Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
133 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Polyhedral Complex Derivation from Piecewise Trilinear Networks (2402.10403v3)

Published 16 Feb 2024 in cs.LG, cs.AI, cs.CV, and cs.GR

Abstract: Recent advancements in visualizing deep neural networks provide insights into their structures and mesh extraction from Continuous Piecewise Affine (CPWA) functions. Meanwhile, developments in neural surface representation learning incorporate non-linear positional encoding, addressing issues like spectral bias; however, this poses challenges in applying mesh extraction techniques based on CPWA functions. Focusing on trilinear interpolating methods as positional encoding, we present theoretical insights and an analytical mesh extraction, showing the transformation of hypersurfaces to flat planes within the trilinear region under the eikonal constraint. Moreover, we introduce a method for approximating intersecting points among three hypersurfaces contributing to broader applications. We empirically validate correctness and parsimony through chamfer distance and efficiency, and angular distance, while examining the correlation between the eikonal loss and the planarity of the hypersurfaces.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (31)
  1. Berzins, A. Polyhedral complex extraction from relu networks using edge subdivision. In International Conference on Machine Learning. PMLR, 2023.
  2. Bruns, H. Das eikonal, volume 21. S. Hirzel, 1895.
  3. TensoRF: Tensorial radiance fields. In European Conference on Computer Vision, pp.  333–350. Springer, 2022.
  4. A volumetric method for building complex models from range images. In Proceedings of the 23rd annual conference on Computer graphics and interactive techniques, pp.  303–312, 1996.
  5. An efficient method of triangulating equi-valued surfaces by using tetrahedral cells. IEICE TRANSACTIONS on Information and Systems, 74(1):214–224, 1991.
  6. Polynomial roots from companion matrix eigenvalues. Mathematics of Computation, 64(210):763–776, 1995.
  7. Friedrich, T. et al. Elementary geometry, volume 43. American Mathematical Soc., 2008.
  8. On transversality of bent hyperplane arrangements and the topological expressiveness of relu neural networks. SIAM Journal on Applied Algebra and Geometry, 6(2):216–242, 2022.
  9. Implicit geometric regularization for learning shapes. arXiv preprint arXiv:2002.10099, 2020.
  10. Complexity of linear regions in deep networks. In International Conference on Machine Learning, pp.  2596–2604. PMLR, 2019.
  11. Mesh optimization. In Proceedings of the 20th annual conference on Computer graphics and interactive techniques, pp.  19–26, 1993.
  12. Splinecam: Exact visualization and characterization of deep network geometry and decision boundaries. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.  3789–3798, 2023.
  13. Tropical algebraic geometry, volume 35. Springer Science & Business Media, 2009.
  14. Katz, E. What is… tropical geometry. Notices of the AMS, 64(4), 2017.
  15. NSML: Meet the MLAAS platform with a real-world case study. arXiv preprint arXiv:1810.09957, 2018.
  16. Analytic marching: An analytic meshing solution from deep implicit surface networks. In International Conference on Machine Learning, pp.  5789–5798. PMLR, 2020.
  17. Learning and meshing from deep implicit surface networks using an efficient implementation of analytic marching. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(12):10068–10086, 2021.
  18. Neuralangelo: High-fidelity neural surface reconstruction. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.  8456–8465, 2023.
  19. Marching cubes: A high resolution 3d surface construction algorithm. ACM SIGGRAPH Computer Graphics, 21(4):163–169, 1987.
  20. Introduction to tropical geometry, volume 161. American Mathematical Society, 2021.
  21. Nerf: Representing scenes as neural radiance fields for view synthesis. Communications of the ACM, 65(1):99–106, 2021.
  22. Instant neural graphics primitives with a multiresolution hash encoding. ACM Transactions on Graphics (ToG), 41(4):1–15, 2022.
  23. On the expressive power of deep neural networks. In international conference on machine learning, pp.  2847–2854. PMLR, 2017.
  24. Bounding and counting linear regions of deep neural networks. In International Conference on Machine Learning, pp.  4558–4566. PMLR, 2018.
  25. Deep marching tetrahedra: a hybrid representation for high-resolution 3d shape synthesis. Advances in Neural Information Processing Systems, 34:6087–6101, 2021.
  26. Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems, 33:7537–7547, 2020.
  27. Attention is all you need. Advances in neural information processing systems, 30, 2017.
  28. Neus: Learning neural implicit surfaces by volume rendering for multi-view reconstruction. arXiv preprint arXiv:2106.10689, 2021.
  29. Bakedsdf: Meshing neural sdfs for real-time view synthesis. arXiv preprint arXiv:2302.14859, 2023.
  30. Tropical geometry of deep neural networks. In International Conference on Machine Learning, pp.  5824–5832. PMLR, 2018.
  31. Meshingnet: A new mesh generation method based on deep learning. In International Conference on Computational Science, pp.  186–198. Springer, 2020.

