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One-bit-matching theorem for ICA, convex-concave programming on polyhedral set, and distribution approximation for

¹Department of Computer Science and Engineering, Chinese University of Hong Kong, Shatin, NT, Hong Kong, PRC. lxu@cse.cuhk.edu.hk

Neural Computation

|January 9, 2007

Summary

This study proves the one-bit-matching conjecture holds even at local optima for independent component analysis. This validates iterative algorithms used for source separation, even when not reaching global solutions.

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Area of Science:

Signal Processing
Statistical Inference

Background:

The one-bit-matching conjecture, crucial for source separation, was previously proven only for global optima in independent component analysis (ICA).
Existing iterative ICA algorithms often converge to local optima, limiting the theoretical support for their success in source separation.

Discussion:

This work provides a novel mathematical proof demonstrating the one-bit-matching conjecture's validity at local optimal solutions.
The proof extends to convex-concave programming on polyhedral sets, offering broader applicability.

Key Insights:

The conjecture holds for local optima, validating many practical iterative source separation algorithms.
New theorems address partial source separation based on kurtosis sign matching and reveal a duality between maximization and minimization for super- and sub-gaussian sources.
A corollary confirms the symmetric orthogonalization method for parallel multi-source separation.

Outlook:

Establishes a link between source separation, combinatorial optimization, and manifold perspectives.
Introduces algorithms with guaranteed convergence and constraint satisfaction for advanced signal processing tasks.