Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices

We present a heuristic for minimizing the rank of a positive semidefinite matrix over a convex set. We use the logarithm of the determinant as a smooth approximation for rank, and locally minimize this function to obtain a sequence of trace minimization problems. We then present a lemma that relates the rank of any general matrix to that of a corresponding positive semidefinite one. Using this, we readily extend the proposed heuristic to handle general matrices. We examine the vector case as a special case, where the heuristic reduces to an iterative l₁-norm minimization technique. As practical applications of the rank minimization problem and our heuristic, we consider two examples: minimum-order system realization with time-domain constraints, and finding lowest-dimension embedding of points in a Euclidean space from noisy distance data.