I've placed some R code on Github that is useful for working with higher-order spectral estimators. These estimators are based on the higher-order singular value decomposition of De Lathauwer et. al. [2000] and are useful when your data exhibit tensor-specific structure, such as having approximately low multilinear rank. This code will allow you to:

• Calculate Stein's unbiased risk estimate (SURE) for all higher-order spectral estimators that are weakly differentiable and satisfy mild integrability conditions.
• Calculate the mode-specific soft-thresholding estimator that minimizes the SURE using a coordinate descent algorithm.
• Iterate through all the possible multilinear ranks of a mean tensor and choose the multilinear rank the minimizes the SURE.
• Calculate a generalized SURE, motivated by generalized cross validation [Sardy, 2012], for all higher-order spectral estimators.
• Calculate the SURE of estimators that apply higher-order spectral shrinkage to sub-tensors of the overall data tensor.
• Calculate the SURE for estimators that individually shrink elements of the core array of the HOSVD of the data tensor.
• Non-parametrically estimate the variance to use in these SURE procedures.

All details of these methods may be found in

Gerard, D., & Hoff, P. (2015). Adaptive higher-order spectral estimators. arXiv preprint arXiv:1505.02114. [Link to arXiv] [Code] [bib]

I also provide a vignette that demonstrates how to fit the truncated HOSVD and the mode-specific soft-thresholding estimators.

You can download the package by typing in R:

 install.packages(c("tensr", "softImpute", "RMTstat", "devtools")) devtools::install_github("dcgerard/hose") 

These scripts were created and are maintained by David Gerard.

Please provide us with questions or comments: David Gerard (dcgerard) or Peter Hoff (pdhoff) -- @uchicago.edu and @uw.edu, respectively.

## References

Gerard, D., & Hoff, P. (2015). Adaptive higher-order spectral estimators. arXiv preprint arXiv:1505.02114. [Link to arXiv] [Code] [bib]

Lieven De Lathauwer, Bart De Moor, and Joos Vandewalle. A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl., 21(4):1253–1278 (electronic), 2000. ISSN 0895-4798. doi: 10.1137/S0895479896305696. [Link]

Sylvain Sardy. Smooth blockwise iterative thresholding: a smooth fixed point estimator based on the likelihood’s block gradient. J. Amer. Statist. Assoc., 107(498):800–813, 2012. ISSN 0162-1459. doi: 10.1080/01621459.2012.664527. [Link]