Statistical properties of large data sets with linear latent features

Abstract

Analytical understanding of how low-dimensional latent features reveal themselves in large-dimensional data is still lacking. We study this by defining a linear latent feature model with additive noise constructed from probabilistic matrices, and analytically and numerically computing the statistical distributions of pairwise correlations and eigenvalues of the correlation matrix. This allows us to resolve the latent feature structure across a wide range of data regimes set by the number of recorded variables, observations, latent features and the signal-to-noise ratio. We find a characteristic imprint of latent features in the distribution of correlations and eigenvalues and provide an analytic estimate for the boundary between signal and noise even in the absence of a clear spectral gap.

20 pages, 6 figures

References in corpus (3)

• Introduction to Random Matrices - Theory and Practice
• Spectrum of deformed random matrices and free probability
• The Perturbative Resolvent Method: spectral densities of random matrix ensembles via perturbation theory

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• Bayesian Renormalization
• Singular Vectors of Sums of Rectangular Random Matrices and Optimal Estimators of High-Rank Signals: The Extensive Spike Model
• Generative random latent features models and statistics of natural images
• Renormalization group for spectral collapse in random matrices with power-law variance profiles
• Distribution of singular values in large sample cross-covariance matrices