We consider the problem of learning a binary classifier from only positive and unlabeled observations (PU learning). Recent studies in PU learning have shown theoretical and empirical performance. However, most existing algorithms may not be suitable for large-scale datasets because they face repeated computations of large Gram matrix or require massive hyperparameter optimization. In this paper, we propose a computationally efficient and theoretically grounded PU learning algorithm. The proposed PU learning algorithm produces a closed-form classifier when the hypothesis space is a closed ball in reproducing kernel Hilbert space. In addition, we establish upper bounds of the estimation error and the excess risk. The obtained estimation error bound is sharper than existing results and the derived excess risk bound has an explicit form, which vanishes as sample sizes increase. To the best of our knowledge, we are the first to explicitly derive the excess risk bound with convergence rate in PU learning. Finally, we conduct extensive numerical experiments using both synthetic and real datasets, demonstrating improved accuracy, scalability, and robustness of the proposed algorithm.
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