Minimax-optimal semi-supervised regression on unknown manifolds

Amit Moscovich, Ariel Jaffe, Nadler Boaz
; Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:933-942, 2017.

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(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.

Cite this Paper


BibTeX
@InProceedings{pmlr-v54-moscovich17a, title = {{Minimax-optimal semi-supervised regression on unknown manifolds}}, author = {Amit Moscovich and Ariel Jaffe and Nadler Boaz}, booktitle = {Proceedings of the 20th International Conference on Artificial Intelligence and Statistics}, pages = {933--942}, year = {2017}, editor = {Aarti Singh and Jerry Zhu}, volume = {54}, series = {Proceedings of Machine Learning Research}, address = {Fort Lauderdale, FL, USA}, month = {20--22 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v54/moscovich17a/moscovich17a.pdf}, url = {http://proceedings.mlr.press/v54/moscovich17a.html}, 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(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.} }
Endnote
%0 Conference Paper %T Minimax-optimal semi-supervised regression on unknown manifolds %A Amit Moscovich %A Ariel Jaffe %A Nadler Boaz %B Proceedings of the 20th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2017 %E Aarti Singh %E Jerry Zhu %F pmlr-v54-moscovich17a %I PMLR %J Proceedings of Machine Learning Research %P 933--942 %U http://proceedings.mlr.press %V 54 %W PMLR %X 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.
APA
Moscovich, A., Jaffe, A. & Boaz, N.. (2017). Minimax-optimal semi-supervised regression on unknown manifolds. Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, in PMLR 54:933-942

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