Constrained Low Rank Approximation of the Hermitian Nonnegative-Definite Matrix

In this paper, we consider a constrained low rank approximation problem: ( ) , min rank X p X X E = ∈Ω − , where E is a given complex matrix, p is a positive integer, and Ω is the set of the Hermitian nonnegative-definite least squares solution to the matrix equation AXA B ∗ = . We discuss the range of p and derive the corresponding explicit solution expression of the constrained low rank approximation problem by matrix decompositions. And an algorithm for the problem is proposed and the numerical example is given to show its feasibility.


Introduction
In the last few years, the structured low rank matrix approximation has been one of the topics of very active research in matrix theory and the applications. We know the empirical data collected in a matrix generally satisfy either the special structure properties or the desirable rank as is expected in the original system. Solving a low rank approximation of a general data matrix is an important task in many disciplines.
The structured low rank approximation problem can be written as follows: given a matrix E, a positive integer p, and a matrix class Ω , find a matrix X sa- which is concluded by M.T. Chu, et al. in 2003, see [1]. The structured low rank approximation problem and applications associated with different constraint set Ω have been extensively studied. Generally speaking, Ω is with linear structure, e.g., symmetric Toeplitz, Hankel, upper Hessenberg, Slyvester, correlation, CP, or banded matrices with fixed bandwidth, etc., which can be referred to [1]- [11]. For examples, in the process of noise removal in signal processing or image enhancement, the underlying covariance matrix with Toeplitz or block Toeplitz structure, see [5] [6]. In the model reduction problem in encoding and filter design, the underlying structure matrix is Hankel, see [7] [12]. In computer algebra, approximating the greatest common divisor of polynomials can be formulated as a low rank approximation problem with Sylvester structure, see [9].
In computing the nearest Euclidean distance, the symmetric nonnegative matrix of rank 5 is a necessary condition for the approximation, see [10]. In the asset portfolio, the structured low rank approximation is about correlation matrix, see [2].

The General Solution to Problem 1
In this section, we consider the general solution of Problem 1 proposed in where the diagonal elements of the diagonal matrix . By (2)-(4), the unitary invariance of Frobenius norm, and assume , we get Then min is solvable. We obtain the following result.
. Suppose the matrix 0 B has s positive eigenvalues, then 0 B has the following decomposition . And the eigendecomposition of ( ) and a diagonal matrix Then the Hermitian nonnegative definite solution to the least squares solution of AXA B * = can be expressed as Proof. According to above analysis, the matrix equation has the form of (3.7). Substituting (3.10) into (3.4), we get By Lemma 2.1, it is equivalent to By Theorem 3.1, for X ∈ Ω defined in Problem 1, X has the form of (3.9) with , The eigenvalue decomposition of 1 Q E Q * has the form of  , Then the solution X in Problem 1 is given as follows from the three cases: 1) If p s = , then 2) If s p s t < ≤ + , then 1 13 where ( ) Proof. According to Theorem 3.1, if X ∈ Ω , then X has the formula of (3.9), where ( )   2) When p s t − > , we take 2 Y is of (3.21). In this case, we obtain the representation of the solution to Problem 1, which has the form of (3.19) with (3.21).

2) If
The proof is completed.

Numerical Examples
We in this section propose an algorithm for finding the solution of Problem 1 and give illustrative numerical example.
We get   Remark The Algorithm 4.1 can be applied for small sizes of matrices in Problem 1. In the process of computation, it only involves once singular value decomposition and four times eigendecompositions. Hence it has good numerical stability.

Conclusion
In this paper, we have studied the constrained low rank approximation problem range. And the algorithm has been presented and the numerical example shows its feasibility.