Low-rank matrix and tensor completion using graph-based regularization
(2021)
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- Abstract
- Matrix and tensor completion arise in many different real-world applications related to the inference or acquisition of data. In this thesis, we investigate matrix and tensor completion in the aspects of models, algorithms and regularization. With respect to models and algorithms, we tackle these problems via low-rank matrix optimization using a least-squares model. The rank-constrained models are known not only for their computational efficiency but also the capability of extracting the most important information in the data that has an intrinsically low rank. By exploiting manifold structures in these rank-constrained models, we develop Riemannian algorithms and analyse the convergence properties of these algorithms. In particular, we study algorithms designed via Riemannian preconditioning and provide novel results that explain their efficiency on the quotient manifold of fixed-rank matrices. We also investigate the usage of a certain graph-based regularizer in the matrix and tensor completion problems. The penalty function in the graph-based regularization is an extension of the Frobenius norm in the problem variable, in the sense that the entries in the variable are not penalized with uniform weights but with respect to the intensities of some pairwise relations encoded in a graph Laplacian matrix. The graph Laplacian-based penalty function promotes matrix or tensor solutions such that the pairwise similarities among its entries conform to the underlying graph structure. Besides traditional graph models, we propose a graph learning algorithm for learning a desired graph structure from data.
- Affiliations
UCLouvain SST/ICTM/INMA - Pôle en ingénierie mathématique
Citations
Dong, S. (2021). Low-rank matrix and tensor completion using graph-based regularization. https://hdl.handle.net/2078.5/110433