Projected Gradient Descent for Spectral Compressed Sensing via Symmetric Hankel Factorization

Khái niệm cốt lõi

Proposing a novel nonconvex projected gradient descent method, SHGD, for spectral compressed sensing via symmetric factorization reduces computation and storage costs compared to previous methods.

Tóm tắt

The article introduces the Symmetric Hankel Projected Gradient Descent (SHGD) method for spectral compressed sensing. It highlights the advantages of using symmetric factorization over asymmetric factorization, reducing computational costs. The proposed method demonstrates superior performance in phase transitions and computational efficiency compared to state-of-the-art methods. The complex symmetric factorization employed is novel and introduces a new factorization ambiguity under complex orthogonal transformation. Extensive numerical simulations validate the effectiveness of SHGD in recovering spectrally sparse signals.

Thống kê

SHGD reduces about half of the computation and storage costs compared to prior methods.
Linear convergence guarantee with O(r^2 log(n)) observations.
Numerical simulations demonstrate superior performance in phase transitions and computation efficiency.

Trích dẫn

"SHGD reduces at least half of operations and storage costs."
"Our main contributions are twofold: A new nonconvex symmetrically factorized gradient descent method named SHGD is proposed."

Thông tin chi tiết chính được chắt lọc từ

Projected Gradient Descent for Spectral Compressed Sensing via Symmetric Hankel Factorization

by Jinsheng Li,... lúc arxiv.org 03-15-2024

https://arxiv.org/pdf/2403.09031.pdf

Yêu cầu sâu hơn

How does the complex symmetric factorization introduced in this work impact other fields beyond signal processing

The introduction of complex symmetric factorization in this work has the potential to impact various fields beyond signal processing. One area where it can be particularly beneficial is in machine learning, specifically in dimensionality reduction techniques like Principal Component Analysis (PCA) and Independent Component Analysis (ICA). By incorporating complex symmetric factorization, these algorithms can potentially achieve better performance and efficiency when dealing with high-dimensional data. Additionally, in computer vision applications such as image recognition and object detection, the use of complex symmetric factorization can lead to improved feature extraction and pattern recognition capabilities.

What are potential limitations or drawbacks of using the SHGD method for spectral compressed sensing

While the Symmetric Hankel Projected Gradient Descent (SHGD) method offers several advantages for spectral compressed sensing, there are also some potential limitations or drawbacks to consider:

Complexity: The analysis and implementation of SHGD may require a higher level of mathematical understanding compared to simpler methods.
Convergence Rate: While SHGD demonstrates linear convergence towards the desired signal under certain conditions, it may not always guarantee faster convergence compared to other optimization algorithms.
Sample Complexity: The sample complexity required for SHGD could be relatively high in certain scenarios, which might limit its applicability in real-world settings with limited data availability.
Sensitivity to Parameters: The performance of SHGD could be sensitive to parameters such as step size and initialization conditions, requiring careful tuning for optimal results.

How can the concept of complex orthogonal transformation be applied in other optimization algorithms or mathematical models

The concept of complex orthogonal transformation introduced in this work can have applications beyond spectral compressed sensing:

Optimization Algorithms: In optimization problems involving non-convex functions or matrix factorizations, incorporating complex orthogonal transformations can help improve convergence rates by introducing additional constraints or regularization terms based on symmetry properties.
Signal Processing: In areas like audio processing or speech recognition where signals exhibit inherent symmetries or periodicities, utilizing complex orthogonal transformations can enhance the efficiency and accuracy of signal reconstruction or denoising algorithms.
Machine Learning Models: Complex orthogonal transformations can be integrated into neural network architectures as a form of regularization or weight constraint mechanism to promote symmetry within hidden layers while training deep learning models effectively.

By leveraging the principles behind complex orthogonal transformation across different domains, researchers and practitioners can explore innovative ways to optimize algorithms and enhance computational efficiency while maintaining robustness against noise or uncertainties present in real-world datasets.