Identifiability of Kronecker-Structured Dictionaries for Tensor Data

Zahra Shakeri, Anand D. Sarwate, Waheed U. Bajwa

Abstract: This paper derives sufficient conditions for local recovery of coordinate dictionaries comprising a Kronecker-structured dictionary that is used for representing Kth-order tensor data. Tensor observations are assumed to be generated from a Kronecker-structured dictionary multiplied by sparse coefficient tensors that follow the separable sparsity model. This paper provides sufficient conditions on the underlying coordinate dictionaries, coefficient and noise distributions, and number of samples that guarantee recovery of the individual coordinate dictionaries up to a specified error, as a local minimum of the objective function, with high probability. In particular, the sample complexity to recover K coordinate dictionaries with dimensions m <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> × p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> up to estimation error ε <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> is shown to be max <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k∈[K]</sub> O(m <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ε <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> ).