Hierarchical Cubical Tensor Decomposition through Low Complexity Orthogonal Transforms

In this work, new approaches are proposed for the 3D decomposition of a cubical tensor of size N × N × N for N = 2n through hierarchical deterministic orthogonal transforms with low computational complexity, whose kernels are based on the Walsh-Hadamard Transform (WHT) and the Complex Hadamard Transform (CHT). On the basis of the symmetrical properties of the real and complex Walsh-Hadamard matrices are developed fast computational algorithms whose computational complexity is compared with that of the famous deterministic transforms: the 3D Fast Fourier Transform, the 3D Discrete Wavelet Transform and the statistical Hierarchical Tucker decomposition. The comparison results show the lower computational complexity of the offered algorithms. Additionally, they ensure the high energy concentration of the original tensor into a small number of coefficients of the so calculated transformed spectrum tensor. The main advantage of the proposed algorithms is the reduction of the needed calculations due to the low number of hierarchical levels compared to the significant number of iterations needed to achieve the required decomposition accuracy based on the statistical methods. The choice of the 3D hierarchical decomposition is defined by the requirements and limitations related to the corresponding application area.