Low-Rank Tensor Networks for Dimensionality Reduction and Large-Scale Optimization Problems: Perspectives and Challenges PART 1 (1609.00893v3)

Abstract: Machine learning and data mining algorithms are becoming increasingly important in analyzing large volume, multi-relational and multi--modal datasets, which are often conveniently represented as multiway arrays or tensors. It is therefore timely and valuable for the multidisciplinary research community to review tensor decompositions and tensor networks as emerging tools for large-scale data analysis and data mining. We provide the mathematical and graphical representations and interpretation of tensor networks, with the main focus on the Tucker and Tensor Train (TT) decompositions and their extensions or generalizations. Keywords: Tensor networks, Function-related tensors, CP decomposition, Tucker models, tensor train (TT) decompositions, matrix product states (MPS), matrix product operators (MPO), basic tensor operations, multiway component analysis, multilinear blind source separation, tensor completion, linear/multilinear dimensionality reduction, large-scale optimization problems, symmetric eigenvalue decomposition (EVD), PCA/SVD, huge systems of linear equations, pseudo-inverse of very large matrices, Lasso and Canonical Correlation Analysis (CCA) (This is Part 1)

• The paper demonstrates how low-rank tensor networks decompose high-order data using Tucker and Tensor Train methods to mitigate the curse of dimensionality.
• It employs structured low-rank approximations that compress complex data sets while preserving essential features for scalable optimization.
• The research highlights challenges in balancing computational efficiency with approximation accuracy in managing multi-modal, high-dimensional data.

Overview of "Low-Rank Tensor Networks for Dimensionality Reduction and Large-Scale Optimization Problems: Perspectives and Challenges"

The paper "Low-Rank Tensor Networks for Dimensionality Reduction and Large-Scale Optimization Problems: Perspectives and Challenges" by Cichocki et al. provides a comprehensive examination of low-rank tensor networks (TN) as effective tools for handling massive-scale data challenges. The authors focus on Tucker and Tensor Train (TT) decompositions as core techniques, emphasizing the utility of these methods in managing the curse of dimensionality, which plagues many high-dimensional datasets in machine learning and data mining.

Key Contributions

1. Tensor Network Paradigms: The paper introduces tensor networks as flexible representations that decompose high-order tensors into interconnected low-order tensors, improving interpretability and computational feasibility. This decomposition is pivotal for working with large multiway arrays that arise in multi-relational, multi-modal datasets.
2. Core Methods - Tucker and TT Decompositions:
   • Tucker Decomposition: This method breaks down tensors into a core tensor multiplied by a set of matrices across each mode. It is particularly effective in capturing complex interactions but can suffer from the curse of dimensionality for very high-order tensors.
   • Tensor Train (TT) Decomposition: Also known as Matrix Product States (MPS), TT decompositions represent tensors as a sequence of interconnected low-rank tensors. The TT format is computationally efficient and provides stable control over approximation errors.
3. Dimensionality Reduction and Optimization: The authors present tensor networks as a promising avenue for dimensionality reduction, offering a structured approach to deal with large volumes of data by leveraging low-rank approximations to compress data while preserving essential information.
4. Challenges and Trade-offs: The paper highlights the challenges in ensuring computational efficiency while maintaining the desired accuracy of approximations. Efficient algorithms for TT decompositions and the need for scalable optimization methods are a focal point.
5. Applications and Implications: Tensor networks are proposed as alternatives or complements to conventional optimization techniques like alternating direction method of multipliers (ADMM) and random coordinate descent (RCD). The ability to solve multi-block data problems by converting them into linked smaller subproblems makes tensor networks valuable in various applications, from quantum physics to machine learning.

Implications for AI and Future Work

• Theoretical Implications: The structured sparsity provided by tensor networks supports efficient computation and storage, making them suitable for foundational models in quantum computation and AI.
• Practical Applications: The potential for tensor networks to manage data efficiently suggests applications in healthcare, finance, and other data-intensive fields.
• Future Directions: The work encourages further development of algorithms that leverage tensor networks for real-time data processing and learning in AI, emphasizing scalability and practicality.

In conclusion, Cichocki et al. present low-rank tensor networks as robust frameworks for modern high-dimensional challenges, offering both theoretical insight and practical tools for large-scale optimization in data-rich environments. The presented methods promise to bridge the gap between manageable computation and rich, high-volume data management.