Manuscript Psy03-B@1 · Psy03-B-v1.tex · section · 8ca3caf9

Protection regularity and complete residuals

Biased-noise and full-state perturbation certificates › Protection regularity and complete residuals · lines 1087–1117

For two coupled states, put

e_{ℓ} = \norm B_{ℓ} - B_{ℓ}^{'}_{op}, ϵ_{ℓ}^{b} = \norm b_{ℓ} - b_{ℓ}^{'}_{2}, v_{ℓ}^{'} = B_{ℓ}^{'} h_{ℓ - 1}^{'} + b_{ℓ}^{'},

and

u = q β b_{bias} (p) - d_{off} (p), u^{'} = q^{'} β^{'} b_{bias} (p^{'}) - d_{off} (p^{'}) .

For $ℓ \leq 3$ , set

r_{ℓ} = q (e_{ℓ} \norm h_{ℓ - 1}^{'}_{2} + ϵ_{ℓ}^{b}) + q \norm φ (v_{ℓ}^{'}) - φ^{'} (v_{ℓ}^{'})_{2} + \abs q - q^{'} \norm φ^{'} (v_{ℓ}^{'})_{2} + \abs q σ - q^{'} σ^{'} \norm G_{ℓ}_{2} + \abs u - u^{'} n_{ℓ} .

Define

A_{ℓ} = max {q \norm B_{ℓ}_{op}, q^{'} \norm B_{ℓ}^{'}_{op}}, ℓ \leq 3, A_{4} = max {\norm B_{4}_{op}, \norm B_{4}^{'}_{op}},

and $r_{4} = e_{4} \norm h_{3}^{'}_{2} + ϵ_{4}^{b}$ .

Displayed formulas in this passage

unlabeled · line 1101
$r_{ℓ} = q (e_{ℓ} \norm h_{ℓ - 1}^{'}_{2} + ϵ_{ℓ}^{b}) + q \norm φ (v_{ℓ}^{'}) - φ^{'} (v_{ℓ}^{'})_{2} + \abs q - q^{'} \norm φ^{'} (v_{ℓ}^{'})_{2} + \abs q σ - q^{'} σ^{'} \norm G_{ℓ}_{2} + \abs u - u^{'} n_{ℓ} .$
unlabeled · line 1090
$e_{ℓ} = \norm B_{ℓ} - B_{ℓ}^{'}_{op}, ϵ_{ℓ}^{b} = \norm b_{ℓ} - b_{ℓ}^{'}_{2}, v_{ℓ}^{'} = B_{ℓ}^{'} h_{ℓ - 1}^{'} + b_{ℓ}^{'},$
unlabeled · line 1096
$u = q β b_{bias} (p) - d_{off} (p), u^{'} = q^{'} β^{'} b_{bias} (p^{'}) - d_{off} (p^{'}) .$
unlabeled · line 1110
$A_{ℓ} = max {q \norm B_{ℓ}_{op}, q^{'} \norm B_{ℓ}^{'}_{op}}, ℓ \leq 3, A_{4} = max {\norm B_{4}_{op}, \norm B_{4}^{'}_{op}},$