Local Identifiability of $\ell_1$-minimization Dictionary Learning: a Sufficient and Almost Necessary Condition

Siqi Wu, Bin Yu; 18(168):1−56, 2018.

Abstract

We study the theoretical properties of learning a dictionary from $N$ signals $\mathbf{x}_i\in \mathbb R^K$ for $i=1,\ldots,N$ via $\ell_1$-minimization. We assume that $\mathbf{x}_i$'s are $i.i.d.$ random linear combinations of the $K$ columns from a complete (i.e., square and invertible) reference dictionary $\mathbf{D}_0 \in \mathbb R^{K\times K}$. Here, the random linear coefficients are generated from either the $s$-sparse Gaussian model or the Bernoulli-Gaussian model. First, for the population case, we establish a sufficient and almost necessary condition for the reference dictionary $\mathbf{D}_0$ to be locally identifiable, i.e., a strict local minimum of the expected $\ell_1$-norm objective function. Our condition covers both sparse and dense cases of the random linear coefficients and significantly improves the sufficient condition by Gribonval and Schnass (2010). In addition, we show that for a complete $\mu$-coherent reference dictionary, i.e., a dictionary with absolute pairwise column inner-product at most $\mu\in[0,1)$, local identifiability holds even when the random linear coefficient vector has up to $O(\mu^{-2})$ nonzero entries. Moreover, our local identifiability results also translate to the finite sample case with high probability provided that the number of signals $N$ scales as $O(K\log K)$.

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