Low Complexity Gaussian Latent Factor Models and a Blessing of Dimensionality
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Learning the structure of graphical models from data is a fundamental problem that typically carries a curse of dimensionality. We consider a special class of Gaussian latent factor models where each observed variable depends on at most one of a set of latent variables. We derive information-theoretic lower bounds on the sample complexity for structure recovery that suggest a blessing of dimensionality. With a fixed number of samples, structure recovery for this class using existing methods deteriorates with increasing dimension. We design a new approach to learning Gaussian latent factor models with low computational complexity that empirically benefits from dimensionality. Our approach relies on an information-theoretic constraint to find parsimonious solutions without adding regularizers or sparsity hyper-parameters. Besides improved structure recovery, we also show that we are able to outperform state-of-the-art approaches for covariance estimation on both synthetic data and on under-sampled, high-dimensional stock market data.
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