Hypercube fits to the multivariate linear model complete the class of Penalized Least Squares (PLS) fits with quadratic penalties. They include submodel Least Squares fits that are limits of PLS fits as penalty weights tend to infinity. Through control of condition number, they improve the numerical stability of PLS fits.

Adaptive hypercube fits that minimize estimated risk can behave asymptotically, as the number of regressors increases, like their oracle counterparts. In particular, suitable adaptive hypercube fits extend to general regression designs the asymptotic risk reduction achieved by multiple Efron-Morris affine shrinkage in balanced orthogonal designs. Reduced risk fits to unbalanced MANOVA designs illustrate.