Clustered lasso

Clustered lasso encourages sparse coefficients and equal signed effects. Every unordered pair contributes once. Unlike SLOPE, fusion compares signed values: equal positive and negative magnitudes do not form one fusion cluster. The constructor takes separate L1 and fusion strengths.

R(\beta) = \lambda_1\sum_j|\beta_j| + \lambda_2\sum_{i<j}|\beta_i - \beta_j|.

Constraint region

Feasible region for clustered lasso; the shaded set includes its boundary.

Top: $\lambda_1 = \lambda_2 = 0.5$ . Bottom: pure fusion, $\lambda_1 = 0$ , $\lambda_2 = 1$ , giving the unbounded strip $|\beta_1 - \beta_2| \leq 1$ . See the plot conventions for the fixed budget and coefficient scale.

Use the penalty

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

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let penalty = ClusteredLasso::new(0.5, 0.25);
    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(())
}

At zero fusion strength, this reduces to lasso. Without the L1 term, a common shift of all coefficients has no penalty, so the constraint region is unbounded along the equality direction. In two dimensions, its geometry equals that of unit-weight chain fused lasso. With more features, all-pairs and adjacent fusion differ. Gaussian and binomial fits support proximal gradient and hybrid. See the clustered lasso guide for cluster labels and logistic fitting controls.

API: shrinkage::ClusteredLasso.