Core Concepts
The core message of this paper is to provide a precise asymptotic characterization of the performance of spectral methods for estimating multiple signals in mixed generalized linear models, and to use this characterization to optimize the design of such spectral estimators.
Abstract
The paper considers a mixed generalized linear model (GLM) setting, where the goal is to learn multiple d-dimensional signals x^* from n unlabeled observations. Each observation comes from exactly one of the signals, but it is not known which one.
The authors focus on spectral methods, which output the top eigenvectors of a suitable data-dependent matrix, as a popular class of estimators for this problem. Despite their wide applicability, the design of spectral methods is typically obtained via heuristic considerations, and the number of samples n needed to guarantee recovery is super-linear in the signal dimension d.
The key contributions of the paper are:
- A master theorem (Theorem 3.1) that characterizes the joint distribution of the linear estimator, the spectral estimator, and the signals in the high-dimensional limit where n and d grow proportionally. This allows the authors to:
- Derive the normalized correlations (overlaps) between the linear/spectral estimators and the signals (Corollaries 3.3 and 3.5).
- Determine the optimal preprocessing functions for the linear and spectral estimators that maximize the overlap with each signal (Propositions 3.4 and 3.6).
- Identify the optimal way to combine the linear and spectral estimators (Corollary 3.2).
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Specialization of the results to two canonical settings: mixed linear regression and mixed phase retrieval (Corollaries 3.7-3.9). The analysis reveals intriguing differences in the performance of the linear and spectral estimators across these two models.
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Numerical simulations demonstrating the advantage of the optimized spectral method over existing designs.
The technical approach combines tools from random matrix theory, free probability, and the theory of approximate message passing algorithms.
Stats
The paper does not contain any explicit numerical data or statistics. The key quantities of interest are the asymptotic overlaps between the estimators and the signals, which are expressed in terms of the model parameters and the preprocessing functions.
Quotes
"Spectral methods are a popular class of estimators which output the top two eigenvectors of a suitable data-dependent matrix. However, despite the wide applicability, their design is still obtained via heuristic considerations, and the number of samples n needed to guarantee recovery is super-linear in the signal dimension d."
"Our characterization exploits a mix of tools from random matrices, free probability and the theory of approximate message passing algorithms."