Tensor networks have emerged as powerful tools for solving large-scale optimization problems in recent years. These networks are capable of handling complicated tensor structures and have proven to be highly effective in various fields. One popular tensor network model is the tensor train (TT) decomposition, which serves as the building block for complex tensor networks. However, the TT decomposition relies heavily on permutations of tensor dimensions and sequential multilinear products, making it challenging to find the optimal TT representation.

What is Tensor Ring Decomposition?

Tensor ring (TR) decomposition is a fundamental tensor decomposition model that represents high-dimensional tensors using circular multilinear products over a sequence of low-dimensional cores. It can be visually interpreted as a cyclic interconnection of third-order tensors, forming a ring-like structure. This unique circular dimensional permutation invariance is achieved by treating the latent cores equivalently and employing the trace operation. In other words, the TR model can be seen as a linear combination of TT decompositions, offering enhanced representation capabilities.

How does Tensor Ring Decomposition differ from Tensor Train (TT) Decomposition?

While both TR decomposition and TT decomposition are tensor network models, they differ in terms of representation and permutation invariance. The TT decomposition relies on strict sequential multilinear products over latent cores, making it highly dependent on permutations of tensor dimensions. On the other hand, the TR decomposition introduces circular dimensional permutation invariance by employing a cyclic interconnection of third-order tensors. This fundamental difference allows the TR model to overcome the challenges associated with finding the optimal TT representation.

What are the Advantages of using Tensor Ring Model?

The tensor ring (TR) model offers several advantages in comparison to other tensor decomposition approaches:

  • Circular dimensional permutation invariance: The TR model’s circular structure provides invariance to permutations of tensor dimensions, which can help simplify optimization problems.
  • Enhanced representation capabilities: TR decomposition can be viewed as a linear combination of TT decompositions, allowing for powerful and generalized representation of high-dimensional tensors.
  • Efficient multilinear algebra: TR representations enable efficient performance of basic multilinear algebra operations, enhancing computational efficiency.
  • Convenient transformation: Classical tensor decompositions can be conveniently transformed into the TR representation, facilitating the integration of existing tensor decomposition approaches.

What Algorithms are used for Optimization of Latent Cores in Tensor Ring Decomposition?

To optimize the latent cores in tensor ring (TR) decomposition, the research article presents four different algorithms:

  1. Sequential SVDs: This algorithm utilizes sequential singular value decompositions to optimize the latent cores and find an optimal TR representation.
  2. ALS scheme: The Alternating Least Squares (ALS) scheme is employed as an optimization algorithm for the latent cores in the TR model.
  3. Block-wise ALS techniques: Researchers explore block-wise ALS techniques as an alternative approach for optimizing the latent cores in TR decomposition.

These algorithms aim to find the best configurations for the latent cores within the TR model, ensuring efficient representation of high-dimensional tensors.

How can Classical Tensor Decompositions be Transformed into Tensor Ring Representation?

The tensor ring (TR) decomposition allows for convenient transformation of classical tensor decompositions into its representation. This transformation provides the benefits of the TR model while integrating with existing decomposition approaches. The TR model offers a generalized and powerful representation, blending the advantages of both classical tensor decompositions and the TR decomposition. The specific transformations and methodologies for this integration are explored in detail in the research article.

By leveraging the circular dimensional permutation invariance and enhanced representation capabilities of the TR model, it becomes possible to optimize various latent core configurations and effectively represent complex tensors.

Overall, the tensor ring (TR) decomposition offers a versatile and efficient approach to solving large-scale optimization problems involving high-dimensional tensors. The research conducted by Zhao, Zhou, Xie, Zhang, and Cichocki demonstrates the advantages of TR decomposition and introduces novel algorithms for optimizing latent cores. These advancements have the potential to greatly impact fields such as machine learning, data analysis, and signal processing.

Source:

Qibin Zhao, Guoxu Zhou, Shengli Xie, Liqing Zhang, and Andrzej Cichocki introduced the tensor ring (TR) decomposition in a research article titled “Tensor Ring Decomposition”.

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