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We study the theoretical properties of learning a dictionary from $N$ signals$\mathbf x_i\in \mathbb R^K$ for $i=1,...,N$ via $l_1$-minimization. We assumethat $\mathbf x_i$'s are $i.i.d.$ random linear combinations of the $K$ columnsfrom a complete (i.e., square and invertible) reference dictionary $\mathbf D_0\in \mathbb R^{K\times K}$. Here, the random linear coefficients are generatedfrom either the $s$-sparse Gaussian model or the Bernoulli-Gaussian model.First, for the population case, we establish a sufficient and almost necessarycondition for the reference dictionary $\mathbf D_0$ to be locallyidentifiable, i.e., a local minimum of the expected $l_1$-norm objectivefunction. Our condition covers both sparse and dense cases of the random linearcoefficients and significantly improves the sufficient condition by Gribonvaland Schnass (2010). In addition, we show that for a complete $\mu$-coherentreference dictionary, i.e., a dictionary with absolute pairwise columninner-product at most $\mu\in[0,1)$, local identifiability holds even when therandom linear coefficient vector has up to $O(\mu^{-2})$ nonzeros on average.Moreover, our local identifiability results also translate to the finite samplecase with high probability provided that the number of signals $N$ scales as$O(K\log K)$.

Local identifiability of $l_1$-minimization dictionary learning: a sufficient and almost necessary condition