The correlation matrix is a fundamental statistic that used in many fields. For example, GroupLens, a collaborative filtering system, uses the correlation between users for predictive purposes. Since the correlation is a natural similarity measure between users, the correlation matrix may be used as the Gram matrix in kernel methods. However, the estimated correlation matrix sometimes has a serious defect: although the correlation matrix is originally positive semidefinite, the estimated one may not be positive semidefinite when not all ratings are observed. To obtain a positive semidefinite correlation matrix, the nearest correlation matrix problem has recently been studied in the fields of numerical analysis and optimization. However, statistical properties are not explicitly used in such studies. To obtain a positive semidefinite correlation matrix, we assume an approximate model. By using the model, an estimate is obtained as the optimal point of an optimization problem formulated with information on the variances of the estimated correlation coefficients. The problem is solved by a convex quadratic semidefinite program. A penalized likelihood approach is also examined. The MovieLens data set is used to test our approach.

Density estimation plays an important and fundamental role in pattern recognition, machine learning, and statistics. In this article, we develop a parametric approach to univariate (or low-dimensional) density estimation based on semidefinite programming (SDP). Our density model is expressed as the product of a nonnegative polynomial and a base density such as normal distribution, exponential distribution, and uniform distribution. When the base density is specified, the maximum likelihood estimation of the polynomial is formulated as a variant of SDP that is solved in polynomial timewith the interior point methods. Since the base density typically contains just one or two parameters, computation of the maximum likelihood estimate reduces to a one- or two-dimensional easy optimization problem with this use of SDP. Thus, the rigorous maximum likelihood estimate can be computed in our approach. Furthermore, such conditions as symmetry and unimodality of the density function can be easily handled within this framework. AIC is used to choose the best model. Through applications to several instances, we demonstrate flexibility of the model and performance of the proposed procedure. Combination with amixture approach is also presented. The proposed approach has possible other applications beyond density estimation. This point is clarified through an application to the maximum likelihood estimation of the intensity function of a nonstationary Poisson process.