Regularization on Graphs with Function-adapted Diffusion Processes

The use of data-adapted kernels has been shown to lead to state-of-the-art results in machine learning tasks, especially in the context of semi-supervised and transductive learning. We introduce a general framework for analysis both of data sets and functions defined on them. Our approach is based on diffusion operators, adapted not only to the intrinsic geometry of the data, but also to the function being analyzed. Among the many possible applications of this framework, we consider two apparently dissimilar tasks: image denoising and classification in a graph transductive setting. We show that these tasks can be tackled within our framework both conceptually and algorithmically. On benchmarks for transductive learning, our results are better than state of the art on most data sets.