Field Matching: an Electrostatic Paradigm to Generate and Transfer Data

Dear reviewer, thank you for reviewing our paper. Below we answer your questions and comments.

(Q1) Only some visual comparisons with PFGM are provided. However, a comparison with RF/FM is necessary. Only toy datasets like 2D point sets and MNIST are provided.

It is worth noting that our main goal is to demonstrate proof-of-concept of our method in the experimental section. We agree that providing additional generating experiments with more complex datasets enhances the understanding of the method's performance. Further scalability of the method is a promising avenue for future research.

Nevertheless following your request, we include $**additional experiments**$ with more complex data such as CIFAR-10. For qualitative analysis, we demonstrate our EFM's results as well as PFGM's. We would like to ask you to get acquainted with Fig. 3 that is available via the anonymous link https://drive.google.com/file/d/1DTbQR_GNah7hVGGnDF822aD96iWxjR-k/view?usp=sharing.

For quantitative analysis, we calculate FID/CMMD scores on test part of the aforementioned datasets and compare our method with PFGM, DDPM, DDIM, GLOW. Firstly, we demonstrate quantitative performance of our method on full Colored MNIST dataset. We see that our method outperforms other approaches on full Colored MNIST, reaching the lowest FID/CMMD.

Secondly, we demonstrate quantitative performance of our method on CIFAR-10 dataset. We see that in image generation on CIFAR-10, our performance is comparable to PFGM. At the same time, we remind that our method is also capable of performing data-to-data transfer while PFGM is not capable of doing that.

In accordance of your request of comparison with FM, we quantitatively and qualitatively demonstrate the performance of our approach with SB-based(DDIB, $\alpha$-DSBM ) approaches as well as GAN-based (Cycle-GAN ) for unpaired data setups on Colored MNIST. Please, familiarize yourself with Fig. 6 via the afore mentioned link. For quantitative analysis, we demonstrate FID and CMMD metrics on Colored MNIST dataset for our and compared methods.

We see that our method demonstrates the highest FID and CMMD among compared approaches.

(Q2) [...] rectified flow and flow matching (RF/FM) [...] the training loss is just a scaled version of RF/FM.

We think that there might be a misunderstanding of our EFM method. In fact, we do not have a lot in common with flow matching (FM), except the fact that we learn an ODE to generate or transfer data. In particular, all the theoretical derivations and motivation totally differ.

• We work in $D+1$ dimensional space and learn a static (non time-dependent) vector field to transfer data, while the field in FM is time-dependent and is in $D$-dimensional space.

• The FM defines the interpolation $tx + (1-t)y$ between data samples $x$ and $y$ from two different data sets to regress a velocity field on $y-x$, where $t$ is sampled from a standard uniform distribution. In fact, the usage of this particular interpolant is principle for their loss construction. In turn, in our case, this interpolation is just a $*technical*$ way to define some inter-plate points $\widetilde{x}$ between data distributions where to approximate Coulomb field $E(\widetilde{x})$ at these points. In principle, we can use any other way to define the intermediate points.

To illustrate the aforementioned fact, we conduct the following 2-dimensional experiment with Swiss roll dataset. In the first case, we define the inter-plate points via the interpolation $tx + (1-t)y$ and approximate Coulomb field there. In the second case, we define uniform cube mesh between plates. The result of experiments demonstrates that the performance does not depend on interpolation. Please, familiarize yourself with the Fig.5 via the main link.

Since the geometry of data distributions is sufficiently difficult in high-dimensional spaces, we use Eq. (19) as the way to define intermediate points just to cover a bigger volume of space.

Concluding remarks. We would be grateful if you could let us know if the explanations we gave are satisfactory. If so, we kindly ask that you consider increasing your rating. We are also open to discussing any other questions you may have.

Dear reviewer, thank you for reviewing our paper. Below we answer to your questions and comments.

(Q1) [...] performance on high-dimensional or complex tasks (real-images) [...] testing on more diverse and complex datasets [...] quantitative results are lacking..

It is worth noting that our main goal is to demonstrate proof-of-concept of our method in the experimental section. We agree that providing additional generating experiments with more complex datasets enhances the understanding of method's performance. Further scalability of the method is a promising avenue for future research.

Nevertheless following your request, we include additional experiments with more complex data such as CIFAR-10. For qualitative analysis, we demonstrate our EFM's results as well as PFGM's performance. We would like to ask you to get acquainted with Fig. 3 that is available via the anonymous link https://drive.google.com/file/d/1DTbQR_GNah7hVGGnDF822aD96iWxjR-k/view?usp=sharing.

For quantitative analysis, we calculate FID and CMMD scores on test part of the aforementioned datasets and compare our method with PFGM, DDPM, DDIM and GLOW. Firstly, we demonstrate quantitative performance of our method on full Colored MNIST dataset. We see that our method outperforms other approaches on full Colored MNIST, reaching the lowest FID and CMMD.

