Minimax Lower Bounds on Dictionary Learning for Tensor Data

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

doi:10.1109/tit.2018.2799931

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arXiv · 1608.02792

Minimax Lower Bounds on Dictionary Learning for Tensor Data

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

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Abstract

This paper provides fundamental limits on the sample complexity of estimating dictionaries for tensor data. The specific focus of this work is on $K$th-order tensor data and the case where the underlying dictionary can be expressed in terms of $K$ smaller dictionaries. It is assumed the data are generated by linear combinations of these structured dictionary atoms and observed through white Gaussian noise. This work first provides a general lower bound on the minimax risk of dictionary learning for such tensor data and then adapts the proof techniques for specialized results in the case of sparse and sparse-Gaussian linear combinations. The results suggest the sample complexity of dictionary learning for tensor data can be significantly lower than that for unstructured data: for unstructured data it scales linearly with the product of the dictionary dimensions, whereas for tensor-structured data the bound scales linearly with the sum of the product of the dimensions of the (smaller) component dictionaries. A partial converse is provided for the case of 2nd-order tensor data to show that the bounds in this paper can be tight. This involves developing an algorithm for learning highly-structured dictionaries from noisy tensor data. Finally, numerical experiments highlight the advantages associated with explicitly accounting for tensor data structure during dictionary learning.

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BibTeX RIS

Zahra Shakeri, Waheed U. Bajwa, Anand D. Sarwate. 2016-08-09. Minimax Lower Bounds on Dictionary Learning for Tensor Data. https://doi.org/10.1109/tit.2018.2799931

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