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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
Summary
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.
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.
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