Out-of-sample extension of graph adjacency spectral embedding

Keith Levin, Fred Roosta, Michael Mahoney, Carey Priebe
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:2975-2984, 2018.

Abstract

Many popular dimensionality reduction procedures have out-of-sample extensions, which allow a practitioner to apply a learned embedding to observations not seen in the initial training sample. In this work, we consider the problem of obtaining an out-of-sample extension for the adjacency spectral embedding, a procedure for embedding the vertices of a graph into Euclidean space. We present two different approaches to this problem, one based on a least-squares objective and the other based on a maximum-likelihood formulation. We show that if the graph of interest is drawn according to a certain latent position model called a random dot product graph, then both of these out-of-sample extensions estimate the true latent position of the out-of-sample vertex with the same error rate. Further, we prove a central limit theorem for the least-squares-based extension, showing that the estimate is asymptotically normal about the truth in the large-graph limit.

Cite this Paper

BibTeX
@InProceedings{pmlr-v80-levin18a, title = {Out-of-sample extension of graph adjacency spectral embedding}, author = {Levin, Keith and Roosta, Fred and Mahoney, Michael and Priebe, Carey}, booktitle = {Proceedings of the 35th International Conference on Machine Learning}, pages = {2975--2984}, year = {2018}, editor = {Dy, Jennifer and Krause, Andreas}, volume = {80}, series = {Proceedings of Machine Learning Research}, month = {10--15 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v80/levin18a/levin18a.pdf}, url = {https://proceedings.mlr.press/v80/levin18a.html}, abstract = {Many popular dimensionality reduction procedures have out-of-sample extensions, which allow a practitioner to apply a learned embedding to observations not seen in the initial training sample. In this work, we consider the problem of obtaining an out-of-sample extension for the adjacency spectral embedding, a procedure for embedding the vertices of a graph into Euclidean space. We present two different approaches to this problem, one based on a least-squares objective and the other based on a maximum-likelihood formulation. We show that if the graph of interest is drawn according to a certain latent position model called a random dot product graph, then both of these out-of-sample extensions estimate the true latent position of the out-of-sample vertex with the same error rate. Further, we prove a central limit theorem for the least-squares-based extension, showing that the estimate is asymptotically normal about the truth in the large-graph limit.} }
Endnote
%0 Conference Paper %T Out-of-sample extension of graph adjacency spectral embedding %A Keith Levin %A Fred Roosta %A Michael Mahoney %A Carey Priebe %B Proceedings of the 35th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2018 %E Jennifer Dy %E Andreas Krause %F pmlr-v80-levin18a %I PMLR %P 2975--2984 %U https://proceedings.mlr.press/v80/levin18a.html %V 80 %X Many popular dimensionality reduction procedures have out-of-sample extensions, which allow a practitioner to apply a learned embedding to observations not seen in the initial training sample. In this work, we consider the problem of obtaining an out-of-sample extension for the adjacency spectral embedding, a procedure for embedding the vertices of a graph into Euclidean space. We present two different approaches to this problem, one based on a least-squares objective and the other based on a maximum-likelihood formulation. We show that if the graph of interest is drawn according to a certain latent position model called a random dot product graph, then both of these out-of-sample extensions estimate the true latent position of the out-of-sample vertex with the same error rate. Further, we prove a central limit theorem for the least-squares-based extension, showing that the estimate is asymptotically normal about the truth in the large-graph limit.
APA
Levin, K., Roosta, F., Mahoney, M. & Priebe, C.. (2018). Out-of-sample extension of graph adjacency spectral embedding. Proceedings of the 35th International Conference on Machine Learning, in Proceedings of Machine Learning Research 80:2975-2984 Available from https://proceedings.mlr.press/v80/levin18a.html.

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