two-dimensional real vector space, which is a representation of the real Clifford Algebra Cl(1,1) (a certain Majorana spinor). Both operations are purely ternary in that they cannot be decomposed into two group-operations on the relevant spaces. The second key component of the method is combining these operations using permutation-symmetry preserving linear combinations. We apply the model to the MovieLens and Fannie Mae datasets, and find that it outperforms the PARAFAC model. We propose some future directions, such as unsupervised-learning.