Multiscale Laplacian Learning

Abstract

Machine learning methods have greatly changed science, engineering, finance, business, and other fields. Despite the tremendous accomplishments of machine learning and deep learning methods, many challenges still remain. In particular, the performance of machine learning methods is often severely affected in case of diverse data, usually associated with smaller data sets or data related to areas of study where the size of the data sets is constrained by the complexity and/or high cost of experiments. Moreover, data with limited labeled samples is a challenge to most learning approaches. In this paper, the aforementioned challenges are addressed by integrating graph-based frameworks, multiscale structure, modified and adapted optimization procedures and semi-supervised techniques. This results in two innovative multiscale Laplacian learning (MLL) approaches for machine learning tasks, such as data classification, and for tackling diverse data, data with limited samples and smaller data sets. The first approach, called multikernel manifold learning (MML), integrates manifold learning with multikernel information and solves a regularization problem consisting of a loss function and a warped kernel regularizer using multiscale graph Laplacians. The second approach, called the multiscale MBO (MMBO) method, introduces multiscale Laplacians to a modification of the famous classical Merriman-Bence-Osher (MBO) scheme, and makes use of fast solvers for finding the approximations to the extremal eigenvectors of the graph Laplacian. We demonstrate the performance of our methods experimentally on a variety of data sets, such as biological, text and image data, and compare them favorably to existing approaches.

References in corpus (10)

• Semi-Supervised Classification with Graph Convolutional Networks
• Spectral Networks and Locally Connected Networks on Graphs
• TopologyNet: Topology based deep convolutional neural networks for biomolecular property predictions
• Laplacian Support Vector Machines Trained in the Primal
• Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening
• Attention-based Graph Neural Network for Semi-supervised Learning
• GMNN: Graph Markov Neural Networks
• A review of mathematical representations of biomolecules
• N-GCN: Multi-scale Graph Convolution for Semi-supervised Node Classification
• Detection and tracking of gas plumes in LWIR hyperspectral video sequence data