Consistent Estimation of Low-Dimensional Latent Structure in High-Dimensional Data

Understand

We consider the problem of extracting a low-dimensional, linear latent variable structure from high-dimensional random variables.

• Specifically, we show that under mild conditions and when this structure manifests itself as a linear space that spans the conditional means, it is possible to consistently recover the structure using only information up to the second moments of these random variables.
• This finding, specialized to one-parameter exponential families whose variance function is quadratic in their means, allows for the derivation of an explicit estimator of such latent structure.
• This approach serves as a latent variable model estimator and as a tool for dimension reduction for a high-dimensional matrix of data composed of many related variables.