Laplacian Spectrum Learning

The eigenspectrum of a graph Laplacian encodes smoothness information over the graph. A natural approach to learning involves transforming the spectrum of a graph Laplacian to obtain a kernel. While manual exploration of the spectrum is conceivable, non-parametric learning methods that adjust the Laplacian’s spectrum promise better performance. For instance, adjusting the graph Laplacian using kernel target alignment (KTA) yields better performance when an SVM is trained on the resulting kernel. KTA relies on a simple surrogate criterion to choose the kernel; the obtained kernel is then fed to a large margin classification algorithm. In this paper, we propose novel formulations that jointly optimize relative margin and the spectrum of a kernel defined via Laplacian eigenmaps. The large relative margin case is in fact a strict generalization of the large margin case. The proposed methods show significant empirical advantage over numerous other competing methods.