Tensor Ring Decomposition

TL;DR

This work introduces tensor ring (TR) decomposition, a generalization of tensor train that uses circular trace-based products of 3rd-order cores to achieve permutation-invariant, efficient representations of high-order tensors. It develops four learning algorithms (TR-SVD, TR-ALS, TR-ALSAR, TR-BALS) to compute TR representations and analyzes the operational properties that enable scalable multilinear algebra on TR cores. The paper clarifies TR's relationships to CP, Tucker, and TT models, showing TR can offer stronger representation power with flexible rank distributions. Experimental results on synthetic data and real datasets (COIL-100, KTH) demonstrate favorable compression and competitive predictive performance, highlighting TR’s practical impact for large-scale tensor analysis.

Abstract

Tensor networks have in recent years emerged as the powerful tools for solving the large-scale optimization problems. One of the most popular tensor network is tensor train (TT) decomposition that acts as the building blocks for the complicated tensor networks. However, the TT decomposition highly depends on permutations of tensor dimensions, due to its strictly sequential multilinear products over latent cores, which leads to difficulties in finding the optimal TT representation. In this paper, we introduce a fundamental tensor decomposition model to represent a large dimensional tensor by a circular multilinear products over a sequence of low dimensional cores, which can be graphically interpreted as a cyclic interconnection of 3rd-order tensors, and thus termed as tensor ring (TR) decomposition. The key advantage of TR model is the circular dimensional permutation invariance which is gained by employing the trace operation and treating the latent cores equivalently. TR model can be viewed as a linear combination of TT decompositions, thus obtaining the powerful and generalized representation abilities. For optimization of latent cores, we present four different algorithms based on the sequential SVDs, ALS scheme, and block-wise ALS techniques. Furthermore, the mathematical properties of TR model are investigated, which shows that the basic multilinear algebra can be performed efficiently by using TR representaions and the classical tensor decompositions can be conveniently transformed into the TR representation. Finally, the experiments on both synthetic signals and real-world datasets were conducted to evaluate the performance of different algorithms.

Figures (5)

• Figure 1: A graphical representation of the tensor ring decomposition
• Figure 2: Highly oscillated functions. The left panel is $f_1(x)=(x+1)sin(100(x+1)^2)$. The middle panel is Airy function: $f_2(x)=x^{-\frac{1}{4}} sin(\frac{2}{3}x^{\frac{3}{2}})$. The right panel is Chirp function $f_3(x)=sin\frac{x}{4}cos(x^2)$.
• Figure 3: The reconstruction of Coil-100 dataset by using TRSVD. The top row shows the original images, while the reconstructed images are shown from the second to sixth rows corresponding to $\epsilon$=0.1, 0.2, 0.3, 0.4, 0.5, respectively.
• Figure 4: Video dataset consists of six types of human actions performed by 25 subjects in four different scenarios. From the top to bottom, six video examples corresponding to each type of actions are shown.
• Figure 5: The six examples of reconstructed video sequences by TR-BALS with $\epsilon=0.27$, $r_{max}=24$, $\bar{r}=13.3$. The classification accuracy is 87.0% by using the cores obtained from TR-BALS.

Theorems & Definitions (14)

• Theorem 2.1
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.4
• proof
• Theorem 3.5
• proof
• Theorem 4.1
• proof