Thank you for recognizing the theoretical contributions and experimental demonstrations. We want to emphasize that our approach represents a paradigmatic shift from time-based diffusion to operator-based interpolation, fundamentally reconceptualizing how multitask generation can be achieved with guaranteed unbiased sampling.

Should you find our reply satisfactory, we kindly invite you to reconsider your rating.

Larger-scale experiments:

Since submission, we have conducted extensive experiments on CelebA (128×128) and FFHQ (256×256) datasets to demonstrate scalability and benchmark our approach against different methods in various image restoration tasks, evaluating the average Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM) on 100 test images from each dataset. To assess the performance of our methodology using Hadamard product-based operators, we employed two types of masking: square masks of sizes $40\times40$ and $80\times80$ with added Gaussian noise of standard deviation 0.05, and random masks covering $70\%$ of image pixels with Gaussian noise of standard deviation 0.01.

The results reported in the table below show that our method consistently ranks either first or second in both reconstruction metrics across all tasks and datasets. Regarding visual quality, our method generates realistic, artifact-free images.

CelebA Dataset (128×128)

AFHQ-Cat Dataset (256×256)

These experiments indicate that our operator-based interpolants maintain their versatility at scale, and confirm that our approach scales effectively while preserving the key advantage of training once and adapting at inference time.

In addition, we have preliminary results on a robotics maze planning experiment, as is done in [5]. In this setting, given starting and terminal coordinates in a maze, the aim is to construct a path between the points along the maze without hitting walls or obstacles. Our method allows the zero-shot generation of this path for any pair of points in the maze without re-training. In our preliminary results, we see a 100% success rate in this task, and are also in direct communication with the authors of [5] to produce their reward metrics, if that is of interest to the reviewers.

Operator selection complexity:

This is indeed a key practical challenge. We have added extensive discussion and empirical guidelines for operator selection, including: (1) stability analysis for different operator classes, (2) empirical scaling laws for training complexity vs. operator diversity, (3) practical heuristics for choosing appropriate operator spaces for different application domains. The CelebA/FFHQ experiments provide concrete examples of operator selection at scale. Importantly, once selected, our operator-based framework provides exact sampling without the approximation biases that plague existing methods.

Implementation details:

We have expanded the algorithmic section with detailed implementation guidelines, stability considerations, and practical tips for different operator types based on our large-scale experiments. Unlike existing multitask approaches that require complex approximation strategies, our mathematically rigorous framework provides clear implementation paths with theoretical guarantees.

Key conceptual contribution:

While existing multitask methods still rely on traditional diffusion in time concepts with heterogeneous processes, our work represents a fundamental reconceptualization - treating all generative tasks as different geometric traversals through the same underlying operator space. This enables theoretically exact, unbiased sampling for all tasks without the systematic errors inherent in current approximation-based approaches.

References:

[1] A. Pokle, M. Muckley, R. Chen, B. Karrer, "Training-free linear image inverses via flows" (TMLR)

[2] H. Ben-Hamu, O. Puny, I. Gat, B. Karrer, U. Singer, Y. Lipman, "D-Flow: Differentiating through Flows for Controlled Generation"

[3] Y. Zhang, P. Yu, Y. Zhu, Y. Chang, F. Gao, Y. Nian Wu, O. Leong, "Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching" (Neurips 2024)

[4] S. Martin, A. Gagneux, P. Hagemann, G. Steidl, "PnP-Flow: Plug-and-Play Image Restoration with Flow Matching" (ICLR 2025)

[5] B. Chen, D. Marti Monso, Y. Du, M. Simchowitz, R. Tedrake, V. Sitzmann "Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" (NeurIPS 2024)

Thank you for the positive evaluation and acceptance recommendation. We want to emphasize that our work represents a fundamental conceptual breakthrough - moving beyond traditional time-based diffusion to operator-based interpolation, with theoretically guaranteed unbiased sampling unlike existing approximation-based methods.

