Adaptive Estimation in Structured Factor Models with Applications to Overlapping Clustering

Understand

This work introduces a novel estimation method, called LOVE, of the entries and structure of a loading matrix A in a sparse latent factor model X = AZ + E, for an observable random vector X in Rp, with correlated unobservable factors Z \in RK, with K unknown, and independent noise E.

• Each row of A is scaled and sparse.
• In order to identify the loading matrix A, we require the existence of pure variables, which are components of X that are associated, via A, with one and only one latent factor.
• Despite the fact that the number of factors K, the number of the pure variables, and their location are all unknown, we only require a mild condition on the covariance matrix of Z, and a minimum of only two pure variables per latent factor to show that A is uniquely defined, up to signed permutations.