A tensor ring decomposition represents a physical state as a sequence of interconnected third-order core tensors with circular invariance. It generalizes the tensor train decomposition, allowing linear parameter scaling while preserving expressiveness and enabling inference on large-scale, high-dimensional data.
ALS variants mitigate intermediate data explosion and improve numerical stability, yielding scalable algorithms (Yuan et al., 2023). Sampling-based ALS and leverage-score sketching further reduce update costs to sublinear in tensor size, and achieve good reliability at low noise levels.
Reliability
Tensor decompositions are important tools for reducing the storage requirements of high-dimensional data. They can help reduce the number of bits required to store a matrix and make it possible to compute efficiently with the underlying matrices. However, the reliability of tensor decompositions depends on the decomposition format used. Several different formats are available, including the canonical polyadic decomposition, Tucker decomposition, hierarchical Tucker decomposition, and tensor train (TT) decomposition. These methods are based on the fundamental geometry of the quotient manifold, and they require knowledge of the ring structure. Therefore, they are not suited for applications in which the rank is fixed or unknown.
In contrast, tensor ring decomposition allows for a more flexible representation of the input tensor by relaxing the rank constraint for cores on the first and last modes. This allows for more flexible and reliable computations than TT, while also providing better accuracy. However, it is essential to understand the limitations of this approach.
The quotient manifold structure of TR is more complicated than that of TT. This is because the ring structure of TR introduces redundancy, which can lead to nonuniqueness of the group action induced by GL. Moreover, the ring structure makes it difficult to prove that TR is a symmetric quotient manifold.
Despite these drawbacks, tensor ring decomposition is a reliable method for computing the eigenvalues of a tensor. It is especially useful for inverse problems such as matrix reconstruction and inverse variational invariance. It can be applied to a variety of fields, including image processing, tensor completion, and eigenvalue estimation. It can also be used to solve problems in fluid dynamics and quantum physics.
In addition, tensor ring decomposition can be used to improve the reliability of computational algorithms that use low-rank representations of data. In particular, it can be used to improve the performance of matrix product states (MPS) in tensors, which are an important component of matrix multiplication algorithms. MPS is a popular technique for matrix multiplication, but it has been found to be inaccurate in some cases. Fortunately, tensor ring decompositions can reduce the size of these matrix product states by combining overlapping multidimensional slices. This can significantly improve the accuracy of MPS in tensors.
Accuracy
Tensor ring decomposition is a cyclic generalization of low-rank tensor factorizations that supports robust algorithms for high-dimensional inference and completion. The TR decomposition allows for linear parameter scaling, while preserving expressiveness and flexibility. The resulting model is widely used for image recovery and nonlocal self-similarity denoising. It also provides new algorithms for challenging tasks in matrix product state tomography and provable learning of pushforward distributions.
Accuracy challenges for TR arise primarily from model selection and computational scaling. Latent regularization and adaptive rank minimization mitigate overfitting, but cannot eliminate it. Adaptive data manifold geometry allows for principled Riemannian optimization, and computational efficiency via block-randomized SGD improves performance on large-scale tensors.
Dimensionality reduction methods reduce the size of the underlying tensor by removing redundant modes from the decomposition. These techniques are essential for large-scale tensor processing. ALS variants with Gram/QR stabilization reduce intermediate data explosion and computational stability, enabling scalable iteration cost N11RN+1=R1