A Multilinear Singular Value Decomposition

Abstract

We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higher-order tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, first-order perturbation effects, etc., are analyzed. We investigate how tensor symmetries affect the decomposition and propose a multilinear generalization of the symmetric eigenvalue decomposition for pair-wise symmetric tensors.

Extended HALS algorithm for nonnegative Tucker decomposition and its applications for multiway analysis and classification

“…In many applications such as dimensionality reduction and feature extraction, several existing Tucker decomposition algorithms consider orthogonality of factors, such as the Higher-Order Singular Value Decomposition (HOSVD) and Higher Order Orthogonal Iteration (HOOI) algorithms [21][22][23][24]. For Nonnegative Tucker Decomposition (NTD), the multiplicative algorithms [1,16,25,26] are natural extensions of multiplicative Nonnegative Matrix Factorization (NMF) algorithms based on minimization of the squared Euclidean distance (Frobenius norm) and the Kullback-Leibler divergence.…”

Sparse non-negative tensor factorization using columnwise coordinate descent

“…Our experiments show the effectiveness of the proposed unbalanced core tensor especially when the target tensor is very unbalanced. We have constructed a collection of basis images for Drosophila gene expression pattern images from stages [11][12]. We plan to analyze the biological significance of the learnt basis images in the future.…”

“…A key issue in applying the tucker model is how to construct the core tensor. Lathauwer et al [12] presented the well-known HOSVD (Higher-Order Singular Value Decomposition) factorization based on the SVD algorithm for matrices. Without a non-negative requirement, it forced all factors to be orthogonal so that the core tensor could be computed through a unique and explicit expression.…”

Correlation-based algorithm for multi-dimensional single-tone frequency estimation

“…Now the task is to find {ω r } from the M = R r=1 M r samples of Y Y Y. By writing g r = e jωr e jωr2 · · · e jωrMr T , to align with the presentation in Section 2 we define the rth unfolding of X X X as the transpose version of [19]:…”

“…It is worthy to point out that the derivation of the weighting matrix and performance analysis are different from those of [18]. Section 3 generalizes the proposed solutions to higher dimensional signals with the use of tensor algebra [19]. Simulation results are included in Section 4 to corroborate the theoretical development and to compare the correlation-based approach with the approximate iterative quadratic ML (AIQML) [15], IMDF [16] and UE [8] algorithms as well as Cramér-Rao lower bound (CRLB) [20].…”