Supplementary Materials Localized Lasso for High-Dimensional Regression

Supplementary Materials Localized Lasso for High-Dimensional Regression Page 1 Supplementary Materials Localized Lasso for High-Dimensional Regression Makoto Yamada1,2, Koh Takeuchi3, Tomoharu Iwata3, John Shawe-Taylor4, Samuel Kaski5 1RIKEN AIP, Japan 2JST, PRESTO 3NTT Communication Science Laboratories, Japan 4University College London, UK 5Aalto University, Finland makoto.yamada@riken.jp, {takeuchi.koh,iwata.tomoharu}@lab.ntt.co.jp j.shawe-taylor@ucl.ac.uk, samuel.kaski@aalto.fi Propositions used for deriving Eq. (4) in main paper Proposition 1 Under rij ≥ 0, rij = rji, rii = 0, we have ∂ ∂vec(W) n ∑ i,j=1 rij wi − wj 2 = 2Fgvec(W), where Fg = Id ⊗ C, [C]i,j = { ∑n j =1 rij vi−vj 2 − rij vi−vj 2 (i = j) −rij vi−vj 2 (i = j) . Proof: Under rij ≥ 0, rij = rji, rii = 0, the derivative of the network regularization term with respect to wk is given as ∂ ∂wk n ∑ i,j=1 rij wi − wj 2 = n ∑ i=1 rik wk − wi wk − wi …