Efficient Robust Matrix Factorization with Nonconvex Penalties
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Robust matrix factorization (RMF) is a fundamental tool with lots of applications. The state-of-art is robust matrix factorization by majorization and minimization (RMF-MM) algorithm. It iteratively constructs and minimizes a novel surrogate function. Besides, it is also the only RMF algorithm with convergence guarantee. However, it can only deal with the convex ℓ_1-loss and does not utilize sparsity when matrix is sparsely observed. In this paper, we proposed robust matrix factorization by nonconvex penalties (RMF-NP) algorithm addressing these two problems. RMF-NP enables nonconvex penalties as the loss, which makes it more robust to outliers. As the surrogate function from RMF-MM no longer applies, we construct a new one and solve it in its dual. This makes the runtime and memory cost of RMF-NP only depends on nonzero elements. Convergence analysis based on the new surrogate function is also established, which shows RMF-NP is guaranteed to produce a critical point. Finally, experiments on both synthetic and real-world data sets demonstrate the superiority of RMF-NP over existing algorithms in terms of recovery performance and runtime.
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