Existence and uniqueness of global solutions to the stochastic heat equation with superlinear drift on an unbounded spatial domain

We prove the existence and uniqueness of global solutions to the semilinear stochastic heat equation on an unbounded spatial domain with forcing terms that grow superlinearly and satisfy an Osgood condition [Formula: see text] along with additional restrictions. For example, consider the forcing [Formula: see text]. A new dynamic weighting procedure is introduced to control the solutions, which are unbounded in space.

Learning from Complementary Labels via Partial-Output Consistency Regularization

In complementary-label learning (CLL), a multi-class classifier is learned from training instances each associated with complementary labels, which specify the classes that the instance does not belong to. Previous studies focus on unbiased risk estimator or surrogate loss while neglect the importance of regularization in training phase. In this paper, we give the first attempt to leverage regularization techniques for CLL. By decoupling a label vector into complementary labels and partial unknown labels, we simultaneously inhibit the outputs of complementary labels with a complementary loss and penalize the sensitivity of the classifier on the partial outputs of these unknown classes by consistency regularization. Then we unify the complementary loss and consistency loss together by a specially designed dynamic weighting factor. We conduct a series of experiments showing that the proposed method achieves highly competitive performance in CLL.