TITLE:
The Equivalence between Orthogonal Iterations and Alternating Least Squares
AUTHORS:
Achiya Dax
KEYWORDS:
Alternating Least Squares (ALS), Orthogonal Iterations, Equivalence Relations, Low-Rank Approximations
JOURNAL NAME:
Advances in Linear Algebra & Matrix Theory,
Vol.10 No.2,
April
30,
2020
ABSTRACT: This note explores the relations between two different methods. The first one is the Alternating Least Squares (ALS) method for calculating a rank-k approximation of a real m×n matrix, A. This method has important applications in nonnegative matrix factorizations, in matrix completion problems, and in tensor approximations. The second method is called Orthogonal Iterations. Other names of this method are Subspace Iterations, Simultaneous Iterations, and block-Power method. Given a real symmetric matrix, G, this method computes k dominant eigenvectors of G. To see the relation between these methods we assume that G = AT A. It is shown that in this case the two methods generate the same sequence of subspaces, and the same sequence of low-rank approximations. This equivalence provides new insight into the convergence properties of both methods.