Minimax-optimal semi-supervised regression on unknown manifolds

Amit Moscovich, Ariel Jaffe, Nadler Boaz

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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(k N 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
Title of host publicationProceedings of the 20th International Conference on Artificial Intelligence and Statistics
EditorsAarti Singh, Jerry Zhu
Number of pages10
StatePublished - 1 Aug 2017
Event20th International Conference on Artificial Intelligence and Statistics, AISTATS 2017 - Fort Lauderdale, United States
Duration: 20 Apr 201722 Apr 2017

Publication series

NameProceedings of Machine Learning Research


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


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