Larger-scale experiments:

Since submission, we have conducted extensive experiments on CelebA (128×128) and FFHQ (256×256) datasets to demonstrate scalability and benchmark our approach against different methods in various image restoration tasks, evaluating the average Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM) on 100 test images from each dataset. To assess the performance of our methodology using Hadamard product-based operators, we employed two types of masking: square masks of sizes $40\times40$ and $80\times80$ with added Gaussian noise of standard deviation 0.05, and random masks covering $70\%$ of image pixels with Gaussian noise of standard deviation 0.01.

The results reported in the table below show that our method consistently ranks either first or second in both reconstruction metrics across all tasks and datasets. Regarding visual quality, our method generates realistic, artifact-free images.

CelebA Dataset (128×128)

AFHQ-Cat Dataset (256×256)

These experiments indicate that our operator-based interpolants maintain their versatility at scale, and confirm that our approach scales effectively while preserving the key advantage of training once and adapting at inference time.

In addition, we have preliminary results on a robotics maze planning experiment, as is done in [5]. In this setting, given starting and terminal coordinates in a maze, the aim is to construct a path between the points along the maze without hitting walls or obstacles. Our method allows the zero-shot generation of this path for any pair of points in the maze without re-training. In our preliminary results, we see a 100% success rate in this task, and are also in direct communication with the authors of [5] to produce their reward metrics, if that is of interest to the reviewers.

Trade-off analysis:

These quantitative results directly address your question about trade-offs. While our multitask model has higher computational overhead during training, it achieves competitive or superior performance compared to specialized methods across tasks while eliminating the need for separate models. Critically, unlike existing multitask approaches that introduce systematic biases through approximations, our method provides exact, unbiased sampling for all tasks.

Invertible operators $\alpha$ and $\beta$:

We appreciate this question, but note that invertibility is not required for all tasks - it primarily helps with computational efficiency by allowing us to train one η field and deduce the other from relation (5), rather than having to train both η₀ and η₁ separately. For our main applications (Hadamard products, Fourier domain operations), invertibility is naturally satisfied.

References:

[1] A. Pokle, M. Muckley, R. Chen, B. Karrer, "Training-free linear image inverses via flows" (TMLR)

[2] H. Ben-Hamu, O. Puny, I. Gat, B. Karrer, U. Singer, Y. Lipman, "D-Flow: Differentiating through Flows for Controlled Generation"

[3] Y. Zhang, P. Yu, Y. Zhu, Y. Chang, F. Gao, Y. Nian Wu, O. Leong, "Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching" (Neurips 2024)

[4] S. Martin, A. Gagneux, P. Hagemann, G. Steidl, "PnP-Flow: Plug-and-Play Image Restoration with Flow Matching" (ICLR 2025)

[5] B. Chen, D. Marti Monso, Y. Du, M. Simchowitz, R. Tedrake, V. Sitzmann "Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" (NeurIPS 2024)

We appreciate the detailed feedback. We want to stress that our work represents a completely novel conceptual paradigm: while existing methods still operate within traditional time-based diffusion frameworks, we fundamentally reconceptualize multitask generation through operator-valued interpolants. Crucially, our method provides theoretically unbiased sampling unlike current approaches which often comes with inherent systematic biases.

Should you find our reply satisfactory, we kindly invite you to reconsider your rating. WE would also be happy to answer any additional questions you have or provide clarification on our replies.

Related work contextualization:

You are correct about relations to diffusion forcing and MuLAN, but our approach represents a paradigmatic shift from temporal diffusion to operator-based interpolation. While existing methods still rely on the concept of diffusion in time with task-specific modifications, we completely reconceptualize the problem. Our framework enables fine-tuning and posterior sampling capabilities these methods cannot handle, and guarantees exact sampling from target distributions, eliminating the approximation errors that compromise existing multitask methods.

