在高维稀疏数据下修正白化方法，提升球形高斯混合模型估计精度
Whitening Spherical Gaussian Mixtures in the Large-Dimensional Regime
- 基于随机矩阵理论推导高维下白化后均值的点积极限
- 提出修正白化矩阵使均值在大维度下渐近正交
- 适用于高维稀疏数据下的潜变量模型估计场景
白化是无监督学习中标准化数据的经典技术，可促进估计任务。其重要应用是通过高阶矩张量分解来估计潜在变量模型。具体而言，白化可使球形高斯混合模型（GMM）的均值正交化，从而使对应的矩张量可正交分解，更易处理。然而，在高维稀疏数据（大维度情形，LDR）下，基于样本协方差构建的标准白化矩阵因谱畸变而失效，导致白化后的均值不再正交。本文利用随机矩阵理论，推导出这些均值点积在LDR下的精确极限，发现其通常非零。作为主要贡献，我们构造了一个修正的白化矩阵，恢复了渐近正交性，从而在球形GMM估计中实现性能提升。
原文摘要 · Abstract (English)
Whitening is a classical technique in unsupervised learning that can facilitate estimation tasks by standardizing data. An important application is the estimation of latent variable models via the decomposition of tensors built from high-order moments. In particular, whitening orthogonalizes the means of a spherical Gaussian mixture model (GMM), thereby making the corresponding moment tensor orthogonally decomposable, hence easier to decompose. However, in the large-dimensional regime (LDR) where data are high-dimensional and scarce, the standard whitening matrix built from the sample covariance becomes ineffective because the latter is spectrally distorted. Consequently, whitened means of a spherical GMM are no longer orthogonal. Using random matrix theory, we derive exact limits for their dot products, which are generally nonzero in the LDR. As our main contribution, we then construct a corrected whitening matrix that restores asymptotic orthogonality, allowing for performance gains in spherical GMM estimation.
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