Intelligent Initialization and Adaptive Thresholding for Iterative Matrix Completion: Some Statistical and Algorithmic Theory forAdaptive-Impute

Abstract

Over the past decade, various matrix completion algorithms have been developed. Thresholded singular value decomposition (SVD) is a popular technique in implementing many of them. A sizable number of studies have shown its theoretical and empirical excellence, but choosing the right threshold level still remains as a key empirical difficulty. This paper proposes a novel matrix completion algorithm which iterates thresholded SVD with theoretically-justified and data-dependent values of thresholding parameters. The estimate of the proposed algorithm enjoys the minimax error rate and shows outstanding empirical performances. The thresholding scheme that we use can be viewed as a solution to a non-convex optimization problem, understanding of whose theoretical convergence guarantee is known to be limited. We investigate this problem by introducing a simpler algorithm, generalized-softImpute, analyzing its convergence behavior, and connecting it to the proposed algorithm.

Artificial intelligence enhanced two-dimensional nanoscale nuclear magnetic resonance spectroscopy

“…Inspired by unobserved entries fill strategy 21,22 and matrix completion based domain adaptation [23][24][25] , we propose to combine the deep learning network with the matrix completion algorithm 15 to develop our DLMC method for efficient NV spectrum map reconstruction. DL can learn very complex non-linear mapping from a partially filled spectrum map to its full-resolution map, with the DL network trained with simulation data; while the traditional matrix completion (MC) method is used for post-processing the DL output map to keep its low-rank property, thus further alleviate the domain shift problem.…”

Asymptotic theory for estimating the singular vectors and values of a partially-observed low rank matrix with noise

“…The literature has established a sizable body of algorithmic research (Rennie and Srebro (2005); Keshavan et al (2009); Cai et al (2010); Mazumder et al (2010); Hastie et al (2014); Cho et al (2015)) and theoretical results (Fazel (2002); Sre-1 This research is supported by NSF grant DMS-1309998 and ARO grant W911NF-15-1-0423. arXiv:1508.05431v2 [stat.ME] 1 May 2016May bro et al (2004; Candès and Recht (2009); Candès and Plan (2010); Keshavan et al (2010); Recht (2011); Gross (2011); Negahban et al (2011); Koltchinskii et al (2011a); Rohde et al (2011); Koltchinskii et al (2011b); Candès and Plan (2011); Negahban and Wainwright (2012); Cai and Zhou (2013); Davenport et al (2014); Chatterjee (2014)).…”