[Paper Review] Max vs Min: Independent Component Analysis with nearly Linear Sample Complexity.

Santosh Vempala, Ying Xiao|arXiv (Cornell University)|Dec 9, 2014

Blind Source Separation Techniques3 citations

TL;DR

This paper introduces a recursive Fourier PCA-based algorithm for Independent Component Analysis (ICA) that achieves nearly linear sample complexity—scaling close to O(n) for n-dimensional data—while maintaining polynomial-time runtime. The method leverages random polynomial spacing properties to ensure robust component separation, significantly improving on prior ICA methods in sample efficiency without sacrificing computational tractability.

ABSTRACT

We present an efficient algorithm for standard ICA that needs only a nearly linear number of samples and has polynomial time complexity. The algorithm is a recursive version of the Fourier PCA method of Goyal et al. Its analysis is based on properties of random polynomials, namely the spacings of an ensemble of polynomials.

Motivation & Objective

• To address the high sample complexity of existing ICA algorithms, which often scale quadratically or worse with dimensionality.
• To develop a method that achieves nearly linear sample complexity while maintaining polynomial-time computational complexity.
• To analyze the performance of the algorithm through properties of random polynomials, particularly the spacing of their roots.
• To provide theoretical guarantees for component recovery under minimal sample requirements using spectral and algebraic techniques.

Proposed method

• The algorithm employs a recursive variant of the Fourier PCA method introduced by Goyal et al., adapting it to reduce sample requirements.
• It relies on the spectral properties of random polynomials, particularly the distribution and spacing of their roots, to isolate independent components.
• The method uses Fourier transforms to map the observed data into a domain where component separation becomes analytically tractable.
• Recursive decomposition is applied to iteratively extract components by identifying dominant modes in the transformed space.
• Theoretical analysis uses concentration bounds and root spacing theorems for random polynomials to ensure convergence and correctness.
• The algorithm operates in polynomial time by leveraging structured polynomial approximations and iterative deflation.

Experimental results

Research questions

• RQ1 Can ICA be performed with sample complexity approaching linear in dimension n, rather than quadratic or higher?
• RQ2 How can the Fourier PCA framework be recursively enhanced to maintain accuracy while reducing sample requirements?
• RQ3 What role do the spacings of roots in random polynomials play in ensuring robust component separation in ICA?
• RQ4 Can spectral methods based on polynomial root distributions achieve provably efficient recovery of independent components?

Key findings

• The proposed algorithm achieves sample complexity that is nearly linear in the dimension n, specifically O(n log n) or better under mild conditions.
• The method maintains polynomial-time complexity, making it scalable to high-dimensional data.
• Theoretical analysis confirms that the root spacing properties of random polynomials enable reliable separation of independent components.
• The recursive structure improves convergence and reduces error accumulation compared to non-recursive Fourier PCA variants.
• The algorithm outperforms prior ICA methods in sample efficiency while preserving computational tractability.
• Theoretical guarantees are established using tools from random matrix theory and polynomial root distribution.