Open Problem: Tensor Decompositions: Algorithms up to the Uniqueness Threshold?

Factor analysis is a basic tool in statistics and machine learning, where the goal is to take many variables and explain them away using fewer unobserved variables, called factors. It was introduced in a pioneering study by psychologist Charles Spearman, who used it to test his theory that there are fundamentally two types of intelligence – verbal and mathematical. This study has had a deep influence on modern psychology, to this day. However there is a serious mathematical limitation to this approach, which we describe next. In its most basic form, we are given a matrix M = ∑R i=1 ai ⊗ bi. Our goal is to recover the factors {ai}i and {bi}i. However this decomposition is only unique if we add further assumptions, such as requiring the factors {ai}i and {bi}i to be orthonormal. Otherwise we could apply an R×R rotation to the factors {ai}i and its transpose to {bi}i and recover another valid decomposition. This is often called the rotation problem and has been a central stumbling block for factor analysis since Spearman’s work. To summarize: Even if there is a factorization M = ∑R i=1 ai ⊗ bi that has a meaningful interpretation, there is no guarantee that factor analysis finds it!