Training sequence size in clustering algorithms and averaging single-particle images

In clustering the training vectors, we may consider an algorithm, which tries to find empirically optimal representative vectors that achieve the empirical minimum to inductively design optimal representative vectors yielding the true optimum. In order to evaluate the performance of the representative vectors, we may observe the empirical minimum with respect to the training ratio, the ratio of the training sequence size to the number of representative vectors. In this paper, the convergence rates of the expectations of the empirical minimum and the validating errors are observed with respect to the training ratio. When enhancing the noisy particle images, which are obtained from the transmission electron microscopy, the theoretical analysis is employed to discuss the performance in conjunction with the overfitting property.