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On Support Recovery with Sparse CCA: Information Theoretic and Computational Limits.

Nilanjana Laha1, Rajarshi Mukherjee2

  • 1Department of Statistics, Texas A&M University, College Station, TX 77843.

IEEE Transactions on Information Theory
|October 16, 2023
PubMed
Summary
This summary is machine-generated.

We explored support recovery in high-dimensional Canonical Correlation Analysis (CCA). Support recovery is possible with low sparsity but impossible with high sparsity, with moderate sparsity showing complex computational trade-offs.

Keywords:
Canonical Correlation AnalysisHigh DimensionLow Degree PolynomialsSupport RecoveryVariable Selection

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Area of Science:

  • Statistics
  • Machine Learning
  • Data Science

Background:

  • Canonical Correlation Analysis (CCA) is a statistical method to find relationships between two sets of variables.
  • High-dimensional data and sparse structures present significant challenges in statistical analysis.
  • Support recovery is crucial for identifying relevant variables in complex datasets.

Purpose of the Study:

  • To investigate asymptotically exact support recovery in high-dimensional and sparse Canonical Correlation Analysis (CCA).
  • To delineate different sparsity regimes and their implications for computational and information-theoretic feasibility of support recovery.
  • To establish conditions for consistent support recovery and explore the limits of polynomial-time algorithms.

Main Methods:

  • Information-theoretic analysis to establish lower bounds for support recovery.
  • Development and analysis of computationally efficient algorithms for support recovery.
  • Utilizing coordinate thresholding methods and the 'Low Degree Polynomial' Conjecture for computational complexity analysis.

Main Results:

  • Identified four distinct sparsity regimes impacting support recovery feasibility.
  • Demonstrated that support recovery is achievable with low sparsity but information-theoretically impossible with high sparsity.
  • Showed polynomial-time recovery is possible in moderate sparsity regimes, but potentially inconsistent in higher moderate sparsity based on the 'Low Degree Polynomial' Conjecture.

Conclusions:

  • The feasibility of support recovery in sparse CCA is highly dependent on the sparsity level.
  • There are fundamental limits to support recovery in high-dimensional settings, influenced by both statistical and computational factors.
  • The study provides a comprehensive understanding of support recovery across different sparsity regimes, guiding future algorithm development.