rrr {MultiwayRegression}

Penalized reduced rank regression for tensors

Description

Fits a linear model to estimate one multi-way array from another, under the restriction that the coefficient array has given PARAFAC rank. By default, estimates are chosen to minimize a least-squares objective; an optional penalty term allows for $L_2$ regularization of the coefficient array.

Usage

rrr(X,Y,R=1,lambda=0,annealIter=0,convThresh=10^(-5), seed=0)

Arguments

X

A predictor array of dimension N x P_1 x ... x P_L.

Y

An outcome array of dimension N x Q_1 x ... X Q_M.

R

Assumed rank of the P_1 x ... x P_L x Q_1 x ... x Q_M coefficient array.

lambda

Ridge ($L_2$) penalty parameter for the coefficient array.

annealIter

Number of tempering iterations to improve initialization

convThresh

Converge threshold for the absolute difference in the objective function between two iterations

seed

Random seed for generation of initial values.

Value

U

List of length L. U[[l]]: P_l x R gives the coefficient basis for the l'th mode of X.

V

List of length M. V[[m]]: Q_m x R gives the coefficient basis for the m'th mode of Y.

B

Coefficient array of dimension P_1 x ... x P_L x Q_1 x ... x Q_M. Given by the CP factorization defined by U and V.

sse

Vector giving the sum of squared residuals at each iteration.

sseR

Vector giving the value of the objective (sse+penalty) at each iteration.

Author(s)

Eric F. Lock

References

Lock, E. F. (2018). Tensor-on-tensor regression. Journal of Computational and Graphical Statistics, 27 (3): 638-647, 2018.

Examples

data(SimData) ##loads simulated X: 100 x 15 x 20 and Y: 100 x 5 x 10 
Results <- rrr(X,Y,R=2)  ##Fit rank 2 model with no regularization
Y_pred <- ctprod(X,Results$B,2)  ##Array of fitted values