Minimax Lower Bounds for Kronecker-Structured Dictionary Learning
This provides fundamental limits for dictionary learning in tensor applications, such as image processing, but is incremental as it extends existing lower bound techniques to a specific structured case.
The paper addresses the sample complexity of estimating Kronecker-structured dictionaries for tensor data by proving a minimax lower bound, showing that the required sample size can be significantly lower than for unstructured data.
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.