group sparsity

Group lasso

β^λ=arg⁡min⁡β∥Y−Xβ∥22+λ∑g=1G∥βIg∥2,\hat{\bm \beta}_\lambda = \arg \min_{\bm \beta} \| \bm Y - \bm X \bm \beta \|_2^2 + \lambda \sum_{g=1}^G \|\bm \beta_{\mathcal{I}_g}\|_2,β^λ=argβmin∥Y−Xβ∥22+λg=1∑G∥βIg∥2,
where Ig\mathcal{I}_gIg is the index set belonging to the gggth group of variables, g=1,…,Gg=1,\ldots,Gg=1,…,G.

This penalty can be viewed as an intermediate between the ℓ1\ell_1ℓ1 and ℓ2\ell_2ℓ2-type penalty.

The ℓ1\ell_1ℓ1-penalty treats the three coordinate directions differently from other directions, and this encourages sparsity in individual coefficients. The ℓ2\ell_2ℓ2-penalty treats all directions equally and does not encourage sparsity. The group lasso encourages sparsity at the factor level.

The estimates have the attractive property of being invariant under groupwise orthogonal reparameterizations (transformations)\textcolor{red}{\text{\small invariant under groupwise orthogonal reparameterizations (transformations)}}invariant under groupwise orthogonal reparameterizations (transformations), like ridge regression.

Group LARS

∥M∥2,1=∑i=1d∥M∥2\|\bm M \|_{2,1}= \sum_{i=1}^d \| \bm M\|_2∥M∥2,1=i=1∑d∥M∥2

Group non-negative garrotte

	group lasso	group LARS	group non-negative garrotte
performance	excellent	comparable
computational efficiency	intensive in large scale problems	quickly	fastest
applicability			sub-optimal when p→np \rightarrow np→n, not applicable when p>np>np>n

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group sparsity

Group lasso

Group LARS

Group non-negative garrotte

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