Response to your official comment
Dear Reviewer, we appreciate your thorough attention to the details of our paper. Please find below the clarifications for the concerns which you raised.
**(1a) Figure 23. In columns 5 and 6, as $w$ increases, the generated chair seems to deviate from the input pattern. In penultimate (11th) and final (12th) columns, the generated chair doesn't seem to closely match the input pattern.**
**(1b) Figure 24. In the 1st column, the generated face at $w=8$ seems to lack the teeth detail. In the 5th column, the representation at $w=1$ appears to be more in line with the input because it does not show the teeth.**
In order to address your concerns about the patterns and details, we additionally explain several points about our results.
- Particular samples are not good representatives of the properties of the entire learned transport map $\hat{T}$. According to the average $\ell^2$ cost between the generated and respective input images, the **average input-output dissimilarity** $\frac{1}{N_{test}}\sum_{n=1}^{N_{test}}\ell^{2}(x_n,\hat{T}(x_n))$ indeed decreases for all the dataset pairs with the increase of the weight $w$, see Table 1 in our paper.
- Moreover, we additionally calculate $\ell^2(x,\hat{T}(x))$ for the **particular face/texture samples** $x$ to which you pointed to. We see that their input-output dissimilarity also decreases with the increase of $w$ in majority of the cases, see the Table below.
| | $w=1$ | $w=2$ | $w=4$ | $w=8$ |
|------------------|-------|-------|-------|-------|
| Fig. 23, col. 5 | 0.635 | 0.526 | 0.522 | 0.482 |
| Fig. 23, col. 6 | 0.524 | 0.378 | 0.323 | 0.277 |
| Fig. 23, col. 11 | 0.798 | 0.724 | 0.610 | 0.591 |
| Fig. 23, col. 12 | 0.410 | 0.420 | 0.291 | 0.226 |
| Fig.24, col. 1 | 0.258 | 0.239 | 0.213 | 0.198 |
| Fig.24, col. 5 | 0.154 | 0.131 | 0.105 | 0.103 |
- It is a generic knowledge that $\ell^2$ cost is mostly sensitive to low-frequency image components (big image parts) rather than high-frequency components (tiny details). For example, as your noted, in the 1st column of Figure 24, IT map for $w=8$ lacks the fine teeth detail. However, here we observe the presence of the long hair (like in the input image). Note that the hair was not generated for smaller weights. In the *textures*$\rightarrow$*chairs* experiment, the situation is also expainable. Here with the increase of $w$, the IT method with $\ell^{2}$ focuses more on the reducing the white background area. At the same time, preservation of the fine details of the texture may become less important. We have partially mentioned this aspect in lines 508-509 of Appendix A (limitations).
To conclude, when using $\ell^{2}$ cost, it is not correct to assess similarity of the images focusing only on fine/tiny image details. In order to focus on such details, one may consider using more advanced cost functions, e.g., perceptual ones from computer vision. While this is a promising research avenue, it is beyond the scope of our paper which focuses on developing the theoretical formulation and the computational algorithm. Still to ensure that our IT algorithm may work with costs beside $\ell^{2}$, we have already tested it with a VGG-based perceptual cost, see Appendix G.1.
**(2) Further enhancement, e.g., using kernel neural OT.**
We agree that many potential modifications to our methodology can be considered. For example, apart from the advanced cost functions mentioned above, one indeed may employ kernel regularization [1]. We note that the goal of the current work is to establish the **basic fundament** to which such modifications can be applied. Studying of all possible modifications is a separate interesting topic which is out of the scope of our paper. At the same time, we have already provided evidence that such modifications are applicable and sometimes useful. In Appendix G.1, there are our experiments with a perceptual cost and in the attached PDF $-$ your requested experiment with kernel regularization.
**Concluding remarks.** We hope that our clarifications have resolved your concerns regarding our method performance and potential enhancement. If the responses above are sufficient, we kindly ask that you consider raising your score.
**References**
[1] Korotin, A. et al. (2023). Kernel neural optimal transport.