Nonnegative 3-way tensor factorization taking in to account possible missing data

This paper deals with the problem of incomplete data i.e. data with missing, unknown or unreliable values, in the polyadic decomposition of a nonnegative three-way tensor. The main advantage of the nonnegativity constraint is that the approximation problem becomes well posed. To tackle simultaneously these two problems, we suggest the use of a weighted least square cost function whose weights are gradually modified through the iterations. Moreover, the nonnegative nature of the loading matrices is taken into account directly in the problem parameterization. Then, the three gradient components can be explicitly derived allowing to efficiently implement the CP decomposition using standard optimization algorithms. In our case, we focus on the conjugate gradient and the BFGS algorithms. Finally, the good behaviour of the proposed approaches and their robustness versus possible model errors is illustrated through computer simulations in the context of data analysis.