Improved Identifiability and Sample Complexity Analysis of Complete Dictionary Learning

It has been recently shown that dictionary learning is identifiable with a complete dictionary, with relatively loose assumptions that the dictionary is invertible and that a geometric body obtained from the sparse coefficient matrix is sufficiently scattered in the hypercube. In addition, if the sparse coefficients are generated from the Bernoulli- Gaussian model, then the number of samples required to guarantee identifiability is
$O(k\log(k))$
. In this paper, we further improve the identifiability results in two-folds. First, we extend the deterministic identifiability results from the real domain in the previous work to the complex domain, hence significantly expand its applications. Second, we consider the generative model for the sparse coefficients from the more specific (also widely adopted) Bernoulli-Gaussian to a much wider class where the nonzero entries can be drawn from any subexponential distributions, and show that the same sample complexity results still holds. The recovery performance is confirmed in a Monte-Carlo simulation where the sparse coefficients are drawn from Bernoulli-Laplacian model.