Nuclear-norm regularization plays a vital role in many learning tasks, such as low-rank matrix recovery (MR), and low-rank representation (LRR). Solving this problem directly can be computationally expensive due to the unknown rank of variables or large-rank singular value decompositions (SVDs). To address this, we propose a proximal Riemannian gradient (PRG) scheme which can efficiently solve trace-norm regularized problems defined on real-algebraic variety @math of real matrices of rank at most @math . Based on PRG, we further present a simple and novel subspace pursuit (SP) paradigm for general trace-norm regularized problems without the explicit rank constraint @math . The proposed paradigm is very scalable by avoiding large-rank SVDs. Empirical studies on several tasks, such as matrix completion and LRR based subspace clustering, demonstrate the superiority of the proposed paradigms over existing methods.
The authors in @cite_42 exploited Riemannian structures and presented a trust-region algorithm to address trace-norm minimizations. The proposed method, denoted by MMBS, alternates between fixed-rank optimization and rank-one updates. However, this method shows slower speed than APG on large-scale problems @cite_42 . The authors in @cite_5 proposed a Grassmannian manifold method to address trace-norm minimizations on a fixed-rank manifold. In general, this method has similar complexity to ScGrassMC that also operates on Grassmannian manifold @cite_0
@cite_0 @cite_5 @cite_42
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The paper addresses the problem of low-rank trace norm minimization. We propose an algorithm that alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is characterized by an efficient factorization that makes the trace norm differentiable in the search space and the computation of duality gap numerically tractable. The search space is nonlinear but is equipped with a Riemannian structure that leads to efficient computations. We present a second-order trust-region algorithm with a guaranteed quadratic rate of convergence. Overall, the proposed optimization scheme converges superlinearly to the global solution while maintaining complexity that is linear in the number of rows and columns of the matrix. To compute a set of solutions efficiently for a grid of regularization parameters we propose a predictor-corrector approach that outperforms the naive warm-restart approach on the fixed-rank quotient manifold. The performance of the proposed algorithm is illustrated on p... We address the numerical problem of recovering large matrices of low rank when most of the entries are unknown. We exploit the geometry of the low-rank constraint to recast the problem as an unconstrained optimization problem on a single Grassmann manifold. We then apply second-order Riemannian trust-region methods (RTRMC 2) and Riemannian conjugate gradient methods (RCGMC) to solve it. A preconditioner for the Hessian is introduced that helps control the conditioning of the problem and we detail preconditioned versions of Riemannian optimization algorithms. The cost of each iteration is linear in the number of known entries. The proposed methods are competitive with state-of-the-art algorithms on a wide range of problem instances. In particular, they perform well on rectangular matrices. We further note that second-order and preconditioned methods are well suited to solve badly conditioned matrix completion tasks. Patch-based low-rank models have shown effective in exploiting spatial redundancy of natural images especially for the application of image denoising. However, two-dimensional low-rank model can not fully exploit the spatio-temporal correlation in larger data sets such as multispectral images and 3D MRIs. In this work, we propose a novel low-rank tensor approximation framework with Laplacian Scale Mixture (LSM) modeling for multi-frame image denoising. First, similar 3D patches are grouped to form a tensor of d-order and high-order Singular Value Decomposition (HOSVD) is applied to the grouped tensor. Then the task of multiframe image denoising is formulated as a Maximum A Posterior (MAP) estimation problem with the LSM prior for tensor coefficients. Both unknown sparse coefficients and hidden LSM parameters can be efficiently estimated by the method of alternating optimization. Specifically, we have derived closed-form solutions for both subproblems. Experimental results on spectral and dynamic MRI images show that the proposed algorithm can better preserve the sharpness of important image structures and outperform several existing state-of-the-art multiframe denoising methods (e.g., BM4D and tensor dictionary learning). The problem of minimizing the rank of a matrix subject to affine constraints has applications in several areas including machine learning, and is known to be NP-hard. A tractable relaxation for this problem is nuclear norm (or trace norm) minimization, which is guaranteed to find the minimum rank matrix under suitable assumptions. In this paper, we propose a family of Iterative Reweighted Least Squares algorithms IRLS-p (with 0 ≤ p ≤ 1), as a computationally efficient way to improve over the performance of nuclear norm minimization. The algorithms can be viewed as (locally) minimizing certain smooth approximations to the rank function. When p = 1, we give theoretical guarantees similar to those for nuclear norm minimization, that is, recovery of low-rank matrices under certain assumptions on the operator defining the constraints. For p < 1, IRLS-p shows better empirical performance in terms of recovering low-rank matrices than nuclear norm minimization. We provide an efficient implementation for IRLS-p, and also present a related family of algorithms, sIRLS-p. These algorithms exhibit competitive run times and improved recovery when compared to existing algorithms for random instances of the matrix completion problem, as well as on the MovieLens movie recommendation data set.