Tensor Networks for Latent Variable Analysis: Novel Algorithms for Tensor Train Approximation.

Area of Science:

• Applied Mathematics
• Machine Learning
• Signal Processing

Background:

• Tensor decompositions are vital in signal processing and machine learning.
• Tensor network (TN) decomposition, while established in physics, is underutilized in data science.
• Existing TN decomposition methods face scalability challenges with large datasets.

Purpose of the Study:

• Introduce novel algorithms for tensor network (TN) decomposition, focusing on the tensor train (TT) variant.
• Enhance the mathematical tractability and scalability of TT decomposition for large-scale data.
• Evaluate the performance of new TT decomposition algorithms in classic signal processing applications.

Main Methods:

• Developed novel iterative algorithms for TT decomposition that update core tensors.
• Considered rigorous cases including given ranks, approximation error, and error bounds.
• Applied algorithms to blind source separation, denoising, and feature extraction.

Main Results:

• Achieved well-balanced TT-decompositions with enhanced mathematical tractability and scalability.
• Demonstrated superior performance compared to widely used truncated TT decomposition algorithms.
• Validated effectiveness in diverse signal processing and machine learning tasks.

Conclusions:

• The proposed novel algorithms offer significant improvements for tensor train decomposition.
• These advancements enable more effective application of TN decompositions to large-scale data.
• The methods show promise for enhancing performance in signal processing and machine learning.