A least-squares strategy is proposed for representing a two-mode proximity matrix as an approximate sum of a small number of matrices that satisfy certain simple order constraints on their entries. The primary class of constraints considered define Q-forms (or anti-Q-forms) for a two-mode matrix, where after suitable and separate row and column reorderings, the entries within each row and within each column are nondecreasing (or nonincreasing) to a maximum (or minimum) and thereafter nonincreasing (or nondecreasing). Several other types of order constraints are also mentioned to show how alternative structures can be considered using the same computational strategy.

There are various optimization strategies for approximating, through the minimization of a least-squares loss function, a given symmetric proximity matrix by a sum of matrices each subject to some collection of order constraints on its entries. We extend these approaches to include components in the approximating sum that satisfy what are called the strongly-anti-Robinson (SAR) or circular strongly-anti-Robinson (CSAR) restrictions. A matrix that is SAR or CSAR is representable by a particular graph-theoretic structure, where each matrix entry is reproducible from certain minimum path lengths in the graph. One published proximity matrix is used extensively to illustrate the types of approximation that ensue when the SAR or CSAR constraints are imposed.