  Can we learn a binary classifier from only positive data, without any
negative data or unlabeled data? We show that if one can equip positive data
with confidence (positive-confidence), one can successfully learn a binary
classifier, which we name positive-confidence (Pconf) classification. Our work
is related to one-class classification which is aimed at "describing" the
positive class by clustering-related methods, but one-class classification does
not have the ability to tune hyper-parameters and their aim is not on
"discriminating" positive and negative classes. For the Pconf classification
problem, we provide a simple empirical risk minimization framework that is
model-independent and optimization-independent. We theoretically establish the
consistency and an estimation error bound, and demonstrate the usefulness of
the proposed method for training deep neural networks through experiments.
