Table 1.

Examples of unstructured and structured penalty terms

Unstructured penalties	J (β)
Lasso (Tibshirani, 1996)	∥β∥1
Ridge (Hoerl and Kennard, 1970)	$\Vert \beta {\Vert }_2^2$
Elastic net (Zou and Hastie, 2005)	$\alpha \Vert \beta {\Vert }_1+(1-\alpha )\Vert \beta {\Vert }_2^2;\alpha \in [0,1]$
Structured penalties	J (β)
Isotropic total variation (Rudin et al., 1992)	∥Dβ∥2,1; matrix D encodes spatial structure
Fused lasso* (Tibshirani et al., 2005)	α∥β∥1 + (1 – α)∥Dβ∥1; α ∈ [0, 1]
Graph net** (Grosenick et al., 2013)	$\alpha \Vert \beta {\Vert }_1+(1-\alpha )\Vert \mathbf{D}\beta {\Vert }_2^2;\alpha \in [0,1]$
Group lasso (Yuan and Lin, 2006)	${\sum }_{g\in \mathcal{G}}\sqrt{p_g}\Vert {\beta }_g{\Vert }_2$; groups $\mathcal{G}$ form a partition of β
Sparse group lasso (Simon et al., 2013)	$\alpha \Vert \beta {\Vert }_1+(1-\alpha ){\sum }_{g\in \mathcal{G}}\sqrt{p_g}\Vert {\beta }_g{\Vert }_2;\alpha \in [0,1]$

^*Also known as anisotropic total variation–l₁

^**Also known as sparse graph Laplacian