Optimal linear estimation under unknown nonlinear transform.

Area of Science:

• Statistics
• Machine Learning
• Signal Processing

Background:

• Linear regression estimates model parameters (β*) from observations (y) using a linear model.
• The single-index model generalizes linear regression by allowing a noisy, quantized, nonlinear, and unknown relationship between the linear predictor (⟨x, β*⟩) and the response (y).
• This model encompasses challenging problems like one-bit compressed sensing.

Purpose of the Study:

• To develop a novel estimation procedure for the single-index model that works under mild conditions on the unknown link function.
• To address parameter estimation in high-dimensional settings (p ≫ n) where the parameter vector (β*) is sparse.
• To establish theoretical guarantees for the proposed estimators, demonstrating their optimality.

Main Methods:

• A novel spectral-based estimation procedure is proposed for the single-index model.
• A two-stage nonconvex framework is introduced to handle sparse parameter estimation in high-dimensional regimes.
• Minimax lower bounds are established for both classical and high-dimensional settings.

Main Results:

• The proposed spectral method successfully recovers the parameter vector (β*) in settings where previous algorithms fail.
• The algorithm requires only mild restrictions on the unknown functional relationship.
• The two-stage framework effectively tackles estimation challenges in high-dimensional scenarios.
• The established lower bounds demonstrate the optimality of the proposed estimators.

Conclusions:

• The novel spectral-based approach provides a robust solution for the single-index model, even with complex link functions.
• The developed framework offers optimal parameter recovery in both low and high-dimensional settings.
• This work advances the field of statistical estimation for generalized linear models and compressed sensing.