Summary

  • The paper introduces a novel computational method to derive polyhedral complexes from piecewise trilinear networks under the eikonal constraint.
  • It employs trilinear interpolation for positional encoding, significantly improving mesh extraction precision and reducing vertex complexity.
  • Empirical assessments using benchmarks like the Stanford bunny demonstrate enhanced efficiency over traditional methods such as marching cubes.

Insights into Piecewise Trilinear Networks and Polyhedral Complex Derivation

Introduction

The visualization of deep neural networks has significantly advanced our understanding of their geometric and functional intricacies. Among various techniques, mesh extraction from neural implicit surfaces has emerged as a potent tool for visualizing and interpreting the geometry underlying these networks. Traditionally, mesh extraction from Continuous Piecewise Affine (CPWA) functions, notably employed in ReLU-based networks, facilitates the delineation of decision boundaries via convex polyhedral complexes. Recent endeavors to mitigate spectral bias through non-linear positional encoding introduced challenges in applying traditional mesh extraction methodologies, thus compelling an exploration into trilinear interpolating methods for positional encoding.

Polyhedral Complex Derivation in Trilinear Spaces

By leveraging trilinear interpolations for positional encoding, this research provides theoretical underpinnings and a practical methodology for mesh extraction from trilinear networks. Through an analytical approach, it demonstrates the transformation of hypersurfaces to planes within trilinear regions under the eikonal constraint, elucidating the geometrical implications of employing trilinear interpolations in neural networks.

A novel method introduced to approximate intersecting points among three hypersurfaces significantly enhances the precision and feasibility of mesh extraction in trilinear spaces. The work extensively validates the methodology's accuracy and simplicity through empirical evaluations, utilizing metrics such as chamfer distance, efficiency, angular distance, and the correlation between eikonal loss and hypersurface planarity.

Theoretical Contributions and Practical Methodology

The transformation of hypersurfaces to planes within trilinear regions underpins the eikonal constraint, showcasing a pivotal theoretical advancement in understanding neural surface representations. This paper proposes an inventive strategy for approximating the determination of intersecting points among three hypersurfaces, paving the way for new applications and research directions.

In practice, the research outlines an algorithm for the derivation of polyhedral complexes from piecewise trilinear networks, encompassing initialization, optimization of sign-vectors, and delineation of piecewise trilinear regions. The methodology asserts itself as both robust and efficient, with a pronounced potential for further optimization, particularly in the extraction of faces and computation of normal vectors.

Empirical Validation and Insights

Empirical assessments underscore the methodology's capability to produce nearly optimal outcomes with a parsimony of vertices, signifying substantial efficiency over traditional sampling-based methods. This is evident in the experiments involving unit spheres and the Stanford bunny, where our method significantly outperforms conventional approaches like marching cubes in terms of chamfer distance and efficiency.

Furthermore, the research meticulously examines the impact of the eikonal constraint on planarity within trilinear regions, offering empirical evidence that underscores the practical efficacy of incorporating such constraints in enhancing mesh extraction accuracy.

Future Directions and Potential Applications

The implications of this research extend beyond the theoretical exploration into practical applications such as real-time rendering and geometric loss computation using extracted geometry. The findings not only contribute to advancing our comprehension of neural network visualization but also open avenues for developing sophisticated visualization tools for deeper and more intricate neural networks.

Looking ahead, this work lays the groundwork for future investigations into optimizing the extraction process, exploring different positional encoding schemes, and expanding the applications of mesh extraction techniques within the domain of neural network visualization and interpretation.

Acknowledgements and Ethical Considerations

This collaborative effort underscores the significance of collective expertise in advancing machine learning research. Utilizing platforms like NAVER Smart Machine Learning (NSML) for experiments, the work adheres to ethical standards, focusing on theoretical and practical advancements within the machine learning community.

In conclusion, this paper marks a significant stride in mesh extraction from neural implicit surface networks, broadening the horizons for visualizing and understanding the complex geometries of deep neural networks through piecewise trilinear networks and the derived polyhedral complexes.

File Document Download Save Streamline Icon: https://streamlinehq.com PDF File Document Download Save Streamline Icon: https://streamlinehq.com Markdown
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets