Journal Article10.1109/TKDE.2018.2882197
Fast and Low Memory Cost Matrix Factorization: Algorithm, Analysis, and Case Study
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TL;DR: This paper proposes a stage-wise matrix factorization algorithm by exploiting manifold optimization techniques and studies two representative cases of low-rank matrix recovery, i.e., collaborative filtering for recommendation and high dynamic range imaging.
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Abstract: Matrix factorization has been widely applied to various applications. With the fast development of storage and internet technologies, we have been witnessing a rapid increase of data. In this paper, we propose new algorithms for matrix factorization with the emphasis on efficiency. In addition, most existing methods of matrix factorization only consider a general smooth least square loss. Differently, many real-world applications have distinctive characteristics. As a result, different losses should be used accordingly. Therefore, it is beneficial to design new matrix factorization algorithms that are able to deal with both smooth and non-smooth losses. To this end, one needs to analyze the characteristics of target data and use the most appropriate loss based on the analysis. We particularly study two representative cases of low-rank matrix recovery, i.e., collaborative filtering for recommendation and high dynamic range imaging. To solve these two problems, we respectively propose a stage-wise matrix factorization algorithm by exploiting manifold optimization techniques. From our theoretical analysis, they are both are provably guaranteed to converge to a stationary point. Extensive experiments on recommender systems and high dynamic range imaging demonstrate the satisfactory performance and efficiency of our proposed method on large-scale real data.
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References
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A Singular Value Thresholding Algorithm for Matrix Completion
TL;DR: This paper develops a simple first-order and easy-to-implement algorithm that is extremely efficient at addressing problems in which the optimal solution has low rank, and develops a framework in which one can understand these algorithms in terms of well-known Lagrange multiplier algorithms.
•Proceedings Article
Probabilistic Matrix Factorization
Andriy Mnih,Ruslan Salakhutdinov +1 more
- 03 Dec 2007
TL;DR: The Probabilistic Matrix Factorization (PMF) model is presented, which scales linearly with the number of observations and performs well on the large, sparse, and very imbalanced Netflix dataset and is extended to include an adaptive prior on the model parameters.
Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
TL;DR: It is shown that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space.
4.4K
•Posted Content
A Singular Value Thresholding Algorithm for Matrix Completion
TL;DR: In this article, a convex relaxation of a rank minimization problem is proposed to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints.
3.6K