Positive semidefinite matrix completion (PSDMC) aims to recover positive semidefinite and low-rank matrices from a subset of entries of a matrix. It is widely applicable in many fields, such as statistic analysis and system control. This task can be conducted by solving the nuclear norm regularized linear least squares model with positive semidefinite constraints. We apply the widely used alternating direction method of multipliers to solve the model and get a novel algorithm. The applicability and efficiency of the new algorithm are demonstrated in numerical experiments. Recovery results show that our algorithm is helpful.
Matrix completion (MC) is the process of recovering the unknown or missing elements of a matrix. Under certain assumptions on the matrix, for example, low-rank or approximately low-rank, the incomplete matrix can be reconstructed very well [
Recently, there have been extensive research on the problems of low-rank matrix completion (LRMC). The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear constraints. However, it is NP-hard (nondeterministic polynomial-time hard) due to the combinatorial nature of the rank function. References [
In practice, the completed matrix is often required to be positive semidefinite. For example, covariance matrix and its inverse: precision matrix of statistic analysis, are both positive semidefinite. Recently, there have been extensive research on high-dimensional covariance matrix estimation. They all motivate the development of positive semidefinite matrix completion (PSDMC). Reference [
Our main contribution in this work is the development of an efficient algorithm for PSDMC. First of all, we present the nuclear norm regularized linear least squares model with positive semidefinite constraints. Because of its robustness, we choose it as the model of PSDMC in this paper. The structure of the model suggests an alternating minimization scheme, which is very suitable for solving large-scale problems. We give an exact ADMM-based algorithm, whose subproblems are solved exactly. We test the new ADMM-based algorithm on two kinds of problems: random matrix completion problems and random low-rank approximation problems. Numerical experiments show that all our proposed algorithm outputs have satisfactory results. The paper is organized as follows. Section
The following notations will be used throughout this paper. Uppercase (lowercase) letters are used for matrices (column vectors). All vectors are column vectors; the subscript
The matrix completion problem of recovering a positive semidefinite low-rank matrix from a subset of its entries is
Let
From the definition of
If the known elements of the matrix
Actually, model (
In this subsection, we present an algorithm developed for model (
The alternating direction method of multipliers for model (
By rearranging the terms of (
By rearranging the terms of (
In short, ADMM applied to model (
From the above considerations, we arrive at Algorithm
( ( ( (
The convergence of ADMM and its variants for convex problems has been studied extensively. The interested reader is referred to [
In this section, we report on the application of our proposed ADMM-based algorithm to a series of matrix problems to demonstrate its ability. To illustrate the performance of our algorithmic approaches combined with different procedures, we test the following two solvers. ADMM-VI; Algorithm ADMM-SDP; Algorithm
We implement our algorithms in MATLAB. All the experiments are performed on a 2.20 GHz Intel Pentium PC with 6.0 GHz of memory and MATLAB 2012b.
We test the above two solvers on random positive semidefinite matrix problems. We do numerical experiments by the following procedure. Firstly, we create a low-rank positive semidefinite matrix
The most important algorithmic parameters in Algorithm
Numerical results of different values of
SR | | | | | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Iter | Rank | Err | Iter | Rank | Err | Iter | Rank | Err | Iter | Rank | Err | |
0.25 | 67 | 297 | | 57 | 20 | | 44 | 20 | | 72 | 20 | |
0.35 | 50 | 274 | | 38 | 20 | | 32 | 20 | | 51 | 20 | |
0.45 | 39 | 194 | | 28 | 20 | | 26 | 20 | | 39 | 20 | |
0.55 | 33 | 137 | | 22 | 20 | | 21 | 20 | | 31 | 20 | |
0.65 | 29 | 97 | | 17 | 20 | | 17 | 20 | | 26 | 20 | |
0.75 | 26 | 55 | | 15 | 20 | | 14 | 20 | | 22 | 20 | |
0.85 | 23 | 20 | | 12 | 20 | | 11 | 20 | | 19 | 20 | |
We can use continuation technique employed in [
The matrix
The computational results of positive semidefinite matrix completion are presented in Table
Numerical results on medium randomly created matrix completion problems.
( | ADMM-VI | ADMM-SDP | ||||
---|---|---|---|---|---|---|
Iter | Rank | Err | Iter | Rank | Err | |
(500, 10, 0.45) | 24 | 10 | | 29 | 10 | |
(500, 10, 0.65) | 16 | 10 | | 19 | 10 | |
(500, 10, 0.85) | 10 | 10 | | 13 | 10 | |
(2000, 10, 0.45) | 27 | 10 | | 26 | 10 | |
(2000, 10, 0.65) | 18 | 10 | | 17 | 10 | |
(2000, 10, 0.85) | 11 | 10 | | 13 | 10 | |
(2000, 20, 0.45) | 27 | 20 | | 28 | 20 | |
(2000, 20, 0.65) | 16 | 20 | | 18 | 20 | |
(2000, 20, 0.85) | 11 | 20 | | 13 | 20 | |
The authors declare that they have no competing interests.
The authors would like to thank Professor Zaiwen Wen for the discussions on matrix completion. They thank Professor Xiaoming Yuan for offering the original codes of ADMM-VI. The work was supported in part by Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents (no. 2015RCJJ056).