Secondly, we demonstrate quantitative performance of our method on CIFAR-10 dataset. We see that in image generation on CIFAR-10, our performance is comparable to PFGM. At the same time, we remind that our method is also capable of performing data-to-data transfer while PFGM is not capable of doing that.

Additionally, we quantitatively and qualitatively compare the performance of our approach with SB-based (DDIB ,$\alpha$-DSBM ) approaches as well as GAN-based (Cycle-GAN ) for unpaired data setups on Colored MNIST. Please, familiarize yourself with Fig.6 via the afore mentioned link. For quantitative analysis, we demonstrate FID and CMMD metrics on Colored MNIST dataset for our and compared methods.

We see that our method demonstrates the highest FID and CMMD among compared approaches.

(Q2) [...] sensitivity to hyper-parameters [...] The inter-plate distance

We conduct additional experiment that deals with the influence $L$ on the performance ( see Fig.4 via the link). The more distance between plates the worse approximation of the field. If $L$ is too small, field is recoverable , but there is "edge effect" and it might has influence on performance.

Concluding remarks. We would be grateful if you could let us know if the explanations we gave are satisfactory. If so, we kindly ask that you consider increasing your rating. We are also open to discussing any other questions you may have.

We thank the reviewer for the valuable comments. Please find the answers to your questions below.

(Q1) [...] quantitative evaluations or comparisons [...] alternative approaches from the literature (e.g., SB-based, GAN-based). CIFAR [...]

Following your request, we include additional experiments with more complex data such as CIFAR-10. For qualitative analysis, we demonstrate our EFM's results as well as PFGM's performance. See Fig.3 via the anonymous link https://drive.google.com/file/d/1DTbQR_GNah7hVGGnDF822aD96iWxjR-k/view?usp=sharing. For quantitative evaluation, we calculate FID and CMMD[1] on test of CIFAR-10 and compare with PFGM, DDPM, DDIM and GLOW. Please see the table with results in the answer to YMMk.We see that in image generation our performance is comparable to PFGM. At the same time, we remind that our method is also capable of performing data-to-data transfer while PFGM is not capable of doing that.

In accordance of your request of comparison with GAN and SB-based methods, we also demonstrate the performance of our approach with SB-based (DDIB,$\alpha$-DSBM ) approaches as well as GAN-based (Cycle-GAN ) for unpaired data setups on Colored MNIST. Please, familiarize yourself with Fig.6 via the afore mentioned link. We also see that our method demonstrates the highest FID and CMMD among compared approaches.

(Q2) The manuscript primarily follows the derivations from [1].

We respectfully disagree. Our significant theoretical advancement compared to PFGM is using the following fundamental property of electric field lines in $D$-dimensional space: field lines starting from the positive charge distribution $\mathbb{P}(\cdot)$ almost surely terminate in the negative charge distribution $\mathbb{Q}(\cdot)$ (Lemma A.7). This property — previously not considered in electrostatic generative models — combined with field flux conservation along current tubes, formally establishes (our Theorem 3.1) that field line trajectories transport samples between $\mathbb{P}$ and $\mathbb{Q}$. This constitutes our main theoretical contribution enabling data-to-data transfer.

(Q3) In line 250, the authors mention that the inference process is summarized in Algorithm 1, but the algorithm only describes the training procedure.

Thanks for pointing that out. The inference algorithm is just a simulation of the learned ODE as described in Section 3. For convenience of the reader, we will add a separate algorithm box with inference procedure in the final version of the paper.

(Q4) I wonder if, in Algorithm 1, t should be sampled from Uniform(0, 1) to ensure proper interpolation between x+ and x-

The uniform sampling $t \sim \text{Uniform}(0,1)$ represents no more than a particular training volume selection strategy for interpolating between $\widetilde{\mathbf{x}}^+$ and $\widetilde{\mathbf{x}}^-$, controlling the density of training points along the interplanar axis. As demonstrated in Fig. 5 (link from Q1) for the Gauss-to-Swiss-roll experiment, both the proposed uniform training volume (Algorithm~1) and conventional cubic lattice initialization produce nearly identical electric field line configurations.

(Q5) How does the batch size influence the stability of the training? [...] batch size of 1?

Theoretically, it is possible to train our method with a batch size equal to 1. However, Monte Carlo integration has a higher variance for a small number of samples. Following the question, we conducted an extra experiment with different batch sizes. We demonstrate that performance almost stops to increase with batch sizes exceeding 64, see Fig. 2 in extra material (link from Q1)

(Q6) I wonder if samples that are on the periphery of the P distribution would not be pushed away and land far away from Q distribution as a result?

This scenario is precluded by Theorem 3.1, which guarantees transport between distributions $\mathbb{P}(\cdot)$ and $\mathbb{Q}(\cdot)$. While peripheral samples exhibit greater curvature compared to central trajectories due to boundary effects, this geometric difference does not prevent their convergence to $\mathbb{Q}(\cdot)$. The continuity of field lines (Lemma A.7) ensures termination on the target distribution almost surely.

