A Note on the Solution of the Nearest Correlation Matrix Problem by Von Neumann Matrix Divergence
The IUP Journal of Computational Mathematics, Vol. V, No. 1, pp. 7-13, March 2012
Posted: 1 Oct 2012
Date Written: October 1, 2012
Abstract
In the extant literature a suggestion has been made to solve the nearest correlation matrix problem by a modified von Neumann approximation. This paper shows that obtaining the nearest positive semi-definite matrix from a given non-positivesemi-definite correlation matrix by such a method is either infeasible or suboptimal. First, if a given matrix is already positive semi-definite, there is no need to obtain any other positive semi-definite matrix closest to it. But when the given matrix is non-positive-semi-definite (Q), then a positive semi-definite matrix closest to it is needed. Then the proposed procedure fails in the absence of log(Q). But if the negative eigenvalue of Q is replaced by a zero/near-zero value, a positive semidefinite matrix is obtained, but it is not nearest to the Q matrix; there are indeed other procedures to obtain better approximation. However, the modified von Neumann approximation method yields results (although sub-optimal) and is, perhaps, one of the fastest methods most suitable to deal with larger matrices. Yet, an alternative algorithm (Fortran program is provided to obtain a positive (semi-) definite matrix that performs (speed as well as accuracy-wise) much better.
Keywords: nearest correlation matrix problem, positive semidefinite, non-positivesemi- definite, von Neumann divergence, Bergman divergence
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