Adaptive Power Method: Eigenvector Estimation from Sampled Data

Seiyun Shin, Han Zhao, Ilan Shomorony
Proceedings of The 34th International Conference on Algorithmic Learning Theory, PMLR 201:1387-1410, 2023.

Abstract

Computing the dominant eigenvectors of a matrix $A$ has many applications, such as principal component analysis, spectral embedding, and PageRank. However, in general, this task relies on the complete knowledge of the matrix $A$, which can be too large to store or even infeasible to observe in many applications, e.g., large-scale social networks. Thus, a natural question is how to accurately estimate the eigenvectors of $A$ when only partial observations can be made by sampling entries from $A$. To this end, we propose the Adaptive Power Method (\textsc{APM}), a variant of the well-known power method. At each power iteration, \textsc{APM} adaptively selects a subset of the entries of $A$ to observe based on the current estimate of the top eigenvector. We show that \textsc{APM} can estimate the dominant eigenvector(s) of $A$ with squared error at most $\epsilon$ by observing roughly $O(n\epsilon^{-2} \log^2 (n/\epsilon))$ entries of an $n\times n$ matrix. We present empirical results for the problem of eigenvector centrality computation on two real-world graphs and show that \textsc{APM} significantly outperforms a non-adaptive estimation algorithm using the same number of observations. Furthermore, in the context of eigenvector centrality, \textsc{APM} can also adaptively allocate the observation budget to selectively refine the estimate of nodes with high centrality scores in the graph.

Cite this Paper

BibTeX
@InProceedings{pmlr-v201-shin23a, title = {{Adaptive Power Method: Eigenvector Estimation from Sampled Data}}, author = {Shin, Seiyun and Zhao, Han and Shomorony, Ilan}, booktitle = {Proceedings of The 34th International Conference on Algorithmic Learning Theory}, pages = {1387--1410}, year = {2023}, editor = {Agrawal, Shipra and Orabona, Francesco}, volume = {201}, series = {Proceedings of Machine Learning Research}, month = {20 Feb--23 Feb}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v201/shin23a/shin23a.pdf}, url = {https://proceedings.mlr.press/v201/shin23a.html}, abstract = {Computing the dominant eigenvectors of a matrix $A$ has many applications, such as principal component analysis, spectral embedding, and PageRank. However, in general, this task relies on the complete knowledge of the matrix $A$, which can be too large to store or even infeasible to observe in many applications, e.g., large-scale social networks. Thus, a natural question is how to accurately estimate the eigenvectors of $A$ when only partial observations can be made by sampling entries from $A$. To this end, we propose the Adaptive Power Method (\textsc{APM}), a variant of the well-known power method. At each power iteration, \textsc{APM} adaptively selects a subset of the entries of $A$ to observe based on the current estimate of the top eigenvector. We show that \textsc{APM} can estimate the dominant eigenvector(s) of $A$ with squared error at most $\epsilon$ by observing roughly $O(n\epsilon^{-2} \log^2 (n/\epsilon))$ entries of an $n\times n$ matrix. We present empirical results for the problem of eigenvector centrality computation on two real-world graphs and show that \textsc{APM} significantly outperforms a non-adaptive estimation algorithm using the same number of observations. Furthermore, in the context of eigenvector centrality, \textsc{APM} can also adaptively allocate the observation budget to selectively refine the estimate of nodes with high centrality scores in the graph.} }
Endnote
%0 Conference Paper %T Adaptive Power Method: Eigenvector Estimation from Sampled Data %A Seiyun Shin %A Han Zhao %A Ilan Shomorony %B Proceedings of The 34th International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2023 %E Shipra Agrawal %E Francesco Orabona %F pmlr-v201-shin23a %I PMLR %P 1387--1410 %U https://proceedings.mlr.press/v201/shin23a.html %V 201 %X Computing the dominant eigenvectors of a matrix $A$ has many applications, such as principal component analysis, spectral embedding, and PageRank. However, in general, this task relies on the complete knowledge of the matrix $A$, which can be too large to store or even infeasible to observe in many applications, e.g., large-scale social networks. Thus, a natural question is how to accurately estimate the eigenvectors of $A$ when only partial observations can be made by sampling entries from $A$. To this end, we propose the Adaptive Power Method (\textsc{APM}), a variant of the well-known power method. At each power iteration, \textsc{APM} adaptively selects a subset of the entries of $A$ to observe based on the current estimate of the top eigenvector. We show that \textsc{APM} can estimate the dominant eigenvector(s) of $A$ with squared error at most $\epsilon$ by observing roughly $O(n\epsilon^{-2} \log^2 (n/\epsilon))$ entries of an $n\times n$ matrix. We present empirical results for the problem of eigenvector centrality computation on two real-world graphs and show that \textsc{APM} significantly outperforms a non-adaptive estimation algorithm using the same number of observations. Furthermore, in the context of eigenvector centrality, \textsc{APM} can also adaptively allocate the observation budget to selectively refine the estimate of nodes with high centrality scores in the graph.
APA
Shin, S., Zhao, H. & Shomorony, I.. (2023). Adaptive Power Method: Eigenvector Estimation from Sampled Data. Proceedings of The 34th International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 201:1387-1410 Available from https://proceedings.mlr.press/v201/shin23a.html.

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