Tensor Networks for Latent Variable Analysis: Higher Order Canonical Polyadic Decomposition.

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Data Science

Background:

• Canonical Polyadic Decomposition (CPD) is a powerful tensor factorization technique.
• Higher-order tensors pose computational challenges for traditional CPD, including high costs and entry permutations.
• Existing compression methods for CPD are limited by the decomposition rank relative to tensor dimensions.

Purpose of the Study:

• To present a novel method for efficient canonical polyadic decomposition (CPD) of higher-order tensors.
• To overcome the computational and permutation limitations of standard CPD for complex tensor data.
• To provide a robust and scalable solution for tensor factorization in advanced applications.

Main Methods:

• A novel tensor network approach utilizing interconnected core tensors (order ≤ 3).
• Development of an exact conversion scheme from core tensors to CPD factor matrices.
• An iterative low-complexity algorithm for estimating factor matrices in inexact CPD scenarios.

Main Results:

• The proposed tensor network method significantly reduces computational cost for higher-order tensor CPD.
• The method effectively mitigates issues of entry permutation inherent in traditional CPD.
• Simulations across diverse scenarios validate the accuracy and efficiency of the novel approach.

Conclusions:

• The novel tensor network-based CPD method offers a computationally efficient and robust alternative for higher-order tensors.
• This approach expands the applicability of CPD to larger and more complex datasets.
• The developed exact and iterative algorithms provide practical tools for tensor factorization research and applications.