Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolvers

Yeqiu Chen, Ziyan Liu, Hong Wang, Lei Liu
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:15702-15733, 2026.

Abstract

Solving large-scale Generalized Eigenvalue Problems (GEPs) is a fundamental yet computationally prohibitive task in science and engineering. As a promising direction, contour integral (CI) methods offer an efficient and parallelizable framework. However, their performance is critically dependent on the selection of integration contours—improper selection without reliable prior knowledge of eigenvalue distribution can incur significant computational overhead and compromise numerical accuracy. To address this challenge, we propose Deepcontour, a novel hybrid framework that integrates a deep learning-based spectral predictor with Kernel Density Estimation (KDE) for principled contour design. Specifically, Deepcontour utilizes its specialized Eigen-Neural-Operator (ENO) to provide rapid spectral distribution priors, driving a KDE module to automatically construct the optimized integration contours, which guide the CI solver to efficiently find the desired eigenvalues. Deepcontour achieves up to a 5.63x speedup across diverse scientific datasets while maintaining strict numerical rigor. By merging the predictive power of deep learning with the numerical rigor of classical solvers, this work establishes an efficient and robust paradigm for solving large-scale GEPs.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-chen26ck, title = {Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolvers}, author = {Chen, Yeqiu and Liu, Ziyan and Wang, Hong and Liu, Lei}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {15702--15733}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/chen26ck/chen26ck.pdf}, url = {https://proceedings.mlr.press/v306/chen26ck.html}, abstract = {Solving large-scale Generalized Eigenvalue Problems (GEPs) is a fundamental yet computationally prohibitive task in science and engineering. As a promising direction, contour integral (CI) methods offer an efficient and parallelizable framework. However, their performance is critically dependent on the selection of integration contours—improper selection without reliable prior knowledge of eigenvalue distribution can incur significant computational overhead and compromise numerical accuracy. To address this challenge, we propose Deepcontour, a novel hybrid framework that integrates a deep learning-based spectral predictor with Kernel Density Estimation (KDE) for principled contour design. Specifically, Deepcontour utilizes its specialized Eigen-Neural-Operator (ENO) to provide rapid spectral distribution priors, driving a KDE module to automatically construct the optimized integration contours, which guide the CI solver to efficiently find the desired eigenvalues. Deepcontour achieves up to a 5.63x speedup across diverse scientific datasets while maintaining strict numerical rigor. By merging the predictive power of deep learning with the numerical rigor of classical solvers, this work establishes an efficient and robust paradigm for solving large-scale GEPs.} }
Endnote
%0 Conference Paper %T Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolvers %A Yeqiu Chen %A Ziyan Liu %A Hong Wang %A Lei Liu %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-chen26ck %I PMLR %P 15702--15733 %U https://proceedings.mlr.press/v306/chen26ck.html %V 306 %X Solving large-scale Generalized Eigenvalue Problems (GEPs) is a fundamental yet computationally prohibitive task in science and engineering. As a promising direction, contour integral (CI) methods offer an efficient and parallelizable framework. However, their performance is critically dependent on the selection of integration contours—improper selection without reliable prior knowledge of eigenvalue distribution can incur significant computational overhead and compromise numerical accuracy. To address this challenge, we propose Deepcontour, a novel hybrid framework that integrates a deep learning-based spectral predictor with Kernel Density Estimation (KDE) for principled contour design. Specifically, Deepcontour utilizes its specialized Eigen-Neural-Operator (ENO) to provide rapid spectral distribution priors, driving a KDE module to automatically construct the optimized integration contours, which guide the CI solver to efficiently find the desired eigenvalues. Deepcontour achieves up to a 5.63x speedup across diverse scientific datasets while maintaining strict numerical rigor. By merging the predictive power of deep learning with the numerical rigor of classical solvers, this work establishes an efficient and robust paradigm for solving large-scale GEPs.
APA
Chen, Y., Liu, Z., Wang, H. & Liu, L.. (2026). Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolvers. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:15702-15733 Available from https://proceedings.mlr.press/v306/chen26ck.html.

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