We propose a new method, parametric embedding (PE), that embeds objects with the class structure into a low-dimensional visualization space. PE takes as input a set of class conditional probabilities for given data points and tries to preserve the structure in an embedding space by minimizing a sum of Kullback-Leibler divergences, under the assumption that samples are generated by a gaussian mixture with equal covariances in the embedding space. PE has many potential uses depending on the source of the input data, providing insight into the classifier's behavior in supervised, semisupervised, and unsupervised settings. The PE algorithm has a computational advantage over conventional embedding methods based on pairwise object relations since its complexity scales with the product of the number of objects and the number of classes. We demonstrate PE by visualizing supervised categorization of Web pages, semisupervised categorization of digits, and the relations of words and latent topics found by an unsupervised algorithm, latent Dirichlet allocation.

This article compares three penalty terms with respect to the efficiency of supervised learning, by using first- and second-order off-line learning algorithms and a first-order on-line algorithm. Our experiments showed that for a reasonably adequate penalty factor, the combination of the squared penalty term and the second-order learning algorithm drastically improves the convergence performance in comparison to the other combinations, at the same time bringing about excellent generalization performance. Moreover, in order to understand how differently each penalty term works, a function surface evaluation is described. Finally, we show how cross validation can be applied to find an optimal penalty factor.

Second-order learning algorithms based on quasi-Newton methods have two problems. First, standard quasi-Newton methods are impractical for large-scale problems because they require N 2 storage space to maintain an approximation to an inverse Hessian matrix ( N is the number of weights). Second, a line search to calculate areasonably accurate step length is indispensable for these algorithms. In order to provide desirable performance, an efficient and reasonably accurate line search is needed. To overcome these problems, we propose a new second-order learning algorithm. Descent direction is calculated on the basis of a partial Broydon-Fletcher-Goldfarb-Shanno (BFGS) update with 2Ns memory space (s « N), and a reasonably accurate step length is efficiently calculated as the minimal point of a second-order approximation to the objective function with respect to the step length. Our experiments, which use a parity problem and a speech synthesis problem, have shown that the proposed algorithm outperformed major learning algorithms. Moreover, it turned out that an efficient and accurate step-length calculation plays an important role for the convergence of quasi-Newton algorithms, and a partial BFGS update greatly saves storage space without losing the convergence performance.