2019 Fiscal Year Final Research Report
Riemannian optimization and applications for high-dimensional large-scale data
| Project/Area Number |
16K00031
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Mathematical informatics
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| Research Institution | Waseda University (2019)
The University of Electro-Communications (2016-2018) |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Keywords | 最適化 / リーマン多様体 / 確率的勾配法 / テンソル / 大規模データ |
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| Outline of Final Research Achievements |
This research project proposed a novel Riemannian manifold preconditioning approach for the tensor completion problem with rank constraint. A novel Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and, considers the structured symmetry that exists in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian optimization on quotient manifolds to develop preconditioned nonlinear conjugate gradient and stochastic gradient descent algorithms for batch and online setups, respectively. Furthermore, a novel Riemannian extensions of the Euclidean stochastic gradient algorithm to a manifold search space have been proposed. The key challenges of averaging, adding, and subtracting multiple gradients are addressed with retraction and vector transport. For this algorithm, a global convergence analysis as well as a local linear rate convergence have been provided.
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| Free Research Field |
最適化理論，機械学習とその応用
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| Academic Significance and Societal Importance of the Research Achievements |
微分多様体上での最適化手法は，従来の凸解析最適化とは全く思想が異なり，近年研究が開始された極めて新たしい手法である．本研究では，最適化目的関数と制約を考慮した新しいリーマン計量を定義し，そこから全く新しい幾何空間（多様体）を提案・構築し，その上で効率的な最適化を試みるという新しい手法である．さらに，オンライン型最適化，分散型最適化への拡張は，また多様体上での確率的勾配法の検討は，新しい試みである．このことから，本研究方式の遂行は，学術的にも例がなく極めて独創的・革新的であり，且つ産業競争力強化にも資する研究内容である．
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