User:Sintel/Dual Weil-Euclidean norm: Difference between revisions
more fixes |
→Relation to other metrics: -stray subscript |
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:<math> | :<math> | ||
\left\langle \alpha, \beta \right\rangle^{\ast} = \alpha G^{-1} \beta^{\mathsf T} | |||
\left\langle \alpha, \beta \right\rangle^{\ast} = \alpha G^{-1} \beta^{\mathsf T} | </math> | ||
:<math> | |||
||\alpha||^{\ast} = \sqrt{\left\langle \alpha,\alpha \right\rangle^{\ast}} = \sqrt{\alpha G^{-1} \alpha^{\mathsf T}} | ||\alpha||^{\ast} = \sqrt{\left\langle \alpha,\alpha \right\rangle^{\ast}} = \sqrt{\alpha G^{-1} \alpha^{\mathsf T}} | ||
</math> | </math> | ||
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<ref>Miller, K. S. (1981). On the Inverse of the Sum of Matrices. Mathematics Magazine, 54(2), 67–72. https://doi.org/10.2307/2690437</ref> | <ref>Miller, K. S. (1981). On the Inverse of the Sum of Matrices. Mathematics Magazine, 54(2), 67–72. https://doi.org/10.2307/2690437</ref> | ||
: If <math>A</math> and <math>A+B</math> are invertible, and <math>B</math> has rank 1, then let <math>g = \text{tr}(BA^{-1})</math>. Then <math>g \neq -1</math> and | ::If <math>A</math> and <math>A+B</math> are invertible, and <math>B</math> has rank 1, then let <math>g = \text{tr}(BA^{-1})</math>. Then <math>g \neq -1</math> and | ||
: <math>(A+B)^{−1}=A^{-1} − \frac{1}{1+g}A^{-1}BA^{-1}</math> | ::<math>(A+B)^{−1}=A^{-1} − \frac{1}{1+g}A^{-1}BA^{-1}</math> | ||
Now identifying <math>A = W^2</math> and <math>B = j^{\mathsf T}j</math>. We can see that | Now identifying <math>A = W^2</math> and <math>B = j^{\mathsf T}j</math>. We can see that | ||
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\end{bmatrix} | \end{bmatrix} | ||
</math>, then | </math>, then | ||
:<math> l = W^{-2}j </math> | |||
:<math> | :<math> | ||
G^{-1} = W^{-2} - \frac{1}{1+g} l^{\mathsf T}l | G^{-1} = W^{-2} - \frac{1}{1+g} l^{\mathsf T}l | ||
</math> | </math> | ||
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:<math> | :<math> | ||
j^{\mathsf T}j \circ W^{-2} = I_n | |||
j^{\mathsf T}j \circ W^{-2} = I_n | </math> | ||
:<math> | |||
g = \text{tr}(BA^{-1}) = \text{tr}(j^{\mathsf T}j W^{-2}) = n | g = \text{tr}(BA^{-1}) = \text{tr}(j^{\mathsf T}j W^{-2}) = n | ||
</math> | </math> | ||
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</math> | </math> | ||
Since the metric is equivalent up to scaling, we multiply by <math>\frac{n}{1+E^ | Since the metric is equivalent up to scaling, we multiply by <math>\frac{n}{1+E^2}</math> to obtain: | ||
:<math> | :<math> | ||
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\hline | \hline | ||
k\cdot j | k\cdot j | ||
\end{bmatrix} | \end{bmatrix} | ||
</math> | |||
:<math> | |||
G(k) = X_k^{\mathsf T} X_k = W^2 + k^2j^{\mathsf T}j | G(k) = X_k^{\mathsf T} X_k = W^2 + k^2j^{\mathsf T}j | ||
</math> | </math> | ||
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:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
nk^2E^2 = 1\\ | nk^2E^2 &= 1\\ | ||
E = \sqrt{\frac{1}{nk^2}}\\ | E &= \sqrt{\frac{1}{nk^2}}\\ | ||
k = \sqrt{\frac{1}{nE^2}} | k &= \sqrt{\frac{1}{nE^2}} | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
== References == | == References == | ||