Large data and zero noise limits of graph-based semi-supervised learning algorithms

• Published in: Applied and computational harmonic analysis Vol. 49; no. 2; pp. 655 - 697
• Main Authors: Dunlop, Matthew M., Slepčev, Dejan, Stuart, Andrew M., Thorpe, Matthew
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.09.2020
• Subjects: Asymptotic consistency; Bayesian inference; Higher-order fractional Laplacian; Kriging; Semi-supervised learning; 49J55; Semi-supervised learning; 62C10; Asymptotic consistency; 62G20; Bayesian inference; Higher-order fractional Laplacian; 62F15; Kriging
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1016/j.acha.2019.03.005

Scalings in which the graph Laplacian approaches a differential operator in the large graph limit are used to develop understanding of a number of algorithms for semi-supervised learning; in particular, the probit algorithm, level set and kriging methods. Both optimization and Bayesian approaches are considered, based around a regularizing quadratic form found from an affine transformation of the Laplacian, raised to a possibly fractional, exponent. Conditions on the parameters defining this quadratic form are identified under which well-defined limiting continuum analogues of the optimization and Bayesian semi-supervised learning problems may be found, thereby shedding light on the design of algorithms in the large graph setting. The large graph limits of the optimization formulations are tackled through Γ-convergence, using the recently introduced TLp metric. The small labeling noise limits of the Bayesian formulations are also identified, and contrasted with pre-existing harmonic function approaches to the problem.