A random algorithm for low-rank decomposition of large-scale matrices with missing entries

11/04/2014
by   Yiguang Liu, et al.
0

A Random SubMatrix method (RSM) is proposed to calculate the low-rank decomposition of large-scale matrices with known entry percentage ρ. RSM is very fast as the floating-point operations (flops) required are compared favorably with the state-of-the-art algorithms. Meanwhile RSM is very memory-saving. With known entries homogeneously distributed in the given matrix, sub-matrices formed by known entries are randomly selected. According to the just proved theorem that subspace related to smaller singular values is less perturbed by noise, the null vectors or the right singular vectors associated with the minor singular values are calculated for each submatrix. The vectors are the null vectors of the corresponding submatrix in the ground truth of the given large-scale matrix. If enough sub-matrices are randomly chosen, the low-rank decomposition is estimated. The experimental results on random synthetical matrices with sizes such as 131072X1024 and on real data sets indicate that RSM is much faster and memory-saving, and, meanwhile, has considerable high precision achieving or approximating to the best.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 7

04/13/2018

On the detection of low rank matrices in the high-dimensional regime

We address the detection of a low rank n× ndeterministic matrix X_0 from...
06/08/2011

Large-Scale Convex Minimization with a Low-Rank Constraint

We address the problem of minimizing a convex function over the space of...
10/26/2017

Optimal Shrinkage of Singular Values Under Random Data Contamination

A low rank matrix X has been contaminated by uniformly distributed noise...
05/27/2021

Entrywise Estimation of Singular Vectors of Low-Rank Matrices with Heteroskedasticity and Dependence

We propose an estimator for the singular vectors of high-dimensional low...
07/21/2019

Low Rank Approximation of a Matrix at Sub-linear Cost

A matrix algorithm performs at sub-linear cost if it uses much fewer flo...
07/07/2021

A Generalized CUR decomposition for matrix pairs

We propose a generalized CUR (GCUR) decomposition for matrix pairs (A, B...
04/30/2021

Spiked Singular Values and Vectors under Extreme Aspect Ratios

The behavior of the leading singular values and vectors of noisy low-ran...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.