  Non-negative matrix factorization (NMF) minimizes the Euclidean distance
between the data matrix and its low rank approximation, and it fails when
applied to corrupted data because the loss function is sensitive to outliers.
In this paper, we propose a Truncated CauchyNMF loss that handle outliers by
truncating large errors, and develop a Truncated CauchyNMF to robustly learn
the subspace on noisy datasets contaminated by outliers. We theoretically
analyze the robustness of Truncated CauchyNMF comparing with the competing
models and theoretically prove that Truncated CauchyNMF has a generalization
bound which converges at a rate of order $O(\sqrt{{\ln n}/{n}})$, where $n$ is
the sample size. We evaluate Truncated CauchyNMF by image clustering on both
simulated and real datasets. The experimental results on the datasets
containing gross corruptions validate the effectiveness and robustness of
Truncated CauchyNMF for learning robust subspaces.
