Iterative projection strategies for the least‐squares fitting of tree structures to proximity data

A least squares optimization strategy is first reviewed and applied to the task of fitting a given collection of symmetric proximity values defined between the objects from one set by a collection of reconstructed proximity values, satisfying a fixed set of constraints, generated from some specified graph‐theoretic structure, such as an ultrametric or an additive tree, selected for representing the objects. Our method uses itcrative projection on to closed convex sets defined by the collection of given constraints characterizing the structural representation specified, and in contrast to least squares optimization methods that impose such constraints through the use of penalty functions, avoids the use of the latter, as well as the implementation of any gradient‐based optimization technique. Secondly, just as various penalty‐function/ gradient‐based optimization techniques have been turned into heuristic search strategies for such particular structures of interest as ultrametrics or additive trees, the use of iterative projection is suggested as a general heuristic search strategy for locating the best structural representations to impose in the first place, where the collection of constraints used may vary over the course of the optimization process. Our evaluation of the expected results uses several data sets previously analysed in the literature. Finally, several other applications of iterative projection as a heuristic optimization technique are discussed, including the consideration of data beyond that of a single symmetric proximity matrix (for example, extensions to two‐mode proximity matrices, i.e. between two distinct object sets, and to three‐way proximity matrices either symmetric or not), and to representations based on sums of matrices where each is constrained separately to conform to some desired representational structure.