Abstract
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.
Highlights
By clustering, such as k-means clustering [1] and hierarchical clustering [2,3], we usually mean dividing N samples, each consisting of p covariate values, into several categories, where N, p ≥ 1.In this paper, we consider biclustering [4] that is an extended notion of clustering
This paper proposes an efficient algorithm with simple subproblems and a fast convergence rate to solve the convex biclustering problem
We proposed a method to find a solution to the convex biclustering problem
Summary
By clustering, such as k-means clustering [1] and hierarchical clustering [2,3], we usually mean dividing N samples, each consisting of p covariate values, into several categories, where N, p ≥ 1. The main challenge of solving the optimization problem is the two fused penalty terms: indecomposable and non-differentiable. It is necessary to propose an efficient algorithm to solve the convex biclustering problem. This paper proposes an efficient algorithm with simple subproblems and a fast convergence rate to solve the convex biclustering problem. Rather than update each variable alternately, like ADMM, we use the augmented Lagrangian method (ALM) to update the primal variables simultaneously In this way, we can transform the optimization problem to be differentiable, solve the problem via an efficient gradient descent method and further simplify the subproblems. Our proposed method does not require as much computation, even when the tuning parameter λ is large, as the existing approaches do, which means that it is easier to obtain biclustering results simultaneously for several λ values. Rp×q, we use ||X||F ∑ip=1 ∑qj=1 to denote |xij| if not the Frobenius specified
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