How does one learn a classifier from data that isn't always fully labelled? For example, say one has corrupted data from the CIFAR-10 dataset (which has labels like bird/automobile/ship/truck). Now this corrupted data (X, Y) pairs and preserves X, while "confusing" a large number of Y pairs by replacing each label with a set of labels the sample's true label is from. So a label "bird" may become "not ship", "automobile" may become "autombile or truck", "ship" may become "ship" (unchanged) etc.
How does one best exploit this information? Is there a loss function that handles these?