Multiscale Laplacian Learning.

Machine learning has greatly influenced many fields, including science. However, despite of the tremendous accomplishments of machine learning, one of the key limitations of most existing machine learning approaches is their reliance on large labeled sets, and thus, data with limited labeled samples remains a challenge. Moreover, the performance of machine learning methods often severely hindered in case of diverse data, usually associated with smaller data sets or data associated with areas of study where the size of the data sets is constrained by high experimental cost and/or ethics. These challenges call for innovative strategies for dealing with these types of data. In this work, the aforementioned challenges are addressed by integrating graph-based frameworks, semi-supervised techniques, multiscale structures, and modified and adapted optimization procedures. This results in two innovative multiscale Laplacian learning (MLL) approaches for machine learning tasks, such as data classification, and for tackling data with limited samples, diverse data, and small data sets. The first approach, multikernel manifold learning (MML), integrates manifold learning with multikernel information and incorporates a warped kernel regularizer using multiscale graph Laplacians. The second approach, the multiscale MBO (MMBO) method, introduces multiscale Laplacians to the modification of the famous classical Merriman-Bence-Osher (MBO) scheme, and makes use of fast solvers. We demonstrate the performance of our algorithms experimentally on a variety of benchmark data sets, and compare them favorably to the state-of-art approaches.