# Chain fused lasso

Chain fused lasso encourages sparse, piecewise constant coefficients in the supplied feature order. Adjacent equal effects form contiguous segments. Weights default to one. `.with_weights(...)` takes exactly one finite, nonnegative weight per edge; a zero weight cuts the chain. 

$$

  R(\beta) = \lambda_1\sum_{j=1}^p|\beta_j|
             + \lambda_2\sum_{j=1}^{p-1}w_j|\beta_{j+1} - \beta_j|.

$$

<a id="constraint-region"></a>

## Constraint region

![Feasible region for chain fused lasso; the shaded set includes its boundary.](<../assets/bed3505b9a79e0ada2627d7fe5f2c0908eb1909befdf4a3c4b8af5790db1d0e5.svg>) 

Top: $\lambda_1 = \lambda_2 = 0.5$, $w_1 = 1$. Bottom: $\lambda_1 = 0$, $\lambda_2 = 1$, $w_1 = 1$, giving an unbounded pure-fusion strip. See the [plot conventions](<./#read-the-constraint-plots>) for the fixed budget and coefficient scale. 

<a id="use-the-penalty"></a>

## Use the penalty

```rust
use shrinkage::{ChainFusedLasso, Penalty, ProximalPenalty};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let penalty = ChainFusedLasso::new(0.5, 0.25).with_weights(vec![1.0]);
    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(())
}
```

Each edge contributes once. Empty and single-feature chains have no edges. Zero fusion recovers lasso. A two-feature chain has one pair, so its unit-weight constraint region coincides with clustered lasso; larger chains only fuse neighbors. Gaussian and binomial fits support [proximal gradient](<../reference/solvers/proximal-gradient>) and [hybrid](<../reference/solvers/hybrid>). See the [chain fused lasso guide](<../models/chain-fused-lasso>) for segment interpretation and detailed fitting controls. 

API: [`shrinkage::ChainFusedLasso`](<https://shrinkage.cc/packages/rust/reference/b88554a6c3f1121915d157c3e7684558bb6e9e9f59dbacd8115eb616c3631ce1>).