A maximum principle argument for the uniform convergence of graph Laplacian regressors
This paper investigates the use of methods from partial differential equations and the Calculus of variations to study learning problems that are regularized using graph Laplacians. Graph Laplacians are a powerful, flexible method for capturing local and global geometry in many classes of learning problems, and the techniques developed in this paper help to broaden the methodology of studying such problems. In particular, we develop the use of maximum principle arguments to establish asymptotic consistency guarantees within the context of noise corrupted, non-parametric regression with samples living on an unknown manifold embedded in $\mathbb{R}^d$. The maximum principle arguments provide a new technical tool which informs parameter selection by giving concrete error estimates in terms of various regularization parameters. A review of learning algorithms which utilize graph Laplacians, as well as previous developments in the use of differential equation and variational techniques to study those algorithms, is given. In addition, new connections are drawn between Laplacian methods and other machine learning techniques, such as kernel regression and k-nearest neighbor methods.
Code (0)
등록된 구현이 없습니다.
Tasks
regressionSimilar Papers 제목 키워드 기반
Optimal PAC Bounds Without Uniform Convergence
In statistical learning theory, determining the sample complexity of realizable binary classification for VC classes was a long-standing open problem. The results of Simon and Hanneke established sharp upper bounds in th…
Binary ClassificationClassificationLearning TheoryConvergence analysis of controlled particle systems arising in deep learning: from finite to infinite sample size
This paper deals with a class of neural SDEs and studies the limiting behavior of the associated sampled optimal control problems as the sample size grows to infinity. The neural SDEs with $N$ samples can be linked to th…
Uniform Convergence Rates for Lipschitz Learning on Graphs
Lipschitz learning is a graph-based semi-supervised learning method where one extends labels from a labeled to an unlabeled data set by solving the infinity Laplace equation on a weighted graph. In this work we prove uni…
Graph-based Generalization Bounds for Learning Binary Relations
We investigate the generalizability of learned binary relations: functions that map pairs of instances to a logical indicator. This problem has application in numerous areas of machine learning, such as ranking, entity r…
Entity ResolutionGeneralization BoundsLink PredictionAbnormal Mutations: Evolution Strategies Don't Require Gaussianity
The mutation process in evolution strategies has been interlinked with the normal distribution since its inception. Many lines of reasoning have been given for this strong dependency, ranging from maximum entropy argumen…