A fast cross-validation algorithm for model selection in kernel ridge regression problems is proposed, which is aiming to further reduce the computational cost of the algorithm proposed by An et al. by eigenvalue decomposition of a Gram matrix.

The goal of dimension reduction is to represent high-dimensional data in a lower-dimensional subspace, while intrinsic properties of the original data are kept as much as possible. An important challenge in unsupervised dimension reduction is the choice of tuning parameters, because no supervised information is available and thus parameter selection tends to be subjective and heuristic. In this paper, we propose an information-theoretic approach to unsupervised dimension reduction that allows objective tuning parameter selection. We employ quadratic mutual information (QMI) as our information measure, which is known to be less sensitive to outliers than ordinary mutual information, and QMI is estimated analytically by a least-squares method in a computationally efficient way. Then, we provide an eigenvector-based efficient implementation for performing unsupervised dimension reduction based on the QMI estimator. The usefulness of the proposed method is demonstrated through experiments.

Image restoration based on Bayesian estimation in most previous studies has assumed that the noise accumulated in an image was independent for each pixel. However, when we take optical effects into account, it is reasonable to expect spatial correlation in the superimposed noise. In this paper, we discuss the restoration of images distorted by noise which is spatially correlated with translational symmetry in the realm of probabilistic processing. First, we assume that the original image can be produced by a Gaussian model based on only a nearest-neighbor effect and that the noise superimposed at each pixel is produced by a Gaussian model having spatial correlation characterized by translational symmetry. With this model, we can use Fourier transformation to calculate system characteristics such as the restoration error and also minimize the restoration error when the hyperparameters of the probabilistic model used in the restoration process coincides with those used in the formation process. We also discuss the characteristics of image restoration distorted by spatially correlated noise using a natural image. In addition, we estimate the hyperparameters using the maximum marginal likelihood and restore an image distorted by spatially correlated noise to evaluate this method of image restoration.