Here is an interesting use of the low rank + sparse decomposition :
Recommendation systems are becoming increasingly important, as evidenced by the popularity of the Netﬂix prize and the sophistication of various online shopping systems. With this increase in interest, a new problem of nefarious or false rankings that compromise a recommendation system’s integrity has surfaced. We consider such purposefully erroneous rankings to be a form of “toxic waste,” corrupting the performance of the underlying algorithm. In this paper, we propose an adaptive reweighted algorithm as a possible approach towards correcting this problem. Our algorithm relies on ﬁnding a low-rank-plus-sparse decomposition of the recommendation matrix, where the adaptation of the weights aids in rejecting the malicious contributions. Simulations suggest that our algorithm converges fairly rapidly and produces accurate results.
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