A Note on the Uniqueness of the Lasso Solution

In this note we show that, if β₁ and β₂ are two distinct solutions to the lasso problem min_β∈Rp∥ y - Xβ∥2^2 + λ∥β∥1 for some n × p matrix X with p > n, then Xβ1 = Xβ2.