insight - Computational Complexity - # Stability of Ghurye-Olkin Characterization of Vector Gaussian Distributions

Core Concepts

The stability of the Ghurye-Olkin characterization of Gaussian vectors is analyzed using a partition of the vectors into equivalence classes defined by their matrix factors. The sum of the vectors in each class is near-Gaussian in the characteristic function domain if the Ghurye-Olkin independence condition is approximately met, and any vector projection is near-Gaussian in the distribution function domain.

Abstract

The paper analyzes the stability of the Ghurye-Olkin (GO) characterization of Gaussian vectors. The key insights are:
The vectors are partitioned into equivalence classes based on their matrix factors.
If the sums of the vectors in each class are approximately independent in the characteristic function domain, then the sum of the vectors in each class is near-Gaussian in the characteristic function domain.
Any vector projection is near-Gaussian in the distribution function domain, regardless of the independence condition.
The proofs use tools that establish the stabilities of the Kac-Bernstein and Cramér theorems in the characteristic function and distribution function domains, respectively.
The results are applied to prove stability theorems for differential entropies of Gaussian vectors and blind source separation of non-Gaussian sources.

Stats

The paper does not contain explicit numerical data or statistics. It focuses on theoretical analysis and proofs.

Quotes

"The stability of the Ghurye-Olkin characterization of Gaussian vectors is analyzed using a partition of the vectors into equivalence classes defined by their matrix factors."
"The sum of the vectors in each class is near-Gaussian in the characteristic function domain if the GO independence condition is approximately met."
"All vectors have the property that any vector projection is near-Gaussian in the distribution function domain."

Key Insights Distilled From

by Mahdi Mahvar... at **arxiv.org** 05-06-2024

Deeper Inquiries

The stability results obtained for Gaussian distributions can be extended to more general classes of random vectors by considering the properties of characteristic functions and their relationships with distribution functions. By analyzing the behavior of characteristic functions under perturbations and dependencies, one can establish stability theorems for a broader range of random vectors. This extension may involve exploring the stability of distributional properties beyond Gaussian distributions, such as heavy-tailed distributions or non-parametric distributions. Additionally, incorporating concepts from multivariate analysis and probability theory can help generalize the stability results to diverse classes of random vectors with varying characteristics.

The near-Gaussian property of vector projections has significant implications for applications like dimensionality reduction and feature extraction. In the context of dimensionality reduction, the near-Gaussian behavior of vector projections implies that the transformed data retains Gaussian-like properties, which can facilitate the application of techniques like Principal Component Analysis (PCA) or Independent Component Analysis (ICA). These methods rely on the assumption of Gaussianity for efficient representation and extraction of relevant features. By ensuring that vector projections exhibit near-Gaussian characteristics, the effectiveness of dimensionality reduction algorithms can be enhanced, leading to more accurate and meaningful representations of the data.

The stability analysis conducted on Gaussian models in the presence of deviations from independence assumptions can provide insights into the robustness of these models under varying conditions. By examining the impact of deviations on the stability of Gaussian distributions, one can assess the resilience of the models to changes in the underlying assumptions. This analysis can help in understanding the limitations of Gaussian models and identifying scenarios where the models may exhibit vulnerabilities or inaccuracies. By quantifying the extent to which deviations affect the stability of Gaussian models, researchers can make informed decisions about the applicability and reliability of these models in real-world scenarios.

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