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Register a new userOn Algorithms for and Computing with the Tensor Ring Decomposition
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Added by
Oscar Mickelin
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Tensor decompositions such as the canonical format and the tensor train format have been widely utilized to reduce storage costs and operational complexities for high-dimensional data, achieving linear scaling with the input dimension instead of exponential scaling. In this paper, we investigate even lower storage-cost representations in the tensor ring format, which is an extension of the tensor train format with variable end-ranks. Firstly, we introduce two algorithms for converting a tensor in full format to tensor ring format with low storage cost. Secondly, we detail a rounding operation for tensor rings and show how this requires new definitions of common linear algebra operations in the format to obtain storage-cost savings. Lastly, we introduce algorithms for transforming the graph structure of graph-based tensor formats, with orders of magnitude lower complexity than existing literature. The efficiency of all algorithms is demonstrated on a number of numerical examples, and in certain cases, we demonstrate significantly higher compression ratios when compared to previous approaches to using the tensor ring format.
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In this work, we study the tensor ring decomposition and its associated numerical algorithms. We establish a sharp transition of algorithmic difficulty of the optimization problem as the bond dimension increases: On one hand, we show the existence of spurious local minima for the optimization landscape even when the tensor ring format is much over-parameterized, i.e., with bond dimension much larger than that of the true target tensor. On the other hand, when the bond dimension is further increased, we establish one-loop convergence for alternating least square algorithm for tensor ring decomposition. The theoretical results are complemented by numerical experiments for both local minimum and one-loop convergence for the alternating least square algorithm.
We propose a new algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this as a system of polynomial equations allows us to leverage recent numerical linear algebra tools from computational algebraic geometry. We describe the complexity of our algorithm in terms of the multigraded regularity of a multihomogeneous ideal. We prove effective bounds for many formats and ranks and conjecture a general formula. These bounds are of independent interest for overconstrained polynomial system solving. Our experiments show that our algorithm can outperform state-of-the-art algebraic algorithms by an order of magnitude in terms of accuracy, computation time, and memory consumption.
Tensor completion can estimate missing values of a high-order data from its partially observed entries. Recent works show that low rank tensor ring approximation is one of the most powerful tools to solve tensor completion problem. However, existing algorithms need predefined tensor ring rank which may be hard to determine in practice. To address the issue, we propose a hierarchical tensor ring decomposition for more compact representation. We use the standard tensor ring to decompose a tensor into several 3-order sub-tensors in the first layer, and each sub-tensor is further factorized by tensor singular value decomposition (t-SVD) in the second layer. In the low rank tensor completion based on the proposed decomposition, the zero elements in the 3-order core tensor are pruned in the second layer, which helps to automatically determinate the tensor ring rank. To further enhance the recovery performance, we use total variation to exploit the locally piece-wise smoothness data structure. The alternating direction method of multiplier can divide the optimization model into several subproblems, and each one can be solved efficiently. Numerical experiments on color images and hyperspectral images demonstrate that the proposed algorithm outperforms state-of-the-arts ones in terms of recovery accuracy.
We introduce the Subspace Power Method (SPM) for calculating the CP decomposition of low-rank even-order real symmetric tensors. This algorithm applies the tensor power method of Kolda-Mayo to a certain modified tensor, constructed from a matrix flattening of the original tensor, and then uses deflation steps. Numerical simulations indicate SPM is roughly one order of magnitude faster than state-of-the-art algorithms, while performing robustly for low-rank tensors subjected to additive noise. We obtain rigorous guarantees for SPM regarding convergence and global optima, for tensors of rank up to roughly the square root of the number of tensor entries, by drawing on results from classical algebraic geometry and dynamical systems. In a second contribution, we extend SPM to compute De Lathauwers symmetric block term tensor decompositions. As an application of the latter decomposition, we provide a method-of-moments for generalized principal component analysis.
In this paper, we consider the tensor completion problem, which has many researchers in the machine learning particularly concerned. Our fast and precise method is built on extending the $L_{2,1}$-norm minimization and Qatar Riyal decomposition (LNM-QR) method for matrix completions to tensor completions, and is different from the popular tensor completion methods using the tensor singular value decomposition (t-SVD). In terms of shortening the computing time, t-SVD is replaced with the method computing an approximate t-SVD based on Qatar Riyal decomposition (CTSVD-QR), which can be used to compute the largest $r left(r>0 right)$ singular values (tubes) and their associated singular vectors (of tubes) iteratively. We, in addition, use the tensor $L_{2,1}$-norm instead of the tensor nuclear norm to minimize our model on account of it is easy to optimize. Then in terms of improving accuracy, ADMM, a gradient-search-based method, plays a crucial part in our method. Numerical experimental results show that our method is faster than those state-of-the-art algorithms and have excellent accuracy.
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