ON LEAST SQUARES EUCLIDEAN DISTANCE MATRIX APPROXIMATION AND COMPLETION DAVID I. CHU * , HUNTER C. BROWN † , AND MOODY T. CHU ‡ Abstract. The Euclidean distance matrix approximation problem as well as the completion problem have received a lot of attention in recent years because of their many important appli- cations. In contrast to the many interesting but often sophisticated algorithms proposed in the literature, this paper offers a relatively straightforward procedure that can tackle both problems by the same framework. The location vectors whose relative distance squares form the entries of the Euclidean distance matrix are used directly as parameters in the least squares formulation. It is shown how both gradient and Hessian of the objective function can be calculated explicitly in block form and, thus, can be assembled block by block according to designated locations. Highly effective conventional optimization techniques can then utilized to obtain the least squares solu- tion. The approach can be applied to a variety of problems arising in biological or engineering applications, including molecular structure analysis, protein folding problem, remote exploration and sensing, and antenna array processing. Key words. distance geometry, least squares approximation, matrix completion, molecular structure, protein folding, conformational analysis. 1. Introduction. The endeavor that, “given the distance and chirality con- strains which define (our state of knowledge of) a mobile molecule, find one or more conformations which satisfy them, or else prove that no such conformations exists,” has been referred to as the Fundamental Problem in Distance Geometry in [7]. The notion of distance geometry, initiated by Menger and Schoenbeng in the 1930’s, has been an area of active research because of its many important applications, includ- ing molecular conformation problems in chemistry [7], multidimensional scaling in behavioral sciences [10, 18], and multivariate analysis in statistics [20]. The article [13] is an excellent reference that expounds the framework of Euclidean distance geometry in general. More extensive discussion on the background and applications of distance geometry can be found in [7]. One of the most basic requisitions in the study of distance geometry is the information of inter-point relationships. Given n particles at locations p 1 ,..., p n in the space R m , the corresponding Euclidean distance matrix Q(p 1 ,..., p n )=[q ij ] is the n × n symmetric and nonnegative matrix whose entry q ij is defined by q ij = ‖p i − p j ‖ 2 , i, j =1,...,n, (1.1) * Department of Biomedical Engineering, Yale University, New Haven, CT 06520-8042. email: david.chu@yale.edu. † Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205. email: hcbrown2@unity.ncsu.edu ‡ Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205. email: chu@math.ncsu.edu. This research was supported in part by the National Science Foundation under grants DMS-9803759 and DMS-0073056. 1