In many real-world classification problems, the labels of training examples are randomly corrupted. Most previous theoretical work on classification with label noise assumes that the two classes are separable, that the label noise is independent of the true class label, or that the noise proportions for each class are known. In this work, we give conditions that are necessary and sufficient for the true class-conditional distributions to be identifiable. These conditions are weaker than those analyzed previously, and allow for the classes to be nonseparable and the noise levels to be asymmetric and unknown. The conditions essentially state that a majority of the observed labels are correct and that the true class-conditional distributions are "mutually irreducible," a concept we introduce that limits the similarity of the two distributions. For any label noise problem, there is a unique pair of true class-conditional distributions satisfying the proposed conditions, and we argue that this pair corresponds in a certain sense to maximal denoising of the observed distributions. Our results are facilitated by a connection to "mixture proportion estimation," which is the problem of estimating the maximal proportion of one distribution that is present in another. We establish a novel rate of convergence result for mixture proportion estimation, and apply this to obtain consistency of a discrimination rule based on surrogate loss minimization. Experimental results on benchmark data and a nuclear particle classification problem demonstrate the efficacy of our approach.
Data noise is present in many machine learning problems domains, some of these are well studied but others have received less attention. In this paper we propose an algorithm for constructing a kernel Fisher discriminant (KFD) from training examples with noisy labels. The approach allows to associate with each example a probability of the label being flipped. We utilise an expectation maximization (EM) algorithm for updating the probabilities. The E-step uses class conditional probabilities estimated as a by-product of the KFD algorithm. The M-step updates the flip probabilities and determines the parameters of the discriminant. We demonstrate the feasibility of the approach on two real-world data-sets.