Authors : Y. Yang (Belgium) , Y. Feng (Belgium), J. Suykens (Belgium)
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Chapter Description
In this chapter, we propose a nonconvex approach by employing an exponential squared type loss, namely, the Welsch loss which will be introduced later. The Welsch loss was originally introduced in robust statistics to form robust estimators in linear models. We will show that it can also work efficiently in matrix completion problems and bring us robust output. Moreover, for the proposed nonconvex matrix completion problems, we propose efficient algorithms, where at each iteration, the algorithms first compute a rank-one matrix, and then update the new trial as a linear combination of the current trial and the newly generated rank-one matrix. The rank-one matrix is related to the left and right singular vectors of the leading singular value of certain matrix, which can be found efficiently by the power method or Lanczos method. Therefore, the whole algorithms are also efficient. We also show the sublinear convergence rate of the proposed algorithms.