Weida Li

Weida Li, Mingxia Liu, Daoqiang Zhang

Recently, Nyström method has proved its prominence empirically and theoretically in speeding up the training of kernel machines while retaining satisfactory performances and accuracy. So far, there are several different approaches proposed to exploit Nyström method in scaling up kernel machines. However, there is no comparative study over these approaches, and they were individually analyzed for specific types of kernel machines. Therefore, it remains a question that the philosophy of which approach is more promising when it extends to other kernel machines. In this work, motivated by the column inclusion property of Gram matrices, we develop a new approach with a clear geometric interpretation for running Nyström-based kernel machines. We show that the other two well-studied approaches can be equivalently transformed to be our proposed one. Consequently, analysis established for the proposed approach also works for these two. Particularly, our proposed approach makes it possible to develop approximation errors in a general setting. Besides, our analysis also manifests the relations among the aforementioned two approaches and another naive one. First, the analytical forms of the corresponding approximate solutions are only at odds with one term. Second, the naive approach can be implemented efficiently by sharing the same training procedure with others. These analytical results lead to the conjecture that the naive approach can provide more accurate approximate solutions than the other two sophisticated approaches. Since our analysis also offers ways for computing the accuracy of these approximate solutions, we run experiments with classification tasks to confirm our conjecture.

Weida Li, Mingxia Liu, Fang Chen et al.

Aggregating multi-subject functional magnetic resonance imaging (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 warrant aligning fMRI data across subjects. However, the existing functional alignment methods cannot handle well various kinds of fMRI datasets today, especially when they are not temporally-aligned, 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 used as a priori 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. To address this issue, we further regularize the framework, so that a novel feasible kernel-based optimization, which permits nonlinear feature extraction, could be theoretically developed. Specifically, a low-dimension assumption is imposed on each new feature space to avoid overfitting caused by the highspatial-low-temporal resolution of fMRI data. Experimental results on five datasets suggest that the proposed method is not only superior to several state-of-the-art methods on temporally-aligned fMRI data, but also suitable for dealing `with temporally-unaligned fMRI data.