(Q7) [...] source code [...] unable to run. [...] functions not included in the provided package

We will fix this in the final code version.

Concluding remarks. We would be grateful if you could let us know if the explanations we gave are satisfactory. If so, we kindly ask that you consider increasing your rating. We are also open to discussing any other questions you may have.

Dear reviewer, thank you for reviewing our paper. Below we answer your questions.

(Q1) [...] Experiments on more complex datasets[..]some quantitative evaluations would also help.

Following your request, we include additional experiments with more complex data such as CIFAR-10 and the requested full color-MNIST dataset. For qualitative analysis, we demonstrate our EFM's results as well as PFGM's performance. We ask you to get acquainted with Fig. 1 and Fig. 3 that are available via the anonymous link https://drive.google.com/file/d/1DTbQR_GNah7hVGGnDF822aD96iWxjR-k/view?usp=sharing. For quantitative analysis, we calculate FID/CMMD scores and compare with another methods. Please, familiarize yourself with the the answer to YMMk.

(Q2) [...] accurate approximation of the field using Monte Carlo integration? [...] In high dimensions, field contributions from distant charges decay rapidly with distance [...]

If we correctly understand your question, your ask about the accurate approximation of $E(\widetilde{x})$ in Eq. (11) for a given $\widetilde{x}$ via Monte Carlo sampling. Since Monte Carlo integration provides an unbiased estimation of an integral and its variance does not depend on the dimensionality, this estimation might be used for accurate approximation of the field. However, the more samples are used in batch for estimation the lower variance. We conduct additional experiments with different batch sizes and its influence on our EFM's performance. Please, familiarize yourself with the Fig. 2 via the main link.

(Q3) [...] dipole effect [...] Some field lines initially travel backward [...] reach the target plate from behind. [...] how can we ensure accurate tracing of outward field lines?

Indeed, each plate emits two distinct sets of field lines. One set is directed towards the second plate and another oriented in the opposite direction. Crucially, the properties of electric field lines (LemmaA.7) ensure that both sets almost surely terminate on the opposing plate. The primary distinction lies in their geometric trajectories: the forward-directed lines exhibit smaller curvature and reach the target plate more efficiently (faster) than their backward-oriented counterparts. From a practical point of view, prioritizing forward-directed trajectories is advantageous for computational efficiency while remaining theoretically sound due to Theorem~3.1, which guarantees almost sure transport from $\mathbb{P}(\cdot)$ to $\mathbb{Q}(\cdot)$.

To address field lines extending beyond the plate boundary ($z > L$) before reaching $\mathbb{Q}(\widetilde{\mathbf{x}}^-)$, one may use the following natural criterion to distinguish the valid termination on $\mathbb{Q}(\cdot)$ from the transient crossings of $z = L$:

$

\begin{cases}
    E_z(z \to L^-) = E_z(z \to L^+) & \implies \text{The line goes away past the distribution}, \\
    E_z(z \to L^-) \neq E_z(z \to L^+) & \implies \text{Valid termination on } \mathbb{Q}(\widetilde{\mathbf{x}}^-).
\end{cases}

$

This criterion derives from:

1. Field continuity along current tubes (Lemma A.3, Corollary A.4),
2. Boundary field behavior (LemmaA.5): Near a plate, the field is determined by the charge of this plate and is directed away from the plate in the case of positive charge (and towards the plate in the case of negative charge, see lemma A.5). Therefore the direction and magnitude of the field must change in the case if we arrive at the target distribution.

Thus, discontinuities in $E_z$ explicitly signal successful transitions to $\mathbb{Q}(\cdot)$, while continuity indicates a need for further integration.

(Q4) matter of parameter L [...]

We conduct additional experiment that deals with the influence $L$ on the performance (see Fig.4 via the link). The more distance between plates the worse approximation of the field. If L is too small, field is recoverable , but there is "edge effect" and it might has influence on performance.

(Q5) PFGM [...] the underlying theory is the same [...]

We respectfully disagree. Our significant theoretical advancement compared to PFGM is using the following fundamental property of electric field lines in $D$-dimensional space. The field lines starting from the positive charge distribution $\mathbb{P}(\cdot)$ almost surely terminate in the negative charge distribution $\mathbb{Q}(\cdot)$ (Lemma A.7). This property — previously not considered in electrostatic generative models — combined with field flux conservation along current tubes, formally establishes (our Theorem 3.1) that field line trajectories transport samples between $\mathbb{P}$ and $\mathbb{Q}$. This constitutes our main theoretical contribution.

(Q6) Is it safe to ignore field lines that start out facing the wrong direction?

Yes, it is safe due to our main theorem. See the answer to the Q3 for details.