We consider biclustering that clusters both samples and features and propose efficient convex biclustering procedures. The convex biclustering algorithm (COBRA) procedure solves twice the standard convex clustering problem that contains a non-differentiable function optimization. We instead convert the original optimization problem to a differentiable one and improve another approach based on the augmented Lagrangian method (ALM). Our proposed method combines the basic procedures in the ALM with the accelerated gradient descent method (Nesterov’s accelerated gradient method), which can attain O(1/k2) convergence rate. It only uses first-order gradient information, and the efficiency is not influenced by the tuning parameter λ so much. This advantage allows users to quickly iterate among the various tuning parameters λ and explore the resulting changes in the biclustering solutions. The numerical experiments demonstrate that our proposed method has high accuracy and is much faster than the currently known algorithms, even for large-scale problems.
We consider learning as an undirected graphical model from sparse data. While several efficient algorithms have been proposed for graphical lasso (GL), the alternating direction method of multipliers (ADMM) is the main approach taken concerning joint graphical lasso (JGL). We propose proximal gradient procedures with and without a backtracking option for the JGL. These procedures are first-order methods and relatively simple, and the subproblems are solved efficiently in closed form. We further show the boundedness for the solution of the JGL problem and the iterates in the algorithms. The numerical results indicate that the proposed algorithms can achieve high accuracy and precision, and their efficiency is competitive with state-of-the-art algorithms.
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