  We consider the problem of learning a dictionary matrix from a number of
observed signals, which are assumed to be generated via a linear model with a
common underlying dictionary. In particular, we derive lower bounds on the
minimum achievable worst case mean squared error (MSE), regardless of
computational complexity of the dictionary learning (DL) schemes. By casting DL
as a classical (or frequentist) estimation problem, the lower bounds on the
worst case MSE are derived by following an established information-theoretic
approach to minimax estimation. The main conceptual contribution of this paper
is the adaption of the information-theoretic approach to minimax estimation for
the DL problem in order to derive lower bounds on the worst case MSE of any DL
scheme. We derive three different lower bounds applying to different generative
models for the observed signals. The first bound applies to a wide range of
models, it only requires the existence of a covariance matrix of the (unknown)
underlying coefficient vector. By specializing this bound to the case of sparse
coefficient distributions, and assuming the true dictionary satisfies the
restricted isometry property, we obtain a lower bound on the worst case MSE of
DL schemes in terms of a signal to noise ratio (SNR). The third bound applies
to a more restrictive subclass of coefficient distributions by requiring the
non-zero coefficients to be Gaussian. While, compared with the previous two
bounds, the applicability of this final bound is the most limited it is the
tightest of the three bounds in the low SNR regime.
