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Collaborative filtering (CF) recommender systems help people discover what they really need in a large set of alternatives by analyzing the preferences of other related users. Recent research has shown that the accuracy of recommendations can be improved significantly by using matrix factorization (MF) models. In particular, a mixed MF model was used by BellKor's Pragmatic Chaos to win the Netflix Prize. On the other hand, system designers must also be concerned about system robustness - the ability of the system to provide good recommendations when the system database is contaminated with some portion of noisy or erroneous data, perhaps maliciously entered by `profile injection' attackers intent on distorting system recommendations. In this paper, we focus on the robustness of MF based CF algorithms (MFCF), which usually transform the prediction of user preferences on items into a least squares problem, solved by gradient descent. As least squares is known to be sensitive to outliers, it is not surprising that MF algorithms are vulnerable to attack. Nevertheless a number of `robust statistics' have been proposed since the 1960's that provide alternative data fitting strategies that are less sensitive to outliers. In particular, in this paper, we propose a least trimmed squares based MF (LTSMF) to help improve the robustness of the least squares based MF (LSMF) models. Least trimmed squares is shown to be more robust than least squares and another popular robust method-M-estimator. Experiments also show that LTSMF outperforms previous robust CF models on both accuracy and robustness.