As an implementation of the Nystr\"{o}m method, Nystr\"{o}m computational regularization (NCR) imposed on kernel classification and kernel ridge regression has proven capable of achieving optimal bounds in the large-scale statistical learning setting, while enjoying much better time complexity. In this study, we propose a Nystr\"{o}m subspace learning (NSL) framework to reveal that all you need for employing the Nystr\"{o}m method, including NCR, upon any kernel SVM is to use the efficient off-the-shelf linear SVM solvers as a black box. Based on our analysis, the bounds developed for the Nystr\"{o}m method are linked to NSL, and the analytical difference between two distinct implementations of the Nystr\"{o}m method is clearly presented. Besides, NSL also leads to sharper theoretical results for the clustered Nystr\"{o}m method. Finally, both regression and classification tasks are performed to compare two implementations of the Nystr\"{o}m method.
Aggregating multi-subject fMRI data is indispensable for generating valid and general inferences from patterns distributed across human brains. The disparities in anatomical structures and functional topographies of human brains call for aligning fMRI data across subjects. However, the existing functional alignment methods cannot tackle various kinds of fMRI datasets today, especially when they are incomplete, i.e., some of the subjects probably lack the responses to some stimuli, or different subjects might follow different sequences of stimuli. In this paper, a cross-subject graph that depicts the (dis)similarities between samples across subjects is taken as prior information for developing a more flexible framework that suits an assortment of fMRI datasets. However, the high dimension of fMRI data and the use of multiple subjects makes the crude framework time-consuming or unpractical. Therefore, we regularize the framework so that a feasible kernel-based optimization, which permits non-linear feature extraction, could be theoretically developed. Specifically, a low-dimension assumption is imposed on each new feature space to avoid overfitting caused by the high-spatial-low-temporal resolution of fMRI data. Empirical studies confirm that the proposed method under both incompleteness and completeness can achieve better performance than other state-of-the-art functional alignment methods under completeness.