Minimax Lower Bounds for Kronecker-Structured Dictionary Learning

TL;DR

This paper establishes fundamental lower bounds on the sample complexity for Kronecker-structured dictionary learning of tensor data, revealing potential efficiency gains over unstructured data methods.

Contribution

It provides the first minimax lower bounds specific to Kronecker-structured dictionary learning for tensor data, highlighting the impact of tensor dimensions and model parameters.

Findings

Lower bounds depend on tensor dimensions and model parameters

Sample complexity can be significantly lower for tensor data

Results apply to both sparse and Gaussian coefficient models

Abstract

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.