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

Coupled parameter comparisons

Stress, margins, and complete perturbation budgets › Coupled parameter comparisons · lines 704–735

Let a primed and an unprimed computation use the same $(X, G_{x}, G_{1}, G_{2}, G_{3})$ . Their input-noise intensities may differ. Put

e_{ℓ} = \norm B_{ℓ} - B_{ℓ}^{'}_{op}, β_{ℓ} = \norm b_{ℓ} - b_{ℓ}^{'}_{2},

and for hidden layers define

v_{ℓ}^{'} = B_{ℓ}^{'} h_{ℓ - 1}^{'} + b_{ℓ}^{'},

r_{ℓ} = a (e_{ℓ} \norm h_{ℓ - 1}^{'}_{2} + β_{ℓ}) + a \norm φ (v_{ℓ}^{'}) - φ^{'} (v_{ℓ}^{'})_{2} + \abs a - a^{'} \norm φ^{'} (v_{ℓ}^{'})_{2} + \abs aσ - a^{'} σ^{'} \norm G_{ℓ}_{2} .

Let

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

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

Displayed formulas in this passage

unlabeled · line 710
$e_{ℓ} = \norm B_{ℓ} - B_{ℓ}^{'}_{op}, β_{ℓ} = \norm b_{ℓ} - b_{ℓ}^{'}_{2},$
unlabeled · line 715
$v_{ℓ}^{'} = B_{ℓ}^{'} h_{ℓ - 1}^{'} + b_{ℓ}^{'},$
unlabeled · line 718
$r_{ℓ} = a (e_{ℓ} \norm h_{ℓ - 1}^{'}_{2} + β_{ℓ}) + a \norm φ (v_{ℓ}^{'}) - φ^{'} (v_{ℓ}^{'})_{2} + \abs a - a^{'} \norm φ^{'} (v_{ℓ}^{'})_{2} + \abs aσ - a^{'} σ^{'} \norm G_{ℓ}_{2} .$
unlabeled · line 726
$A_{ℓ} = max {a \norm B_{ℓ}_{op}, a^{'} \norm B_{ℓ}^{'}_{op}}, ℓ \leq 3, A_{4} = max {\norm B_{4}_{op}, \norm B_{4}^{'}_{op}},$