Thank you for your response. We would will address your points in the following way.
- If so, I recommend the authors to reconsider the following phrases in the abstract and conclusion: "Using category theory, we give a no-go theorem." "We used tools from category theory to produce a no-go theorem"
We appreciate the advice, and will follow it. We will replace
"Using category theory, we give a no-go theorem"
with
"Using analytical arguments, we give a no-go theorem framed with category theory."
and replace
"We used tools from category theory to produce a no-go theorem"
with
"We give a no-go theorem framed with category theory"
in the abstract and conclusion.
- It would be much impactful and significant if the authors could more directly point out any incorrectness of the proof or inappropriateness of the assumption in the previous studies.
There are several papers which use continuous functions (either as elements of infinite dimensional function spaces or metric spaces) to model images or signal and apply statistical methods and invertible neural networks or maps modeling diffeomorphisms. Often in these papers one derives theoretical results in the continuous models and presents numerical results using a finite dimensional approximations. In this process the errors are caused by the discretization and the effect of changing the dimension of the approximate models. We believe that our work meaningfully addresses these questions as applied to injective/bijective neural operators, an important architecture. We hope that our paper inspires further study these points. We can include citations to the following papers, related to these issues.
The below papers which combine neural networks and approximation of diffeomorphisms, as applied to imaging.
- Elena Celledoni · Helge Glöckner · Jørgen N. Riseth, Alexander Schmeding
Deep neural networks on diffeomorphism groups for
optimal shape reparametrization.
BIT Numerical Mathematics (2023) 63:50
- GradICON: Approximate Diffeomorphisms via Gradient Inverse Consistency
Lin Tian · Hastings Greer · François-Xavier Vialard · Roland Kwitt · Raúl San José Estépar · Richard Jarrett Rushmore · Nikolaos Makris · Sylvain Bouix · Marc Niethammer
West Building Exhibit Halls ABC 153
The below papers combine invertible neural networks and statistical models, especially for solving inverse problems (including imaging problems).
- Alexander Denker , Maximilian Schmidt , Johannes Leuschner and Peter Maass
Conditional Invertible Neural Networks for Medical Imaging. Journal of Imaging 2021, 7(11), 243
- Ardizzone, L.; Kruse, J.; Rother, C.; Köthe, U. Analyzing Inverse Problems with Invertible Neural Networks. In Proceedings of
the 7th International Conference on Learning Representations (ICLR 2019), New Orleans, LA, USA, 6–9 May 2019.
- Anantha Padmanabha, G.; Zabaras, N. Solving inverse problems using conditional invertible neural networks. J. Comput. Phys.
2021, 433, 110194
- Denker, A.; Schmidt, M.; Leuschner, J.; Maass, P.; Behrmann, J. Conditional Normalizing Flows for Low-Dose Computed
Tomography Image Reconstruction. In Proceedings of the ICML Workshop on Invertible Neural Networks, Normalizing Flows,
and Explicit Likelihood Models, Vienna, Austria, 18 July 2020.
- Hagemann, P.; Hertrich, J.; Steidl, G. Stochastic Normalizing Flows for Inverse Problems: A Markov Chains Viewpoint.
SIAM/ASA Journal on Uncertainty QuantificationVol. 10, Iss. 3 (2022) 10.1137
- Papamakarios, G.; Nalisnick, E.T.; Rezende, D.J.; Mohamed, S.; Lakshminarayanan, B. Normalizing Flows for Probabilistic
Modeling and Inference. Journal of Machine Learning Research 22 (2021) 1-64