[Paper Review] A rate optimal procedure for sparse signal recovery under dependence

Jun Li, Ping‐Shou Zhong|arXiv (Cornell University)|Oct 10, 2014

Sparse and Compressive Sensing Techniques19 references3 citations

TL;DR

This paper proposes the Dependence-Assisted Thresholding and Excising (DATE) procedure for sparse signal recovery in high-dimensional settings with dependent random vectors. By leveraging cross-variable dependence, DATE achieves a faster optimal convergence rate for the marginal false non-discovery rate (mFNR) compared to independent data, and it controls the marginal false discovery rate (mFDR) at a pre-specified level, demonstrating improved signal identification under dependence.

ABSTRACT

The paper considers the problem of identifying the sparse different components between two high dimensional means of column-wise dependent random vectors. We show that the dependence can be utilized to lower the identification boundary for signal recovery. Moreover, an optimal convergence rate for the marginal false non-discovery rate (mFNR) is established under the dependence. The convergence rate is faster than the optimal rate without dependence. To recover the sparse signal bearing dimensions, we propose a Dependence-Assisted Thresholding and Excising (DATE) procedure, which is shown to be rate optimal for the mFNR with the marginal false discovery rate (mFDR) controlled at a pre-specified level. Simulation studies and case study are given to demonstrate the performance of the proposed signal identification procedure.

Motivation & Objective

• To address the challenge of identifying sparse signals in high-dimensional data where variables are column-wise dependent, common in genetic and omics studies.
• To investigate how dependence among variables affects the signal identification boundary, particularly whether dependence can lower the threshold for successful signal recovery.
• To establish the optimal convergence rate for the marginal false non-discovery rate (mFNR) under dependence, showing it is faster than under independence.
• To develop a practical, rate-optimal procedure that controls mFDR while minimizing mFNR in the presence of dependence.
• To validate the method through simulations and a real-world case study on differentially expressed genes in chromosome X.

Proposed method

• Proposes the DATE (Dependence-Assisted Thresholding and Excising) procedure, which integrates dependence structure into signal detection via estimated covariance matrices.
• Uses a thresholding rule based on test statistics that accounts for the dependence structure, improving detection power over standard methods.
• Employs a data-driven estimation of the covariance matrix Σ to model the dependence among variables, enabling accurate signal identification.
• Controls the marginal false discovery rate (mFDR) at a pre-specified level using a modified Benjamini-Hochberg-type procedure adapted to dependence.
• Derives the optimal convergence rate for mFNR under dependence, showing it is strictly faster than the rate under independence.
• Applies the method to high-dimensional two-sample mean comparison, modeling data as X_ij = μ_i + ε_ij with ε_ij i.i.d. N(0, Σ_i), and focuses on δ = μ_1 - μ_2.

Experimental results

Research questions

• RQ1 How does dependence among high-dimensional variables affect the signal identification boundary in sparse signal recovery?
• RQ2 Can the dependence structure be leveraged to achieve a faster optimal convergence rate for the marginal false non-discovery rate (mFNR) compared to independent data?
• RQ3 What is the optimal convergence rate for mFNR under general dependence, and how does it compare to the independent case?
• RQ4 Is there a practical procedure that achieves this optimal rate while controlling mFDR at a pre-specified level?
• RQ5 How does the proposed DATE procedure compare to existing methods like BH in terms of mFDR, mFNR, and true positive detection in finite samples?

Key findings

• The signal identification boundary under dependence is strictly lower than under independence, meaning signals can be recovered with weaker effect sizes when dependence is accounted for.
• The optimal convergence rate for the marginal false non-discovery rate (mFNR) under dependence is faster than the rate achieved under independence, demonstrating a theoretical advantage of dependence exploitation.
• The DATE procedure achieves the optimal convergence rate for mFNR while controlling mFDR at a pre-specified level, making it rate optimal in the dependent setting.
• Simulation studies show that DATE maintains low mFDR (around 0.03–0.05) and mFNR (around 0.005–0.007) across various sparsity levels and signal strengths, outperforming the BH procedure.
• In the case study on chromosome X, DATE identified 39 differentially expressed genes at FDR = 0.001, compared to 27 by BH, with 22 genes common to both, indicating higher sensitivity.
• The method is robust to misspecification of the covariance matrix, as shown by the performance of DATE with estimated Ω̂ (DATE_Ω̂) closely matching that of DATE with true Σ (DATE_Ω).