Weighted singular value decomposition for folded matrices

Abstract

Multilinear arrays have attracted an increasing tendency in especially the last two decades. Treating the data in many science and engineering areas like signal processing, computer vision, neuroscience started to use these type of data even though the data were transformed to vectors or matrices in order to efficiently use linear algebraic tools. One of the main goals has been to decompose the given multilinear array data to sums of some products of less variate arrays. This has been accomplished although the uniqueness could have not been achieved because of certain features. Nevertheless the obtained form which was basically developed by Lathauwer has been based on unfolding and folding of multilinear arrays and then using standard linear algebra. Some reductive array decomposition methods have also been developed quite recently by our group members without using a folding or an unfolding of arrays. This work tries to extend this recently proposed idea, without expecting any reduction in the resulting structure via consecutive application of the method, and, does not exclude the possibility of using folding and unfolding.