Earlier Version

Approximate message passing with spectral initialization for generalized linear models

Conference Paper | Published | English
Mondelli, MarcoISTA ; Venkataramanan, Ramji
Banerjee, Arindam; Fukumizu, Kenji
Series Title
Proceedings of Machine Learning Research
We consider the problem of estimating a signal from measurements obtained via a generalized linear model. We focus on estimators based on approximate message passing (AMP), a family of iterative algorithms with many appealing features: the performance of AMP in the high-dimensional limit can be succinctly characterized under suitable model assumptions; AMP can also be tailored to the empirical distribution of the signal entries, and for a wide class of estimation problems, AMP is conjectured to be optimal among all polynomial-time algorithms. However, a major issue of AMP is that in many models (such as phase retrieval), it requires an initialization correlated with the ground-truth signal and independent from the measurement matrix. Assuming that such an initialization is available is typically not realistic. In this paper, we solve this problem by proposing an AMP algorithm initialized with a spectral estimator. With such an initialization, the standard AMP analysis fails since the spectral estimator depends in a complicated way on the design matrix. Our main contribution is a rigorous characterization of the performance of AMP with spectral initialization in the high-dimensional limit. The key technical idea is to define and analyze a two-phase artificial AMP algorithm that first produces the spectral estimator, and then closely approximates the iterates of the true AMP. We also provide numerical results that demonstrate the validity of the proposed approach.
Publishing Year
Date Published
Proceedings Title
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics
The authors would like to thank Andrea Montanari for helpful discussions. M. Mondelli was partially supported by the 2019 Lopez-Loreta Prize. R. Venkataramanan was partially supported by the Alan Turing Institute under the EPSRC grant EP/N510129/1.
AISTATS: Artificial Intelligence and Statistics
Conference Location
Virtual, San Diego, CA, United States
Conference Date
2021-04-13 – 2021-04-15
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arXiv 2010.03460

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