E [ inf ∥ δ ∥ ≤ ϵ y f ( x + δ ) ] , \mathbf{E}\left[\inf_{\|\delta\|\leq\epsilon}yf(x+\delta)\right], E [ ∥ δ ∥ ≤ ϵ in f ​ y f ( x + δ ) ] , The robust usefulness of a linear feature admits an elegant decomposition This E [ inf ∥ δ ∥ ≤ ϵ y f ( x + δ ) ] = E [ y f ( x ) + inf ∥ δ ∥ ≤ ϵ y f ( δ ) ] = E [ y f ( x ) + inf ∥ δ ∥ ≤ ϵ y a T δ ∥ a ∥ Σ ] = E [ y f ( x ) + inf ∥ δ ∥ ≤ ϵ a T δ ∥ a ∥ Σ ] = E [ y f ( x ) ] − ϵ ∥ a ∥ ∗ ∥ a ∥ Σ \begin{aligned} \mathbf{E}\left[\inf_{\|\delta\|\leq\epsilon}yf(x+\delta)\right] & =\mathbf{E}\left[yf(x)+\inf_{\|\delta\|\leq\epsilon}yf(\delta)\right]\\ & =\mathbf{E}\left[yf(x)+\inf_{\|\delta\|\leq\epsilon}y\frac{a^{T}\delta}{\|a\|_{\Sigma}}\right]\\ &…