Distance Matrix Completion by Numerical Optimization

10 to a distance matrix|if a completion does exist, then typically we want to nd it. Although the numerical optimization approach is not deenitive in the sense of Johnson and Tarazaga 8], it is constructive in the sense that it produces a sequence of dissimilarity matrices that converges to a completion. Beyond its ability to construct (approximate) completions, the compelling advantage of the numerical optimization approach is its generality. First, graph-theoretic characterizations of com-pletability have usually supposed that it has already been determined that the partial dissimilarity matrix 0 is a partial distance matrix, i.e. a matrix such that every complete principal submatrix is a distance matrix. Our approach requires no such veriication, which may be quite expensive for large problems. Second, graph-theoretic characterizations of completability have not addressed problems that restrict the dimension of the connguration X for which D(X) completes 0. Our approach easily incorporates this restriction, which is crucial in such important applications as molecular confor-mation. The price that we pay for this generality is a loss of convexity, which means that we are not assured that methods for nding local solutions will necessarily nd global solutions. However, the results in Section 5 suggest that local methods often do nd global solutions. Third, graph-theoretic characterizations of completability have supposed that the known elements of the partial dissimilarity matrix 0 are xed, whereas our approach allows these elements to vary subject to bound constraints. Such variation is natural when 0 is obtained by measurement, as in molecular conformation problems. Finally, Johnson and Tarazaga 8] have noted that graph-theoretic approaches to the distance matrix and positive semideenite completion problems are closely related yet substantially diierent. In contrast, the numerical optimization approach to distance matrix completion is trivially modiied for positive semideenite completion by the simple device of discarding the linear mapping and minimizing F 3 directly. In conclusion, we regard the numerical optimization approach to distance matrix completion as complementary to, rather than competitive with, the graph-theoretic characterizations of com-pletability obtained by other researchers. Each approach has its respective strengths and weaknesses. for organizing that forum and inviting me to participate in it, and Timothy Havel for his comments on an earlier version of this report. 9 Evidently, extremely small values of the objective function are required to recover good approximations of the true squared distances. This suggests that the objective function has very little curvature near solutions and …