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

Unique Sharp Local Minimum in $\ell_1$-minimization Complete Dictionary Learning