It could also be used to held out runs for example.

The reviewers found that this paper was useful for the field, offering a new method for aligning multi-site fMRI data, with some disagreement. One of the main concerns was about the clarity of the paper which greatly impacts its usefulness. Please improve the quality of the writing for the final draft. There is a major typo in the paper: everywhere XX T is mentioned, I think you mean X TX (the identify matrices are also wrongly subscripted). In fact R1's comment is right, XX T is likely not singular given V T.

Multi-voxel pattern analysis (MVPA) learns predictive models from task-based functional magnetic resonance imaging (fMRI) data, for distinguishing when subjects are performing different cognitive tasks -- e.g., watching movies or making decisions. MVPA works best with a well-designed feature set and an adequate sample size. However, most fMRI datasets are noisy, high-dimensional, expensive to collect, and with small sample sizes. Further, training a robust, generalized predictive model that can analyze homogeneous cognitive tasks provided by multi-site fMRI datasets has additional challenges. This paper proposes the Shared Space Transfer Learning (SSTL) as a novel transfer learning (TL) approach that can functionally align homogeneous multi-site fMRI datasets, and so improve the prediction performance in every site. It then uses TL to map these site-specific features to a site-independent shared space in order to improve the performance of the MVPA.