2020 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 | preconditioning / randomized / consistent / linear systems |
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| Outline of Annual Research Achievements |
In order to accelerate the convergence of iterative methods and save the storage
requirement, inner iterations can be applied as a preconditioner inside the Krylov subspace methods instead of applying preconditioning matrices explicitly. Such techniques are often called inner-outer iteration methods. The implicit preconditioning techniques are widely used in numerical linear algebra, optimization and machine learning areas.
Recently, We propose using greedy and randomized Kaczmarz inner-iterations as preconditioners for the right-preconditioned flexible GMRES method to solve consistent linear systems, with a parameter tuning strategy for adjusting the number of inner iterations and the relaxation parameter. We also present theoretical justifications of the right-preconditioned flexible GMRES for solving consistent linear systems. Numerical experiments on overdetermined and underdetermined linear systems show that the proposed method is superior to the GMRES method preconditioned by normal error successive overrelaxation inner iterations in terms of total CPU time.
In addition, we consider solving large sparse symmetric singular linear systems. We first introduce an algorithm for right preconditioned minimum residual (MINRES) and prove that its iterates converge to the preconditioner weighted least squares solution without breakdown for an arbitrary right‐hand‐side vector and an arbitrary initial vector even if the linear system is singular and inconsistent.
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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
As planned, the research achievements are accepted by the SIAM Journal on Scientific Computing and will be published online soon. At the next stage, we will continue to research on the randomized preconditioning technique and randomized algorithms for matrix/tensor based optimization problem.
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| Strategy for Future Research Activity |
1. We will consider the solution of large scale tensor ring decomposition via randonmized algorithm, which select the updated tensor core at each iteration step by rank based probability distribution. We will main focus on the theoretical analysis of the tentative algorithms.
2. The randomized preconditioning technique will be further explored by sketch and project framework. The implicit preconditioning will be achieved by the inner iteration rather than the traditionatiol matrix decomposition methods.
3. We will present the research work on the domestic and international online conference and hope to receive valuable feedback from the experts.
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| Causes of Carryover |
The main amount to be used next fiscal year is:
- Academic annual membership fee for JSIAM (Japan Society of Industrial and Applied Mathematics) and SIAM;
- budget for numerical experiment related softwares and hardwares.
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Research Products
(5 results)
