Generalizable Singular Value Decomposition for Ill-posed Datasets

We demonstrate that statistical analysis of ill-posed data sets is subject to a bias, which can be observed when projecting indepen(cid:173) dent test set examples onto a basis defined by the training exam(cid:173) ples. Because the training examples in an ill-posed data set do not fully span the signal space the observed training set variances in each basis vector will be too high compared to the average vari(cid:173) ance of the test set projections onto the same basis vectors. On basis of this understanding we introduce the Generalizable Singu(cid:173) lar Value Decomposition (GenSVD) as a means to reduce this bias by re-estimation of the singular values obtained in a conventional Singular Value Decomposition, allowing for a generalization perfor(cid:173) mance increase of a subsequent statistical model. We demonstrate that the algorithm succesfully corrects bias in a data set from a functional PET activation study of the human brain. An ill-posed data set has more dimensions in each example than there are examples.