Metric unfolding: Data requirement for unique solution & clarification of Schönemann's algorithm

Abstract

The object of this paper is to clarify Schönemann's unfolding algorithm and, in particular, to make it clear that the equations numbered (3.2) in Schönemann's [1970] article, which define Schönemann's solutions, are not a complete set of restraints for the purpose of defining metric unfoldings. Namely, Schönemann has transformed the original equations which define an unfolding to a set of linear and non-linear equations of which he uses only the linear equations to define his solutions. Given infallible data (solution(s) exist) Schönemann's solutions will include the correct solutions. If enough data are available so that there are enough linear equations to uniquely determine a single solution, then Schönemann's solution will coincide with the correct solution. LetP andQ denote the number of elements in the two sets of points, the interset distances of which are specified by the data in the unfolding problem. Letm denote the dimensionality of the Euclidean space into which these points are to be imbedded. If only the linear equations, numbered (18) herein, are to be used, then Schönemann gives the following data requirement for the solution to be uniquely determined: Max {P − 1,Q − 1} ≥m(m + 3)/2. If the full set of linear and nonlinear equations (18–20) are used, then the amount of data required for a solution to be locally unique is relaxed toP +Q − 1 ≥m(m + 3)/2. Both of these results assume that the equations are independent, which has not been proved.