Manuscript Psy03-A@1 · Psy03-A.tex · corollary · b30eb6e9

Corollary: Stress and input-noise path bound

Stress, margins, and complete perturbation budgets › A general path lemma and the damped certificate · lines 604–627 · proof included or linked

Psy03-A@1#cor:stress, Psy03-A@1#eq:pathbound

For the same network, damping, and activation, coupled with the same clean input,

\norm z_{ξ, σ} - z_{0, 0}_{2} \leq ξ L_{x} \norm G_{x}_{2} + σ r = 1 \sum^{3} a^{4 - r} P_{r} \norm G_{r}_{2},

where

L_{x} = a^{3} ℓ = 1 \prod^{4} \norm B_{ℓ}_{op}, P_{r} = ℓ = r + 1 \prod^{4} \norm B_{ℓ}_{op} .

Proof. At the input $d_{0} = ξ \norm G_{x}_{2}$ . Each hidden noise contributes $e_{r} = aσ \norm G_{r}_{2}$ . Apply the lemma.

Displayed formulas in this passage

Psy03-A@1#eq:pathbound · line 608
$\norm z_{ξ, σ} - z_{0, 0}_{2} \leq ξ L_{x} \norm G_{x}_{2} + σ r = 1 \sum^{3} a^{4 - r} P_{r} \norm G_{r}_{2},$
unlabeled · line 616
$L_{x} = a^{3} ℓ = 1 \prod^{4} \norm B_{ℓ}_{op}, P_{r} = ℓ = r + 1 \prod^{4} \norm B_{ℓ}_{op} .$