Estimating Change Points in High-Dimensional Linear Regression using Approximate Message Passing

Core Concepts

The authors propose an Approximate Message Passing (AMP) algorithm to efficiently estimate the signals and change point locations in high-dimensional linear regression. They provide an exact asymptotic characterization of the algorithm's performance and show how to quantify uncertainty in the estimates.

Abstract

The content discusses the problem of localizing change points in high-dimensional linear regression. The authors propose an Approximate Message Passing (AMP) algorithm to estimate both the signals and the change point locations.
Key highlights:

The AMP algorithm can exploit any prior information on the signal, noise, and change points.
The authors provide an exact asymptotic characterization of the algorithm's estimation performance in the high-dimensional limit.
The AMP algorithm enables uncertainty quantification in the form of an efficiently computable approximate posterior distribution, whose asymptotic form is also characterized.
The authors validate the theory through numerical experiments on synthetic data and images, demonstrating the favorable performance of their estimators.
The content is structured as follows:

Introduction to the problem of change point detection in high-dimensional linear regression.
Preliminaries on the model assumptions and notation.
Description of the AMP algorithm and the main theoretical results:

State evolution characterization of the AMP iterates
Choosing the denoising functions to optimize performance
Change point estimation and uncertainty quantification

Experimental results on synthetic data and images, comparing AMP to other state-of-the-art algorithms.

Stats

The content does not provide any specific numerical values or statistics. It focuses on the theoretical analysis and algorithmic development.

Quotes

None.

Key Insights Distilled From

Inferring Change Points in High-Dimensional Linear Regression via Approximate Message Passing

by Gabriel Arpi... at arxiv.org 04-12-2024

https://arxiv.org/pdf/2404.07864.pdf

Deeper Inquiries

How can the AMP algorithm be extended to handle more general covariate distributions beyond the i.i.d. Gaussian assumption

The AMP algorithm can be extended to handle more general covariate distributions beyond the i.i.d. Gaussian assumption by incorporating the concept of universality. Recent research has shown that AMP algorithms can exhibit universality, meaning that their performance is robust to changes in the distribution of the covariates. By leveraging this universality property, the AMP algorithm can be adapted to work with a broader class of covariate distributions, such as rotationally invariant designs. This extension would involve adjusting the denoising functions and state evolution parameters to accommodate the specific characteristics of the new covariate distribution while maintaining the overall framework of the AMP algorithm.

What are the potential limitations of the change point estimation approach based on the approximate MAP estimate, and how could it be improved

The change point estimation approach based on the approximate MAP estimate may have limitations in scenarios where the signal configuration is complex or the noise level is high. In such cases, the estimator may struggle to accurately identify the true change points, leading to suboptimal performance. To improve this approach, several strategies can be implemented:

Incorporating Prior Information: Utilizing additional prior knowledge about the signal structure or change point locations can enhance the estimation accuracy.
Enhancing Denoising Functions: Developing more sophisticated denoising functions that capture the specific characteristics of the data can improve the estimation performance.
Adopting Adaptive Strategies: Implementing adaptive algorithms that adjust their parameters based on the data characteristics can lead to more robust and accurate change point estimation.

What are the connections between the proposed AMP-based approach and Bayesian methods for change point detection in time series data

The proposed AMP-based approach for change point detection shares several connections with Bayesian methods commonly used in time series data analysis:

Posterior Inference: Both approaches involve posterior inference to estimate the change points or signal configurations. While the AMP algorithm provides an approximate posterior distribution based on the state evolution predictions, Bayesian methods offer a more rigorous probabilistic framework for uncertainty quantification.
Model Flexibility: Bayesian methods allow for the incorporation of complex prior distributions and model structures, enabling more flexibility in capturing the underlying data generating process. The AMP algorithm, on the other hand, focuses on efficient estimation in high-dimensional settings with specific assumptions on the signal and noise distributions.
Computational Efficiency: The AMP algorithm is known for its computational efficiency and scalability in high-dimensional problems, making it suitable for large datasets. Bayesian methods, while providing a principled framework, can be computationally intensive, especially for complex models with high-dimensional data.

By understanding these connections, researchers can leverage the strengths of both approaches to develop more robust and efficient methods for change point detection in various applications.