hose: Higher-Order Spectral Estimators

Summary

hose is a package designed for working with higher-order spectral estimators, which were first introduced in Gerard and Hoff (2017). These estimators are based on the higher-order singular value decomposition of De Lathauwer, De Moor, and Vandewalle (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 (Josse and Sardy 2016), 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.

The main functions are:

• get_c(): Pre-format the data before applying mode-specific singular value shrinkage.
• tensor_var_est(): Estimate the variance of the data from multiple options.
• soft_coord(): Estimate the underlying low-rank mean tensor via soft-thresholding.

Citation

If you find these methods useful, please cite

Gerard, David, and Peter Hoff. 2017. “Adaptive Higher-Order Spectral Estimators.” Electron. J. Statist. 11 (2). The Institute of Mathematical Statistics; the Bernoulli Society: 3703–37. https://doi.org/10.1214/17-EJS1330.

Or, using BibTex:

@ARTICLE{gerard2017adaptive,
    AUTHOR = {David Gerard and Peter Hoff},
     TITLE = {Adaptive higher-order spectral estimators},
   JOURNAL = {Electron. J. Statist.},
  FJOURNAL = {Electronic Journal of Statistics},
      YEAR = {2017},
    VOLUME = {11},
    NUMBER = {2},
     PAGES = {3703-3737},
      ISSN = {1935-7524},
       DOI = {10.1214/17-EJS1330},
      SICI = {1935-7524(2017)11:2<3703:AHOSE>2.0.CO;2-Q},
}

Installation

You can install from CRAN in the usual way:

install.packages("hose")

Or, to install the latest (unstable) version, run the following code in R:

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

Vignette

I’ve provided a vignette demonstrating the methods available in hose. You can find it here. Or you can build the vignette on install with

install.packages("devtools")
devtools::install_github("dcgerard/hose", build_vignettes = TRUE)

and access the vignette by running the following code in R:

utils::vignette("sure_example", package = "hose")

References

De Lathauwer, L., B. De Moor, and J. Vandewalle. 2000. “A Multilinear Singular Value Decomposition.” SIAM Journal on Matrix Analysis and Applications 21 (4): 1253–78. https://doi.org/10.1137/S0895479896305696.

Gerard, David, and Peter Hoff. 2017. “Adaptive Higher-Order Spectral Estimators.” Electron. J. Statist. 11 (2): 3703–37. https://doi.org/10.1214/17-EJS1330.

Josse, Julie, and Sylvain Sardy. 2016. “Adaptive Shrinkage of Singular Values.” Statistics and Computing 26 (3): 715–24. https://doi.org/10.1007/s11222-015-9554-9.