In this paper we analytically investigate the generalization performance of learning using correlated inputs in the framework of on-line learning with a statistical mechanical method. We consider a model composed of linear perceptrons with Gaussian noise. First, we analyze the case of the gradient method. We analytically clarify that the larger the correlation among inputs is or the larger the number of inputs is, the stricter the condition the learning rate should satisfy is, and the slower the learning speed is. Second, we treat the block orthogonal projection learning as an alternative learning rule and derive the theory. In a noiseless case, the learning speed does not depend on the correlation and is proportional to the number of inputs used in an update. The learning speed is identical to that of the gradient method with uncorrelated inputs. On the other hand, when there is noise, the larger the correlation among inputs is, the slower the learning speed is and the larger the residual generalization error is.

In many practical situations in NN learning, training examples tend to be supplied one by one. In such situations, incremental learning seems more natural than batch learning in view of the learning methods of human beings. In this paper, we propose an incremental learning method in neural networks under the projection learning criterion. Although projection learning is a linear learning method, achieving the above goal is not straightforward since it involves redundant expressions of functions with over-complete bases, which is essentially related to pseudo biorthogonal bases (or frames). The proposed method provides exactly the same learning result as that obtained by batch learning. It is theoretically shown that the proposed method is more efficient in computation than batch learning.