Discrete Measure
General Measure
$$ \rho_{\alpha} = \frac{\textrm{d}\alpha}{\textrm{d}x} $$
Given a Cost Matrix \( \mathbf{C}_{i,j}\) where \( i\in [[n]] \), \( j\in[[m]] \)
Assuming \( n = m \), the optimal assignment problem
seeks for a bijection \( \sigma \) in the set \( Perm(n) \) solving
$$ \min_{\sigma \in Perm(n)} \frac{1}{n} \sum_{i=1}^{n} \mathbf{C}_{i, \sigma(i)} $$
$$ \min_{\sigma \in Perm(n)} \frac{1}{n} \sum_{i=1}^{n} \mathbf{C}_{i, \sigma(i)} $$
0 | 1 | 2 | 3 | 4 | 5 |
4 | 1 | 3 | 2 | 0 | 5 |
\( \sigma_1(i) \)
3 | 0 | 5 | 4 | 2 | 1 |
\( \sigma_2(i) \)
\( i \)
Find the best index combination between two measures such that cost is minimized