Algorithms for Lipschitz Learning on Graphs
This work addresses efficient graph-based regression for applications in machine learning and data analysis, offering incremental improvements in algorithmic speed and regularization methods.
The paper tackles the problem of computing smooth extensions of functions on graphs from given vertex values, focusing on minimal Lipschitz extensions as limits of p-Laplacian regularization. It presents algorithms that achieve expected linear time for minimal extensions and expected time O~(m n) for absolutely minimal extensions, with practical variants and polynomial-time regularization capabilities.
We develop fast algorithms for solving regression problems on graphs where one is given the value of a function at some vertices, and must find its smoothest possible extension to all vertices. The extension we compute is the absolutely minimal Lipschitz extension, and is the limit for large $p$ of $p$-Laplacian regularization. We present an algorithm that computes a minimal Lipschitz extension in expected linear time, and an algorithm that computes an absolutely minimal Lipschitz extension in expected time $\widetilde{O} (m n)$. The latter algorithm has variants that seem to run much faster in practice. These extensions are particularly amenable to regularization: we can perform $l_{0}$-regularization on the given values in polynomial time and $l_{1}$-regularization on the initial function values and on graph edge weights in time $\widetilde{O} (m^{3/2})$.