Coupled tensor models for probability mass function estimation: Part I, principles and algorithms

In this article, a probability mass function (PMF) estimation method called partial coupled tensor factorization of 3D marginals or PCTF3D is proposed. To tame the inherent PMF estimation curse of dimensionality, PCTF3D’s principle is to couple 3-dimensional data projections – seen as order-3 tensors – to obtain a low-rank tensor approximation of the PMF. The contribution of PCTF3D relies on partial coupling which consists in choosing a limited subset of 3D marginals. While PMF estimation is possible with all marginals, coupling only a subset of marginals like in PCTF3D permits to reduce the computational burden without losing significant estimation performance. A key concept of PCTF3D is the choice of marginals to be coupled: this problem is formulated and studied with hypergraphs. This Part I paper introduces the algorithmic framework of PCTF3D: optimization problem, coupling strategies, numerical experiments and a real data application of PCTF3D. On the other hand, the Part II paper studies coupled tensor uniqueness properties of the model introduced by PCTF3D.