User:Sintel/Generator optimization: Difference between revisions

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== Least squares solution ==

If we want more than <math>n</math> intervals to be just, the result is an overconstrained system, and no solutions can be found.

However, we can instead obtain an approximate solution~~, the~~ minimizes the squared error.

However, we can instead obtain an approximate solution that minimizes the squared error.

If <math>V</math> is now a <math>k \times m</math> matrix (<math>m \gt n</math>), then the error for these intervals is:

If <math>V</math> is now a <math>k \times m</math> matrix (where <math>m \gt n</math>), then the error for these intervals is:

$$

gMV - jV

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Note that since we are working with row vectors, the norm is defined such that <math>\left\| x \right\|^2 = xx^{\mathsf T}</math>.

Differentiating with respect to <math>g~~^{\mathsf T}~~</math>:

Differentiating with respect to <math>g</math>:

$$

\begin{gather}

& \frac{\partial }{\partial g~~^{\mathsf T}~~} \| gMV - jV \|^2 \\

& \frac{\partial }{\partial g} \| gMV - jV \|^2 \\

= & \frac{\partial }{\partial g~~^{\mathsf T}~~} (gMV - jV)(gMV - jV)^{\mathsf T} \\

= & \frac{\partial }{\partial g} (gMV - jV)(gMV - jV)^{\mathsf T} \\

= & \frac{\partial }{\partial g~~^{\mathsf T}~~} (gMVV^{\mathsf T}M^{\mathsf T}g^{\mathsf T} - 2jVV^{\mathsf T}M^{\mathsf T}g^{\mathsf T} + jVV^{\mathsf T}j^{\mathsf T}) \\

= & \frac{\partial }{\partial g} (gMVV^{\mathsf T}M^{\mathsf T}g^{\mathsf T} - 2jVV^{\mathsf T}M^{\mathsf T}g^{\mathsf T} + jVV^{\mathsf T}j^{\mathsf T}) \\

= & 2gMVV^{\mathsf T}M^{\mathsf T} - 2jVV^{\mathsf T}M^{\mathsf T} = 0

\end{gather}

@@ Line 23: / Line 23: @@
 == Least squares solution ==
 If we want more than <math>n</math> intervals to be just, the result is an overconstrained system, and no solutions can be found.
-However, we can instead obtain an approximate solution, the minimizes the squared error.
+However, we can instead obtain an approximate solution that minimizes the squared error.
-If <math>V</math> is now a <math>k \times m</math> matrix (<math>m \gt n</math>), then the error for these intervals is:
+If <math>V</math> is now a <math>k \times m</math> matrix (where <math>m \gt n</math>), then the error for these intervals is:
 $$
 gMV - jV
@@ Line 34: / Line 34: @@
 Note that since we are working with row vectors, the norm is defined such that <math>\left\| x  \right\|^2 = xx^{\mathsf T}</math>.
-Differentiating with respect to <math>g^{\mathsf T}</math>:
+Differentiating with respect to <math>g</math>:
 $$
 \begin{gather}
-& \frac{\partial }{\partial g^{\mathsf T}}  \|   gMV - jV   \|^2 \\
+& \frac{\partial }{\partial g}  \|   gMV - jV   \|^2 \\
-= & \frac{\partial }{\partial g^{\mathsf T}}  (gMV - jV)(gMV - jV)^{\mathsf T}  \\
+= & \frac{\partial }{\partial g}  (gMV - jV)(gMV - jV)^{\mathsf T}  \\
-= & \frac{\partial }{\partial g^{\mathsf T}} (gMVV^{\mathsf T}M^{\mathsf T}g^{\mathsf T}  - 2jVV^{\mathsf T}M^{\mathsf T}g^{\mathsf T} + jVV^{\mathsf T}j^{\mathsf T}) \\
+= & \frac{\partial }{\partial g} (gMVV^{\mathsf T}M^{\mathsf T}g^{\mathsf T}  - 2jVV^{\mathsf T}M^{\mathsf T}g^{\mathsf T} + jVV^{\mathsf T}j^{\mathsf T}) \\
 = & 2gMVV^{\mathsf T}M^{\mathsf T}  - 2jVV^{\mathsf T}M^{\mathsf T} = 0
 \end{gather}