On homogeneous least-squares problems and the inconsistency introduced by mis-constraining

The term "homogeneous least-squares" refers to models of the form Ya≈0, where Y is some data matrix, and a is an unknown parameter vector to be estimated. Such problems are encountered, e.g., when modeling auto-regressive (AR) processes. Naturally, in order to apply a least-squares (LS) solution to such models, the parameter vector a has to be somehow constrained in order to avoid the trivial solution a=0. Usually, the problem at hand leads to a "natural" constraint on a. However, it will be shown that the use of some commonly applied constraints, such as a quadratic constraint, can lead to inconsistent estimates of a. An explanation to this apparent discrepancy is provided, and the remedy is shown to lie with a necessary modification of the LS criterion, which is specified for the case of Gaussian model-errors. As a result, the modified LS minimization becomes a highly non-linear problem. For the case of quadratic constraints in the context of AR modeling, the resulting minimization involves the solution of an equation reminiscent of a "secular equation". Numerically appealing solutions to this equation are discussed.