  Dictionary learning is the problem of estimating the collection of atomic
elements that provide a sparse representation of measured/collected signals or
data. This paper finds fundamental limits on the sample complexity of
estimating dictionaries for tensor data by proving a lower bound on the minimax
risk. This lower bound depends on the dimensions of the tensor and parameters
of the generative model. The focus of this paper is on second-order tensor
data, with the underlying dictionaries constructed by taking the Kronecker
product of two smaller dictionaries and the observed data generated by sparse
linear combinations of dictionary atoms observed through white Gaussian noise.
In this regard, the paper provides a general lower bound on the minimax risk
and also adapts the proof techniques for equivalent results using sparse and
Gaussian coefficient models. The reported results suggest that the sample
complexity of dictionary learning for tensor data can be significantly lower
than that for unstructured data.
