Analytical solution for the joint max-disparity partition problem

Determine whether an analytical solution exists for the joint max-disparity partition problem, in which the binary partition and the shared rank-d principal-component projection are jointly selected to maximise the difference in groups’ average reconstruction losses.

Background

The paper formulates the max-disparity partition problem as selecting a binary partition of the rows of a data matrix so as to maximise the absolute difference between the groups’ average reconstruction losses under a shared rank-d PCA projection. The shared projection depends on the partition because it is computed from the joint data assigned to both groups, while each group’s reference loss is determined by its own optimal rank-d PCA subspace.

The dependence between the partition and the projection makes the optimisation combinatorial and difficult to solve directly. The paper characterises the problem as NP-hard in general and consequently develops practical heuristic methods, including a greedy Fiduccia–Mattheyses-inspired procedure, a fixed-projection sorting baseline, and simulated annealing. An analytical solution to the joint optimisation remains unresolved in the cited passage.