Supplement : Proofs and Technical Details for “ The Solution Path of the Generalized Lasso ”

In this document we give supplementary details to the paper " The Solution Path of the Generalized Lasso ". We use the prefix " GL " when referring to equations, sections, etc. in the original paper, as in equation (GL-1) or Section GL-1 (this stands for Generalized Lasso). 1 Proof of the boundary lemma We prove the boundary lemma when D = D 1d , but first we give a helpful lemma. Lemma 1. Let T λ denote the function that truncates outside of the interval [−λ, λ]: T λ (x) =      −λ if x < −λ x if |x| ≤ λ λ if x > λ. Proof. Suppose without a loss of generality that λ 0 > λ. We enumerate the possible cases: The remaining cases follow by symmetry. Proof of the boundary lemma. Our approach for the proof is a little unusual: we consider the use of coordinate descent to find the solutionû λ , starting at the pointû λ0 as an initial guess. Because the coordinate updates are especially simple, we can track how the iterates change, and hence we can guarantee thatû λ andû λ0 are close together. Namely, we show thatû λ0 − ˆ u λ ∞ ≤ λ 0 − λ, which implies the desired result. First we describe the coordinate descent updates for finding the solutionû λ of the dual (GL-13), when D = D 1d. We note that any limit point of the coordinate descent algorithm is indeed a solution by Theorem 4.1 of [1]. We take u (0) = ˆ u λ0 as an initial guess, and cycle through the coordinates in 1