The singular value decomposition and long and short spaces of noisy matrices

Using geometrical, algebraic, and statistical arguments, it is clarified why and when the singular value decomposition is successful in so-called subspace methods. First the concepts of long and short spaces are introduced, and a fundamental asymmetry in the consistency properties of the estimates is discussed. The model, which is associated with the short space, can be estimated consistently, but the estimates of the original data, which follow from the long space, are always inconsistent. An expression is found for the asymptotic bias in terms of canonical angles, which can be estimated from the data. This allows all equivalent reconstructions of the original signals to be described as a matrix ball, the center of which is the minimum variance estimate. Remarkably, the canonical angles also appear in the optimal weighting that is used in weighted subspace fitting approaches. The results are illustrated with a numerical simulation. A number of examples are discussed