Risk, prediction and selection answer different questions

The estimable projection

For a fixed design with more variables than observations, the coefficient target in the risk calculation is the projection of the regression vector onto the row space of the supplied design. hr_risk() computes exact Gaussian coordinate moments and total squared-error risk for that target, using the supplied design without automatic centering or standardization.

library(softridge)
set.seed(26)
X <- matrix(rnorm(20 * 30), 20, 30)
beta <- c(1, -1, rep(0, 28))
risk <- hr_risk(X, beta, sigma = 1, k = 2, gamma = 0.05)
c(total_projection_risk = risk$risk,
  sum_coordinate_risks = sum(risk$mse))

## total_projection_risk  sum_coordinate_risks 
##             0.7678852             0.7678852

Fixed-design prediction risk

hr_sure() uses ridge-weighted degrees of freedom for the fixed-design mean prediction target. The stated unbiasedness requires fixed tuning and known error variance. A fixed numeric scale is supported; a response-estimated scale is a different statistical object.

New covariates and support recovery

Out-of-sample PE evaluates prediction at new covariates. MCC evaluates recovery of the original coefficient support. Neither is interchangeable with projection coefficient risk. The independent study records all three targets separately and uses mltools::mcc() with its native zero-denominator convention.

Related foundations

• Shao and Deng (2012), estimation in high-dimensional linear models through a projection and hard-thresholded ridge: https://arxiv.org/abs/1206.0847.
• Hagiwara, scaled soft thresholding in orthogonal settings: https://arxiv.org/abs/1601.08002 and https://arxiv.org/abs/2104.09703.
• Zou and Hastie (2005), elastic-net rescaling: https://doi.org/10.1111/j.1467-9868.2005.00503.x.

These are antecedents, not claims that thresholding or scaling was first introduced in this software. The companion manuscript distinguishes its fixed-parameter results from empirical tuning and scaling.

A fixed-design analytic illustration

For the 20-by-30 design above, seed 26, fixed penalty 2 and error standard deviation 1, this curve evaluates the exact formula at 31 fixed thresholds. It is an illustration for one specified target and design, not an empirical performance study. Source: tools/build-gallery.R.