Optimization: algorithms and consistent approximations
Optimization: algorithms and consistent approximations
Optimization by Vector Space Methods
Optimization by Vector Space Methods
An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
Convex Optimization
Enhanced Line Search: A Novel Method to Accelerate PARAFAC
SIAM Journal on Matrix Analysis and Applications
A comparison of algorithms for fitting the PARAFAC model
Computational Statistics & Data Analysis
Tensor Decompositions and Applications
SIAM Review
Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation
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Computing the minimal polyadic decomposition (also often referred to as canonical decomposition or sometimes Parafac) amounts to finding the global minimum of a coercive polynomial in many variables. In the case of arrays with nonnegative entries, the low-rank approximation problem is well posed. In addition, due to the large dimension of the problem, the decomposition can be rather efficiently calculated with the help of preconditioned nonlinear conjugate gradient algorithms, as subsequently shown, if equipped with an algebraic calculation of the globally optimal stepsize in low dimension. Other algorithms are also studied (gradient and quasi-Newton approaches) for comparisons. Two versions of each algorithm are considered: the enhanced line search version (ELS), and the backtracking version alternating with ELS. Computer simulations are provided and demonstrate the good behavior of these algorithms dedicated to nonnegative arrays, compared to others put forward in the literature. Finally, applications in the context of data analysis illustrate various algorithms. The main advantage of the suggested approach is to explicitly take into account the nonnegative nature of the loading matrices in the problem parameterization, instead of enforcing positive entries by projection. According to the experiments we have run, such an approach also happens to be more robust with respect to possible modeling errors.