  The label noise transition matrix $T$, reflecting the probabilities that true
labels flip into noisy ones, is of vital importance to model label noise and
design statistically consistent classifiers. The traditional transition matrix
is limited to model closed-set label noise, where noisy training data has true
class labels within the noisy label set. It is unfitted to employ such a
transition matrix to model open-set label noise, where some true class labels
are outside the noisy label set. Thus when considering a more realistic
situation, i.e., both closed-set and open-set label noise occurs, existing
methods will undesirably give biased solutions. Besides, the traditional
transition matrix is limited to model instance-independent label noise, which
may not perform well in practice. In this paper, we focus on learning under the
mixed closed-set and open-set label noise. We address the aforementioned issues
by extending the traditional transition matrix to be able to model mixed label
noise, and further to the cluster-dependent transition matrix to better
approximate the instance-dependent label noise in real-world applications. We
term the proposed transition matrix as the cluster-dependent extended
transition matrix. An unbiased estimator (i.e., extended $T$-estimator) has
been designed to estimate the cluster-dependent extended transition matrix by
only exploiting the noisy data. Comprehensive synthetic and real experiments
validate that our method can better model the mixed label noise, following its
more robust performance than the prior state-of-the-art label-noise learning
methods.
