1. Approximate message passing with spectral initialization for generalized linear models*
- Author
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Marco Mondelli, Ramji Venkataramanan, and Apollo - University of Cambridge Repository
- Subjects
Statistics and Probability ,FOS: Computer and information sciences ,Paper ,Computer Science - Machine Learning ,4902 Mathematical Physics ,Mathematics::Complex Variables ,Computer Science - Information Theory ,Information Theory (cs.IT) ,Mathematics - Statistics Theory ,Statistical and Nonlinear Physics ,Machine Learning (stat.ML) ,Statistics Theory (math.ST) ,Machine Learning (cs.LG) ,machine learning ,Statistics - Machine Learning ,learning theory ,Machine Learning 2022 ,FOS: Mathematics ,49 Mathematical Sciences ,Statistics, Probability and Uncertainty ,message-passing algorithms ,statistical inference - Abstract
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., Comment: 38 pages, 5 figures, AISTATS 2021
- Published
- 2022
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