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

被引:33
作者
Dunlop, Matthew M. [1 ]
Slepcev, Dejan [2 ]
Stuart, Andrew M. [1 ]
Thorpe, Matthew [3 ]
机构
[1] CALTECH, Comp & Math Sci, Pasadena, CA 91125 USA
[2] Carnegie Mellon Univ, Dept Math Sci, Pittsburgh, PA 15213 USA
[3] Univ Cambridge, Dept Appl Math & Theoret Phys, Cambridge CB3 0WA, England
基金
美国国家科学基金会;
关键词
Semi-supervised learning; Bayesian inference; Higher-order fractional Laplacian; Asymptotic consistency; Kriging; CONVERGENCE; LAPLACIAN; INTERPOLATION; MINIMIZATION; MANIFOLDS;
D O I
10.1016/j.acha.2019.03.005
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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 G-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. (C) 2019 Elsevier Inc. All rights reserved.
引用
收藏
页码:655 / 697
页数:43
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