DC Algorithm for Estimation of Sparse Gaussian Graphical Models

Sparse estimation for Gaussian graphical models is a crucial technique for making the relationships among numerous observed variables more interpretable and quantifiable. Various methods have been proposed, including graphical lasso, which utilizes the $\ell_1$ norm as a regularization term, as well as methods employing non-convex regularization terms. However, most of these methods approximate the $\ell_0$ norm with convex functions. To estimate more accurate solutions, it is desirable to treat the $\ell_0$ norm directly as a regularization term. In this study, we formulate the sparse estimation problem for Gaussian graphical models using the $\ell_0$ norm and propose a method to solve this problem using the Difference of Convex functions Algorithm (DCA). Specifically, we convert the $\ell_0$ norm constraint into an equivalent largest-$K$ norm constraint, reformulate the constrained problem into a penalized form, and solve it using the DC algorithm (DCA). Furthermore, we designed an algorithm that efficiently computes using graphical lasso. Experimental results with synthetic data show that our method yields results that are equivalent to or better than existing methods. Comparisons of model learning through cross-validation confirm that our method is particularly advantageous in selecting true edges.

Prediction of hierarchical time series using structured regularization and its application to artificial neural networks

This paper discusses the prediction of hierarchical time series, where each upper-level time series is calculated by summing appropriate lower-level time series. Forecasts for such hierarchical time series should be coherent, meaning that the forecast for an upper-level time series equals the sum of forecasts for corresponding lower-level time series. Previous methods for making coherent forecasts consist of two phases: first computing base (incoherent) forecasts and then reconciling those forecasts based on their inherent hierarchical structure. With the aim of improving time series predictions, we propose a structured regularization method for completing both phases simultaneously. The proposed method is based on a prediction model for bottom-level time series and uses a structured regularization term to incorporate upper-level forecasts into the prediction model. We also develop a backpropagation algorithm specialized for application of our method to artificial neural networks for time series prediction. Experimental results using synthetic and real-world datasets demonstrate the superiority of our method in terms of prediction accuracy and computational efficiency.