Large Data and Zero Noise Limits of Graph-Based Semi-Supervised Learning Algorithms
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Abstract
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.
Bibliographical metadata
Original language | English |
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Pages (from-to) | 655-697 |
Number of pages | 43 |
Journal | Applied and Computational Harmonic Analysis |
Volume | 49 |
Issue number | 2 |
Early online date | 4 Apr 2019 |
DOIs | |
Publication status | Published - Sep 2020 |