Go to TMLR homepagePolyhedral Embeddings and Realizations of Orientable and Non-Orientable Cubical Surfaces using Reinforcement Learning
Abstract: Finding realizations in $\mathbb{R}^3$ of polyhedral maps on compact connected surfaces is considered a hard problem in discrete geometry because of the lack of general solution methods. Heuristic approaches have been proven efficient in finding polyhedral embeddings of orientable vertex-minimal surfaces of genus $g$ by minimizing their intersection length; however, they can still be challenging to implement due to large configuration spaces, and can struggle avoiding local minima.
This article studies closed connected cubical surfaces; surfaces made from a collection of faces of a $5$-dimensional cube. The author proposes a Reinforcement Learning (RL) algorithm to minimize the number of face intersections of orientable and non-orientable cubical surfaces through $5$-dimensional rotations or modifications on the perspective projection distances; yielding immersions that are perspective projections of a unitary $5$-dimensional cube. Polyhedral embeddings of orientable cubical surfaces of genus $g=1,2$ and realizations of the Projective Plane and the Klein Bottle with the smallest possible number of face intersections are obtained. The agent's optimal strategy is visualized using three-dimensional animations.
Submission Length: Long submission (more than 12 pages of main content)
Previous TMLR Submission Url: https://openreview.net/forum?id=aT6VkKtM5U&referrer=%5Bthe%20profile%20of%20Manuel%20Est%C3%A9vez%5D(%2Fprofile%3Fid%3D~Manuel_Est%C3%A9vez1)
Changes Since Last Submission: Dear Reviewer.
Thanks for the observations and recommendations, I think they have substantially helped on the improvement of the article. Here a small summary of the modifications in this new version based on the previous requested changes.
1. Most of the preliminaries are now in the Appendix, except the first part defining cubical complexes and surfaces. I still think the section is relevant to define which kind of cubical complexes build a cubical surface and why less than 3 intersections (for a non-orientable) is not achievable. The background Section had substantial changes. First, the background explains the face-intersection minimisation process without having to carry all the notation defined in the RL-formulation. This helps the reader understand the geometric part of the problem without having to go through the implementation first. Second, I added an animation sequence of a cubical projective plane , this time specifying which action is being applied (in renders). Finally, I explain the relevance of the realisations found by the RL agent, in particular orientable surfaces with g=1,2,5, and the non-orientable Projective Plane and Klein Bottle. Here I focused on some well known models that have been studied for continuous closed surfaces and how do they relate with our quadrilateral counterparts.
2. As you noted, one can’t guarantee minimality for the RL method for many cases. But for the orientable with 0 face intersections and the non-orientable with 3 the realisations found are minimal in terms of face intersections. A proof is presented in Appendix A1 for non orientable realisations, and in Table 2 the “Minimal” column shows which of the realisations satisfy minimality.
3. The new models and renders added in Background Section, together with their importance and comparison with known realisations enhance the discussion.
4. I added the optimisation sequence of the projective plane with the corresponding 5-d rotations or camera modifications, I hope this helps to visualise the geometry of the problem. Also, the models and animations whose links can be found in supplements can be of great help understanding how these linear transformations modify the object in 3-d.
5. I replaced some python visualisations with renders, however there is one that I would love to keep. I intend that the sequence in page 13 focuses on how the intersection lines are disappearing. I tried some examples of this process in renders but modifying the lines themself is a little bit challenging and it is very difficult to understand. But with this matplotlib visualisation more information can be shown without turning confusing, the blue intersection lines are easy to distinguish, the 1-skeleton of the cube in gray is too and in red the contour of the surface. This is the reason I insisted on keeping this figure, it gives a lot of information in a neat way.
6. I think the reference to the realisation by Ziegler (page 6) can be a good comparison of this method with the known realisations. But, to adapt the vertex-wise methodology from Hougardy et.al would be a deep change in the RL algorithm here presented and a complete reformulation of the paper; in fact, as stated in further work I think this is the direction this work should take in the future.
7, 8. I find the RL-formulation diagram very useful to understand the stages of the minimisation process, however in the previous version it did work because it was carrying more notation and explaining technicalities that could also be consulted in the pseudo-code. So in this version it keeps only the important information to understand the formulation and the colour code helps separating by “kind of action taken” and in which block is the RL agent working. Also the py images were replaced for renders.
I want to thank again for your carefull review, and hope this modifications enhance the understanding of the paper.
Best wishes
Assigned Action Editor: ~Florian_Shkurti1
Submission Number: 5341
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