Elastic net

Elastic net combines sparse selection with quadratic shrinkage. Its rounded diamond retains corners on the coordinate axes. Supply the two strengths independently with ElasticNet::new(l1, l2). Setting either strength to zero recovers ridge or lasso, respectively.

Constraint region

$\lambda_1 = 0.5$ and $\lambda_2 = 1$: $0.5\|\beta\|_1 + 0.5\|\beta\|_2^2 \leq 1$. See the plot conventions for the fixed budget and coefficient scale.

Use the penalty

use shrinkage::{ElasticNet, Penalty, ProximalPenalty};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let penalty = ElasticNet::from_alpha(0.5, 0.7);
    let coefficients = [2.0, -1.0];
    println!("Penalty: {}", penalty.value(&coefficients)?);
    let mut shrunk = [0.0; 2];
    penalty.prox(&coefficients, 1.0, &mut shrunk)?;
    println!("Proximal result: {shrunk:?}");
    Ok(())
}

Alternatively, ElasticNet::from_alpha(lambda, alpha) uses $\lambda_1 = \lambda\alpha$ and $\lambda_2 = \lambda(1 - \alpha)$, where $0 \leq \alpha \leq 1$. The proximal map soft-thresholds at $s\lambda_1$ and then divides by $1 + s\lambda_2$. Fits support proximal gradient and IRLS.

API: shrinkage::ElasticNet.