User:Sintel/Dual Weil-Euclidean norm: Difference between revisions

@@ Line 103: / Line 103: @@
 $$
 \begin{aligned}
-G_b(E_k) &= \frac{W^{-2}}{jW^{-2}j^{\mathsf T}} (1+E^2_k) -  \frac{W^{-2}j^{\mathsf T}jW^{-2}}{(jW^{-2}j^{\mathsf T})^2} \\
+G_b(E) &= \frac{W^{-2}}{jW^{-2}j^{\mathsf T}} (1+E^2) -  \frac{W^{-2}j^{\mathsf T}jW^{-2}}{(jW^{-2}j^{\mathsf T})^2} \\
-&= \frac{W^{-2}}{n} (1+E^2_k) -  \frac{l^{\mathsf T}l}{n^2}
+&= \frac{W^{-2}}{n} (1+E^2) -  \frac{l^{\mathsf T}l}{n^2}
 \end{aligned}
 $$
@@ Line 111: / Line 111: @@
 $$
-G^{\prime}_b(E_k) = W^{-2} - \frac{1}{n(1+E^2_k)}l^{\mathsf T}l
+G^{\prime}_b(E) = W^{-2} - \frac{1}{n(1+E^2)}l^{\mathsf T}l
 $$
-Again, this is equivalent to <math>G^{-1}</math>, when we pick <math>E_k = \sqrt{\frac{n+1}{n} - 1}</math>
+Again, this is equivalent to <math>G^{-1}</math>, when we pick <math>E = \sqrt{\frac{n+1}{n} - 1}</math>
-==== References ====
+== Generalized norm ==
+For some parameter <math>k</math>, set:
+$$
+X_k = \begin{bmatrix}
+W \\
+\hline
+k\cdot j
+\end{bmatrix}\\
+G(k) =  X_k^{\mathsf T} X_k = W^2 + k^2j^{\mathsf T}j
+$$
+Going through the same derivation, we find:
+$$
+G^{-1}(k) = W^{-2} - \frac{k^2}{n+1} l^{\mathsf T}l
+$$
+And the relation to parametric badness is <math>E = \sqrt{\frac{n+1}{nk^2} - 1}</math> or <math>k = \sqrt{\frac{n+1}{n}(1+E^2)}</math>
+== References ==