MODEL ORDER SELECTION FOR THE SINGULAR-VALUE DECOMPOSITION AND THE DISCRETE KARHUNEN-LOEVE TRANSFORM USING A BAYESIAN-APPROACH

Bayesian model order selection is considered in relation to the singul ar value decomposition (SVD) and the discrete Karhunen-Loeve transform (DKLT). There are many applications of the SVD and DKLT where it is n ecessary to discard some of the small singular values that may represe nt corrupted signal information. Often this task is performed heuristi cally or in an ad hoc manner. The Bayesian approach to model order sel ection involves the determination of the evidence or the conditional p osterior probability of the model structure given the data; this frame work allows the relative probabilities of all possible candidate model s to be compared explicitly. Applied to the SVD, the evidence formulat ion enables the number of nonzero singular values (and hence the effec tive rank) of a singular or ill-conditioned matrix to be determined an alytically. For the DKLT, the evidence allows the determination of the optimal number of basis vectors to choose for the signal reconstructi on. In addition, the Bayesian method allows prior information such as physical smoothness constraints to be incorporated directly into the p roblem specification. Derivations of the evidence formulae are include d along with results that illustrate the usefulness of the method.