Stable Spectral Learning Based on Schur Decomposition

Nikos Vlassis Adobe, Nicolo Colombo LCSB Univ of Luxembourg
Proceedings of the 31st Conference on Uncertainty in Artificial Intelligence, PMLR R13:596-603, 2015.

Abstract

Spectral methods are a powerful tool for inferring the parameters of certain classes of probability distributions by means of standard eigenvalue-eigenvector decompositions. Spectral algorithms can be orders of magnitude faster than log-likelihood based and related iterative methods, and, thanks to the uniqueness of the spectral decomposition, they enjoy global optimality guarantees. In practice, however, the applicability of spectral methods is limited due to their sensitivity to model misspecification, which can cause instability issues in the case of non-exact models. We present a new spectral approach that is based on the Schur triangularization of a family of nearly-commuting matrices, and we carry out the corresponding theoretical analysis. Our main result is a theoretical bound on the estimation error, which is shown to depend directly on the model misspecification error and inversely on an eigenvalue separation gap. Numerical experiments show that the proposed method is more stable, and performs better in general, than the classical spectral approach based on direct matrix diagonalization.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR13-adobe15a, title = {Stable Spectral Learning Based on Schur Decomposition}, author = {Adobe, Nikos Vlassis and Luxembourg, Nicolo Colombo LCSB Univ of}, booktitle = {Proceedings of the 31st Conference on Uncertainty in Artificial Intelligence}, pages = {596--603}, year = {2015}, editor = {Meila, Marina and Heskes, Tom}, volume = {R13}, series = {Proceedings of Machine Learning Research}, month = {12--16 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r13/main/assets/adobe15a/adobe15a.pdf}, url = {https://proceedings.mlr.press/r13/adobe15a.html}, abstract = {Spectral methods are a powerful tool for inferring the parameters of certain classes of probability distributions by means of standard eigenvalue-eigenvector decompositions. Spectral algorithms can be orders of magnitude faster than log-likelihood based and related iterative methods, and, thanks to the uniqueness of the spectral decomposition, they enjoy global optimality guarantees. In practice, however, the applicability of spectral methods is limited due to their sensitivity to model misspecification, which can cause instability issues in the case of non-exact models. We present a new spectral approach that is based on the Schur triangularization of a family of nearly-commuting matrices, and we carry out the corresponding theoretical analysis. Our main result is a theoretical bound on the estimation error, which is shown to depend directly on the model misspecification error and inversely on an eigenvalue separation gap. Numerical experiments show that the proposed method is more stable, and performs better in general, than the classical spectral approach based on direct matrix diagonalization.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T Stable Spectral Learning Based on Schur Decomposition %A Nikos Vlassis Adobe %A Nicolo Colombo LCSB Univ of Luxembourg %B Proceedings of the 31st Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2015 %E Marina Meila %E Tom Heskes %F pmlr-vR13-adobe15a %I PMLR %P 596--603 %U https://proceedings.mlr.press/r13/adobe15a.html %V R13 %X Spectral methods are a powerful tool for inferring the parameters of certain classes of probability distributions by means of standard eigenvalue-eigenvector decompositions. Spectral algorithms can be orders of magnitude faster than log-likelihood based and related iterative methods, and, thanks to the uniqueness of the spectral decomposition, they enjoy global optimality guarantees. In practice, however, the applicability of spectral methods is limited due to their sensitivity to model misspecification, which can cause instability issues in the case of non-exact models. We present a new spectral approach that is based on the Schur triangularization of a family of nearly-commuting matrices, and we carry out the corresponding theoretical analysis. Our main result is a theoretical bound on the estimation error, which is shown to depend directly on the model misspecification error and inversely on an eigenvalue separation gap. Numerical experiments show that the proposed method is more stable, and performs better in general, than the classical spectral approach based on direct matrix diagonalization. %Z Reissued by PMLR on 04 October 2026.
APA
Adobe, N.V. & Luxembourg, N.C.L.U.o.. (2015). Stable Spectral Learning Based on Schur Decomposition. Proceedings of the 31st Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R13:596-603 Available from https://proceedings.mlr.press/r13/adobe15a.html. Reissued by PMLR on 04 October 2026.

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