2016/05/17 by Zahra Shakeri, Waheed U. Bajwa, Anand D. Sarwate
Computer Science · Earth and Planetary Sciences · Engineering · Mathematics · #Blind Source Separation Techniques #Focus (optics) #Gaussian #Generative model #Geophysical and Geoelectrical Methods #Kronecker delta #Kronecker product #Minimax #Sparse and Compressive Sensing Techniques #Sparse approximation #Tensor (intrinsic definition) #Upper and lower bounds #cs.IT #cs.LG #math.IT #stat.ML
paper · pdf · doi:10.1109/isit.2016.7541479
published as Proc. IEEE Intl. Symp. Information Theory, Barcelona, Spain, Jul. 10-15, 2016, pp. 1148-1152 · 5 pages, 1 figure. To appear in 2016 IEEE International Symposium on Information Theory
arxiv created 2016/05/17 · openalex created_date 2016/06/24 · openalex publication_date 2016/07/01 · arxiv updated 2018/03/06 · openalex updated_date 2026/08/06
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.