Minimax-optimal semi-supervised regression on unknown manifolds

Amit Moscovich, Ariel Jaffe, Boaz Nadler

Research output: Contribution to conferencePaperpeer-review

Abstract

We consider semi-supervised regression when the predictor variables are drawn from an unknown manifold. A simple two step approach to this problem is to: (i) estimate the manifold geodesic distance between any pair of points using both the labeled and unlabeled instances; and (ii) apply a k nearest neighbor regressor based on these distance estimates. We prove that given sufficiently many unlabeled points, this simple method of geodesic kNN regression achieves the optimal finite-sample minimax bound on the mean squared error, as if the manifold were known. Furthermore, we show how this approach can be efficiently implemented, requiring only O(kN log N) operations to estimate the regression function at all N labeled and unlabeled points. We illustrate this approach on two datasets with a manifold structure: indoor localization using WiFi fingerprints and facial pose estimation. In both cases, geodesic kNN is more accurate and much faster than the popular Laplacian eigenvector regressor.

Original languageEnglish
StatePublished - 2017
Externally publishedYes
Event20th International Conference on Artificial Intelligence and Statistics, AISTATS 2017 - Fort Lauderdale, United States
Duration: 20 Apr 201722 Apr 2017

Conference

Conference20th International Conference on Artificial Intelligence and Statistics, AISTATS 2017
Country/TerritoryUnited States
CityFort Lauderdale
Period20/04/1722/04/17

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