2021 Fiscal Year Research-status Report
Efficient Numerical Solution for Constrained Tensor Ring Decomposition: A Theoretical Convergence Analysis and Applications
| Project/Area Number |
20K19749
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| Research Institution | The Institute of Statistical Mathematics |
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Principal Investigator |
鄭 寧 統計数理研究所, 統計的機械学習研究センター, 特任研究員 (60859122)
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| Project Period (FY) |
2020-04-01 – 2023-03-31
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| Keywords | tensor learning / randomized algorithm / graph regularization / linear system / iterative methods |
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| Outline of Annual Research Achievements |
The main research achievements can be summarized into two parts. First, we present a novel graph regularized nonnegative tensor ring decomposition for multiway representation learning. Tensor ring (TR) decomposition is a powerful tool for exploiting the low-rank nature of multiway data and has demonstrated great potential in a variety of important applications. The proposed models impose nonnegativity on the core tensors and the latter is additionally able to capture manifold geometry information of tensor data, and significantly extend the applications of TR decomposition for nonnegative multiway representation learning. Accelerated proximal gradient based methods are derived for solving the tensor networks. The experimental result demonstrate that the proposed algorithms can extract parts-based basis with rich colors and rich lines from tensor objects that provide more interpretable and meaningful representation, and hence yield better performance than the state-of-the-art tensor based methods in clustering and classification tasks. This work has been accepted by IEEE Transactions on Cybernetics. Second, a class of restarted randomized surrounding methods are presented for solving the linear equations. Theoretical analysis prove that the proposed method converges under the randomized row selection rule and the expectation convergence rate is also addressed. Numerical experiments further demonstrate that the proposed algorithms outperform the existing method for over-determined and under-determined linear equation, as well as in the application of image processing.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
The research work is continued as initially planned and the achievement and progress are significant. We propose the nonnegative tensor ring decomposition (NTR), which expresses the information of each dimension of tensor data by corresponding 3rd-order core tensors. The NTR can extract the parts-based basis with rich colors and rich lines of tensor objects, which can provide more interpretable and meaningful representation for physical signals. We also combine the graph regularization with NTR to develop GNTR, which perfectly inherits the advantages of NTR and enables the extracted data representation to preserve the manifold geometry information for tensor data. The experimental results demonstrated the effectiveness of our proposed algorithm. The parts-based basis extracted of our algorithms is rich colors and rich lines that provide more interpretable and meaningful representation for physical signals.
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| Strategy for Future Research Activity |
Based on the previous work on tensor network and efficient randomized algorithms, we consider the tensor based linear discrete ill-posed inverse problem on point cloud. Reconstruction of a continuous surface of two-dimensional manifold from its raw, discrete point cloud observation is a long-standing problem in computer vision and graphics research. The problem is technically ill-posed, and becomes more difficult considering that various sensing imperfections would appear in the point clouds obtained by practical depth scanning. We will propose optimization model and impose necessary constraints to capture the geometrical information, such as graph structure, nonnegativity, sparsity, etc. In addition, we will consider randomized algorithms on constrained lp-lq problem with nonnegative constraints.
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| Causes of Carryover |
The budget will be used for necessary numerical experiments related computational products, attending conference and workshop fee, and other academic activities. We will also invite related international and domestic experts to present talks and to cooperate on related topics.
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Research Products
(2 results)
