Recently, a new framework, Fredholm learning, was proposed for semisupervised learning problems based on solving a regularized Fredholm integral equation. It allows a natural way to incorporate unlabeled data into learning algorithms to improve their prediction performance. Despite rapid progress on implementable algorithms with theoretical guarantees, the generalization ability of Fredholm kernel learning has not been studied. In this letter, we focus on investigating the generalization performance of a family of classification algorithms, referred to as Fredholm kernel regularized classifiers. We prove that the corresponding learning rate can achieve ( is the number of labeled samples) in a limiting case. In addition, a representer theorem is provided for the proposed regularized scheme, which underlies its applications.

Regularization is a well-recognized powerful strategy to improve the performance of a learning machine and l q regularization schemes with are central in use. It is known that different q leads to different properties of the deduced estimators, say, l 2 regularization leads to a smooth estimator, while l 1 regularization leads to a sparse estimator. Then how the generalization capability of l q regularization learning varies with q is worthy of investigation. In this letter, we study this problem in the framework of statistical learning theory. Our main results show that implementing l q coefficient regularization schemes in the sample-dependent hypothesis space associated with a gaussian kernel can attain the same almost optimal learning rates for all . That is, the upper and lower bounds of learning rates for l q regularization learning are asymptotically identical for all . Our finding tentatively reveals that in some modeling contexts, the choice of q might not have a strong impact on the generalization capability. From this perspective, q can be arbitrarily specified, or specified merely by other nongeneralization criteria like smoothness, computational complexity or sparsity.