A Note on Solution of the Nearest Correlation Matrix Problem by von Neumann Matrix Divergence

In the extant literature a suggestion has been made to solve the nearest correlation matrix problem by a modified von Neumann approximation. In this paper it has been shown that obtaining the nearest positive semi-definite matrix from a given non-positive-semi-definite correlation matrix by such 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. When the given matrix is non-positive-semi-definite (Q), then only we seek a positive semi-definite matrix closest to it. Then the proposed procedure fails as we cannot find log(Q). But, if we replace negative eigenvalues of Q by a zero/near-zero values, we obtain a positive semi-definite matrix, 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 method most suitable to dealing with larger matrices. Yet, we provide an alternative algorithm (and a Fortran program) to obtain a positive (semi-)definite matrix that performs (speed as well as accuracy-wise) much better