Advanced Tensor Decompositions in Big Data Analytics: Mathematical Methods and Applications

1. Introduction 2. Fundamental Concepts of Tensor Decompositions 2.1 Tensor Definitions and Properties 2.2 Introduction to Tensor Algebra 2.3 Overview of Decomposition Techniques 3. Mathematical Foundations 3.1 Multi-Linear Algebra 3.2 Eigenvalues and Eigenvectors in Tensors 3.3 Optimization Methods for Tensors 4. Advanced Decomposition Methods 4.1 CP (CANDECOMP/PARAFAC) Decomposition 4.2 Tucker Decomposition 4.3 Tensor Train Decomposition 4.4 Hierarchical Tucker Decomposition 5. Tensor Decompositions in Big Data Analytics 5.1 Scalability of Tensor Algorithms 5.2 Tensor Decomposition for Big Data Challenges 5.3 Handling High Dimensionality in Data 6. Applications in Machine Learning 6.1 Dimensionality Reduction Techniques 6.2 Tensor-Based Feature Extraction 6.3 Improvement in Learning Algorithms 7. Case Studies and Practical Applications 7.1 Decomposition for Medical Imaging 7.2 Applications in Graph Analytics 7.3 Tensor Analysis in Social Networks 7.4 Use Cases in Recommendation Systems 8. Future Directions and Research Opportunities 8.1 Emerging Trends in Tensor Analytics 8.2 Integration with Other Technologies 8.3 Open Challenges and Potential Solutions