DEtection TRansformer (DETR)
0 | 1 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 |
- A
- B
- \(\varnothing\)
- \(\varnothing\)
- \(\varnothing\)
$$ \begin{aligned}
\min_{\sigma \in \text{Perm}(N_p)} & \frac{1}{n} \sum_{i=1}^{N_p} C_{i,\sigma(i)}
\end{aligned} $$
$$ \begin{aligned}
\min_{\mathbf{P} \in \{0,1\}^{N_p \times N_g} } & < \mathbf{P},\mathbf{C} > \\
s.t. & \sum_{j} P_{ij} = 1 \quad \forall i \\
& \sum_{i} P_{ij} = 1 \quad \forall j
\end{aligned} $$
Unbalanced OT Lenaic Chizat, et al. (2018)
Optimal Transport Leonid Kantorovich (Nobel Prize 1975)
0 | 1 | 0 | 0 | 0 |
0 | 0 | 0.33 | 0.33 | 0.33 |
0 | 0 | 0.33 | 0.33 | 0.33 |
1 | 0 | 0 | 0 | 0 |
0 | 0 | 0.33 | 0.33 | 0.33 |
- A
- B
- \(\varnothing\)
- \(\varnothing\)
- \(\varnothing\)
$$ \begin{aligned}
\min_{\mathbf{P} \in \mathbb{R}_{+}^{N_p \times N_g} } < \mathbf{P},\mathbf{C} >
& - \epsilon \mathbf{H}(\mathbf{P}) \\
& + \textcolor{red}{10^{-3}} \mathrm{KL}\left(\boldsymbol{P} \mathbf{1}_{N_g} \| \boldsymbol{\alpha}\right) \\
& + \textcolor{red}{10^{3}} \mathrm{KL}\left(\mathbf{1}_{N_p}^{\top} \boldsymbol{P} \| \boldsymbol{\beta}\right) \\
\end{aligned} $$
$$ \begin{aligned}
\min_{\mathbf{P} \in \mathbb{R}_{+}^{N_p \times N_g} } < \mathbf{P},\mathbf{C} >
& - \epsilon \mathbf{H}(\mathbf{P}) \\
& + \textcolor{red}{10^{3}} \mathrm{KL}\left(\boldsymbol{P} \mathbf{1}_{N_g} \| \boldsymbol{\alpha}\right) \\
& + \textcolor{red}{10^{-3}} \mathrm{KL}\left(\mathbf{1}_{N_p}^{\top} \boldsymbol{P} \| \boldsymbol{\beta}\right) \\
\end{aligned} $$
$$ \begin{aligned}
\min_{\mathbf{P} \in \mathbb{R}_{+}^{N_p \times N_g} } < \mathbf{P},\mathbf{C} >
& - \epsilon \mathbf{H}(\mathbf{P}) \\
& + \tau_1 \mathrm{KL}\left(\boldsymbol{P} \mathbf{1}_{N_g} \| \boldsymbol{\alpha}\right) \\
& + \tau_2 \mathrm{KL}\left(\mathbf{1}_{N_p}^{\top} \boldsymbol{P} \| \boldsymbol{\beta}\right) \\
\end{aligned} $$
$$ \begin{aligned}
\min_{\mathbf{P} \in \mathbb{R}_{+}^{N_p \times N_g} } & < \mathbf{P},\mathbf{C} > \textcolor{red}{- \epsilon \mathbf{H}(\mathbf{P})} \\
s.t. & \sum_{j} P_{ij} = \alpha_i \quad \forall i \\
& \sum_{i} P_{ij} = \beta_j \quad \forall j
\end{aligned} $$
\( \textcolor{red}{\mathbf{H}(\mathbf{P}) \stackrel{} = -\sum_{i, j} \mathbf{P}_{i, j}\left(\log \left(\mathbf{P}_{i, j}\right)-1\right)} \)
$$ \begin{aligned}
\min_{\mathbf{P} \in \mathbb{R}_{+}^{N_p \times N_g}} & < \mathbf{P},\mathbf{C} > \\
s.t. & \sum_{j} P_{ij} = \textcolor{red}{\alpha_i} \quad \forall i \\
& \sum_{i} P_{ij} = \textcolor{red}{\beta_j} \quad \forall j
\end{aligned} $$
$$ \begin{aligned}
\min_{\mathbf{P} \in \textcolor{red}{\mathbb{R}_{+}^{N_p \times N_g}} } & < \mathbf{P},\mathbf{C} > \\
s.t. & \sum_{j} P_{ij} = 1 \quad \forall i \\
& \sum_{i} P_{ij} = 1 \quad \forall j
\end{aligned} $$