User:Sintel/Generator optimization: Difference between revisions
No edit summary |
m grammar |
||
| Line 56: | Line 56: | ||
== Weighted norm == | == Weighted norm == | ||
It can be beneficial to introduce weights for each prime. For example, in Tenney-Euclidean tuning, lower primes are prioritized slightly. We can introduce a diagonal weighting matrix <math>W</math>. For technical reasons, the norm is taken for <math>W^2</math> so that <math>\left\| x \right\|_{W^2}^2 = xW^2x^{\mathsf T}</math>. | It can be beneficial to introduce weights for each prime. For example, in Tenney-Euclidean tuning, lower primes are prioritized slightly. We can introduce a (usually diagonal) weighting matrix <math>W</math>. For technical reasons, the norm is taken for <math>W^2</math> so that <math>\left\| x \right\|_{W^2}^2 = xW^2x^{\mathsf T}</math>. | ||
This results in a weighted least squares problem: | This results in a weighted least squares problem: | ||
| Line 62: | Line 62: | ||
\underset{g}{\text{minimize}} \ \| gMV - jV \|_{W^2}^2 | \underset{g}{\text{minimize}} \ \| gMV - jV \|_{W^2}^2 | ||
$$ | $$ | ||
We can write this | We can write this as an ordinary least squares problem by setting: | ||
$$ | $$ | ||
\begin{gather} | \begin{gather} | ||