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Local low-rank approach to nonlinear matrix completion
EURASIP Journal on Advances in Signal Processing ( IF 1.9 ) Pub Date : 2021-02-12 , DOI: 10.1186/s13634-021-00717-7
Ryohei Sasaki , Katsumi Konishi , Tomohiro Takahashi , Toshihiro Furukawa

This paper deals with a problem of matrix completion in which each column vector of the matrix belongs to a low-dimensional differentiable manifold (LDDM), with the target matrix being high or full rank. To solve this problem, algorithms based on polynomial mapping and matrix-rank minimization (MRM) have been proposed; such methods assume that each column vector of the target matrix is generated as a vector in a low-dimensional linear subspace (LDLS) and mapped to a pth order polynomial and that the rank of a matrix whose column vectors are dth monomial features of target column vectors is deficient. However, a large number of columns and observed values are needed to strictly solve the MRM problem using this method when p is large; therefore, this paper proposes a new method for obtaining the solution by minimizing the rank of the submatrix without transforming the target matrix, so as to obtain high estimation accuracy even when the number of columns is small. This method is based on the assumption that an LDDM can be approximated locally as an LDLS to achieve high completion accuracy without transforming the target matrix. Numerical examples show that the proposed method has a higher accuracy than other low-rank approaches.



中文翻译:

非线性矩阵完成的局部低秩方法

本文讨论了矩阵完成的问题,其中矩阵的每个列向量都属于低维可分流形(LDDM),目标矩阵是高秩或全秩。为了解决这个问题,提出了基于多项式映射和矩阵秩最小化(MRM)的算法。此类方法假设目标矩阵的每个列向量均作为低维线性子空间(LDLS)中的向量生成,并映射到p阶多项式,并且其列向量为的第d个单项特征的矩阵的秩目标列向量不足。然而,需要大量的列和观察到的值,以使用此方法时严格解决MRM问题p大;因此,本文提出了一种在不变换目标矩阵的情况下通过最小化子矩阵的秩来获得解的新方法,从而即使在列数较小的情况下也可以获得较高的估计精度。此方法基于以下假设:LDDM可以局部近似为LDLS,以实现高完成精度而无需变换目标矩阵。数值算例表明,该方法比其他低秩方法具有更高的精度。

更新日期:2021-02-12
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