Section 3.4 clarity:

We have completely rewritten this section to address your concerns. Inference adaptation refers to the problem of optimizing the path ($\alpha_t$, $\beta_t$) used during generation to achieve specific objectives, such as minimizing computational cost, maximizing sample quality, or satisfying user constraints. This can be done in two ways: (1) offline optimization where we pre-compute optimal paths for different scenarios using objectives like Wasserstein length minimization in Eq. (22), and (2) online adaptation where paths are dynamically adjusted during generation based on intermediate results or user feedback.

The "bridge distributions" concept is now formally defined as finding paths ($\alpha_t$, $\beta_t$) that optimally transport probability mass from an initial distribution $\mu_{\alpha_0,\beta_0}$ to a target distribution $\mu_{\alpha_1,\beta_1}$, where optimality is measured by criteria such as the Wasserstein distance or other transport costs. Unlike fixed diffusion schedules, our framework allows these transport paths to be optimized post-training for specific applications.

Concrete examples include: (1) adaptive mask scheduling for inpainting where $\alpha_t$ is dynamically updated based on reconstruction quality, (2) multi-resolution generation where paths are optimized to generate coarse features first, then fine details, and (3) interactive editing where user feedback guides path selection in real-time.

Posterior sampling limitation:

While you correctly note the quadratic structure requirement, this represents a large and practically important class, particularly in fine-tuning applications (Gaussian priors, L2 regularization, quadratic rewards). The key insight is that to exploit this class fully, it is important to use operator-based (as opposed to scalar) α and β. The unified treatment allows the same trained model to handle both arbitrary inpainting AND posterior sampling with theoretical guarantees, which previous methods cannot achieve without retraining or biased approximations.

Larger-scale experiments:

Since submission, we have conducted extensive experiments on CelebA (128×128) and FFHQ (256×256) datasets to demonstrate scalability and benchmark our approach against different methods in various image restoration tasks, evaluating the average Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM) on 100 test images from each dataset. To assess the performance of our methodology using Hadamard product-based operators, we employed two types of masking: square masks of sizes $40\times40$ and $80\times80$ with added Gaussian noise of standard deviation 0.05, and random masks covering $70\%$ of image pixels with Gaussian noise of standard deviation 0.01.

The results reported in the table below show that our method consistently ranks either first or second in both reconstruction metrics across all tasks and datasets. Regarding visual quality, our method generates realistic, artifact-free images.

CelebA Dataset (128×128)

AFHQ-Cat Dataset (256×256)

These experiments indicate that our operator-based interpolants maintain their versatility at scale, and confirm that our approach scales effectively while preserving the key advantage of training once and adapting at inference time.

In addition, we have preliminary results on a robotics maze planning experiment, as is done in [5]. In this setting, given starting and terminal coordinates in a maze, the aim is to construct a path between the points along the maze without hitting walls or obstacles. Our method allows the zero-shot generation of this path for any pair of points in the maze without re-training. In our preliminary results, we see a 100% success rate in this task, and are also in direct communication with the authors of [5] to produce their reward metrics, if that is of interest to the reviewers.

Technical corrections:

All typos and formatting errors have been corrected.

References:

[1] A. Pokle, M. Muckley, R. Chen, B. Karrer, "Training-free linear image inverses via flows" (TMLR)

[2] H. Ben-Hamu, O. Puny, I. Gat, B. Karrer, U. Singer, Y. Lipman, "D-Flow: Differentiating through Flows for Controlled Generation"

[3] Y. Zhang, P. Yu, Y. Zhu, Y. Chang, F. Gao, Y. Nian Wu, O. Leong, "Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching" (Neurips 2024)

[4] S. Martin, A. Gagneux, P. Hagemann, G. Steidl, "PnP-Flow: Plug-and-Play Image Restoration with Flow Matching" (ICLR 2025)

[5] B. Chen, D. Marti Monso, Y. Du, M. Simchowitz, R. Tedrake, V. Sitzmann "Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" (NeurIPS 2024)