Abstract: As a method of expanding a higher-order tensor data to tensor products of vectors we have proposed the Third-order Orthogonal Tensor Product Expansion (3OTPE) that did similar expansion as Higher-Order Singular Value Decomposition (HOSVD). In this paper we provide a computation algorithm to improve our previous method, in which SVD is applied to the matrix that constituted by the contraction of original tensor data and one of the expansion vector obtained. The residual of the improved method is smaller than the previous method, truncating the expanding tensor products to the same number of terms. Moreover, the residual is smaller than HOSVD when applying to color image data. It is able to be confirmed that the computing time of improved method is the same as the previous method and considerably better than HOSVD.
Abstract: In digital signal processing it is important to
approximate multi-dimensional data by the method called rank
reduction, in which we reduce the rank of multi-dimensional data from
higher to lower. For 2-dimennsional data, singular value
decomposition (SVD) is one of the most known rank reduction
techniques. Additional, outer product expansion expanded from SVD
was proposed and implemented for multi-dimensional data, which has
been widely applied to image processing and pattern recognition.
However, the multi-dimensional outer product expansion has behavior
of great computation complex and has not orthogonally between the
expansion terms. Therefore we have proposed an alterative method,
Third-order Orthogonal Tensor Product Expansion short for 3-OTPE.
3-OTPE uses the power method instead of nonlinear optimization
method for decreasing at computing time. At the same time the group
of B. D. Lathauwer proposed Higher-Order SVD (HOSVD) that is
also developed with SVD extensions for multi-dimensional data.
3-OTPE and HOSVD are similarly on the rank reduction of
multi-dimensional data. Using these two methods we can obtain
computation results respectively, some ones are the same while some
ones are slight different. In this paper, we compare 3-OTPE to
HOSVD in accuracy of calculation and computing time of resolution,
and clarify the difference between these two methods.