Fast Spectrally Sparse Signal Reconstruction via Jacobi-Preconditioned Gradient Descent

Jian-Feng Cai, Xueyang Quan, Yang Wang, Jiaxi Ying
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:10669-10704, 2026.

Abstract

Spectrally sparse signal reconstruction arises in a wide range of applications and can be formulated as a low-rank Hankel matrix completion problem. We develop a Jacobi-preconditioned gradient descent method that preserves the low per-iteration complexity of first-order algorithms while achieving linear convergence at a rate independent of the condition number. By introducing a generator that maps factor-based iterates to matrix space, we establish equivalence with manifold-based methods, enabling direct convergence analysis while avoiding the need to define distances under complex-symmetric factorization ambiguity. Extensive experiments demonstrate that the proposed algorithm outperforms state-of-the-art methods in both iteration count and computational time across a broad range of problem settings.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-cai26j, title = {Fast Spectrally Sparse Signal Reconstruction via Jacobi-Preconditioned Gradient Descent}, author = {Cai, Jian-Feng and Quan, Xueyang and Wang, Yang and Ying, Jiaxi}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {10669--10704}, 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/cai26j/cai26j.pdf}, url = {https://proceedings.mlr.press/v306/cai26j.html}, abstract = {Spectrally sparse signal reconstruction arises in a wide range of applications and can be formulated as a low-rank Hankel matrix completion problem. We develop a Jacobi-preconditioned gradient descent method that preserves the low per-iteration complexity of first-order algorithms while achieving linear convergence at a rate independent of the condition number. By introducing a generator that maps factor-based iterates to matrix space, we establish equivalence with manifold-based methods, enabling direct convergence analysis while avoiding the need to define distances under complex-symmetric factorization ambiguity. Extensive experiments demonstrate that the proposed algorithm outperforms state-of-the-art methods in both iteration count and computational time across a broad range of problem settings.} }
Endnote
%0 Conference Paper %T Fast Spectrally Sparse Signal Reconstruction via Jacobi-Preconditioned Gradient Descent %A Jian-Feng Cai %A Xueyang Quan %A Yang Wang %A Jiaxi Ying %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-cai26j %I PMLR %P 10669--10704 %U https://proceedings.mlr.press/v306/cai26j.html %V 306 %X Spectrally sparse signal reconstruction arises in a wide range of applications and can be formulated as a low-rank Hankel matrix completion problem. We develop a Jacobi-preconditioned gradient descent method that preserves the low per-iteration complexity of first-order algorithms while achieving linear convergence at a rate independent of the condition number. By introducing a generator that maps factor-based iterates to matrix space, we establish equivalence with manifold-based methods, enabling direct convergence analysis while avoiding the need to define distances under complex-symmetric factorization ambiguity. Extensive experiments demonstrate that the proposed algorithm outperforms state-of-the-art methods in both iteration count and computational time across a broad range of problem settings.
APA
Cai, J., Quan, X., Wang, Y. & Ying, J.. (2026). Fast Spectrally Sparse Signal Reconstruction via Jacobi-Preconditioned Gradient Descent. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:10669-10704 Available from https://proceedings.mlr.press/v306/cai26j.html.

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