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Abstract: We consider the problem of reconstructing a low rank matrix from noisyobservations of a subset of its entries. This task has applications instatistical learning, computer vision, and signal processing. In thesecontexts -noise- generically refers to any contribution to the data that isnot captured by the low-rank model. In most applications, the noise level islarge compared to the underlying signal and it is important to avoidoverfitting. In order to tackle this problem, we define a regularized costfunction well suited for spectral reconstruction methods. Within a random noisemodel, and in the large system limit, we prove that the resulting accuracyundergoes a phase transition depending on the noise level and on the fractionof observed entries. The cost function can be minimized using OPTSPACE amanifold gradient descent algorithm. Numerical simulations show that thisapproach is competitive with state-of-the-art alternatives.

Autor: Raghunandan H. Keshavan, Andrea Montanari


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