Abstract
The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the optimization problem on the set of bounded-rank matrices. We propose a Riemannian rank-adaptive method, which consists of fixed-rank optimization, rank increase step and rank reduction step. We explore its performance applied to the low-rank matrix completion problem. Numerical experiments on synthetic and real-world datasets illustrate that the proposed rank-adaptive method compares favorably with state-of-the-art algorithms. In addition, it shows that one can incorporate each aspect of this rank-adaptive framework separately into existing algorithms for the purpose of improving performance.
Highlights
The low-rank matrix completion problem has been extensively studied in recent years; see the survey [11]
In the rest of this section, we introduce these features in detail. 3.2 Riemannian optimization on Ms Very recently, the Riemannian gradient method with non-monotone line search and Barzilai–Borwein (BB) step size [2] has been shown to be efficient in various applications; see [6,7,8]
We evaluate the performance of Riemannian rank-adaptive method (RRAM) on low-rank matrix completion with real-world datasets
Summary
The low-rank matrix completion problem has been extensively studied in recent years; see the survey [11]. A Riemannian rank-adaptive method for low-rank optimization has been proposed in [19], and problem (2) can be viewed as a specific application. This rank-adaptive algorithm mainly consists of two steps: Riemannian optimization on the fixed-rank manifold and adaptive update of the rank. We propose a new Riemannian rank-adaptive method (RRAM); see Algorithm 1. – We demonstrate the effectiveness of the proposed method applied to low-rank matrix completion The numerical experiments on synthetic and realworld datasets illustrate that the proposed rank-adaptive method is able to find the ground-truth rank and compares favorably with other state-of-the-art algorithms in terms of computational efficiency.
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