  We study the problem of globally recovering a dictionary from a set of
signals via $\ell_1$-minimization. We assume that the signals are generated as
i.i.d. random linear combinations of the $K$ atoms from a complete reference
dictionary $D^*\in \mathbb R^{K\times K}$, where the linear combination
coefficients are from either a Bernoulli type model or exact sparse model.
First, we obtain a necessary and sufficient norm condition for the reference
dictionary $D^*$ to be a sharp local minimum of the expected $\ell_1$ objective
function. Our result substantially extends that of Wu and Yu (2015) and allows
the combination coefficient to be non-negative. Secondly, we obtain an explicit
bound on the region within which the objective value of the reference
dictionary is minimal. Thirdly, we show that the reference dictionary is the
unique sharp local minimum, thus establishing the first known global property
of $\ell_1$-minimization dictionary learning. Motivated by the theoretical
results, we introduce a perturbation-based test to determine whether a
dictionary is a sharp local minimum of the objective function. In addition, we
also propose a new dictionary learning algorithm based on Block Coordinate
Descent, called DL-BCD, which is guaranteed to have monotonic convergence.
Simulation studies show that DL-BCD has competitive performance in terms of
recovery rate compared to many state-of-the-art dictionary learning algorithms.
