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Parallelised Differentiable Straightest Geodesics for 3D Meshes

Hippolyte Verninas, Caner Korkmaz, Stefanos Zafeiriou, Tolga Birdal, Simone Foti

Abstract

Machine learning has been progressively generalised to operate within non-Euclidean domains, but geometrically accurate methods for learning on surfaces are still falling behind. The lack of closed-form Riemannian operators, the non-differentiability of their discrete counterparts, and poor parallelisation capabilities have been the main obstacles to the development of the field on meshes. A principled framework to compute the exponential map on Riemannian surfaces discretised as meshes is straightest geodesics, which also allows to trace geodesics and parallel-transport vectors as a by-product. We provide a parallel GPU implementation and derive two different methods for differentiating through the straightest geodesics, one leveraging an extrinsic proxy function and one based upon a geodesic finite differences scheme. After proving our parallelisation performance and accuracy, we demonstrate how our differentiable exponential map can improve learning and optimisation pipelines on general geometries. In particular, to showcase the versatility of our method, we propose a new geodesic convolutional layer, a new flow matching method for learning on meshes, and a second-order optimiser that we apply to centroidal Voronoi tessellation. Our code, models, and pip-installable library (digeo) are available at: circle-group.github.io/research/DSG.

Parallelised Differentiable Straightest Geodesics for 3D Meshes

Abstract

Machine learning has been progressively generalised to operate within non-Euclidean domains, but geometrically accurate methods for learning on surfaces are still falling behind. The lack of closed-form Riemannian operators, the non-differentiability of their discrete counterparts, and poor parallelisation capabilities have been the main obstacles to the development of the field on meshes. A principled framework to compute the exponential map on Riemannian surfaces discretised as meshes is straightest geodesics, which also allows to trace geodesics and parallel-transport vectors as a by-product. We provide a parallel GPU implementation and derive two different methods for differentiating through the straightest geodesics, one leveraging an extrinsic proxy function and one based upon a geodesic finite differences scheme. After proving our parallelisation performance and accuracy, we demonstrate how our differentiable exponential map can improve learning and optimisation pipelines on general geometries. In particular, to showcase the versatility of our method, we propose a new geodesic convolutional layer, a new flow matching method for learning on meshes, and a second-order optimiser that we apply to centroidal Voronoi tessellation. Our code, models, and pip-installable library (digeo) are available at: circle-group.github.io/research/DSG.
Paper Structure (48 sections, 28 equations, 26 figures, 3 tables, 9 algorithms)

This paper contains 48 sections, 28 equations, 26 figures, 3 tables, 9 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Riemannian Geometry
  4. Differentiable Straightest Geodesics
  5. Straightest geodesics
  6. Differentiation
  7. Differentiation via Extrinsic Proxy (EP)
  8. Differentiation via Geodesic Finite Differences (GFD)
  9. Parallelisation on the GPU
  10. Experimental Evaluation & Applications
  11. Accuracy & Performance
  12. Adaptive Geodesic Convolutions (AGC)
  13. Problem & Setup
  14. Approach
  15. Results
  16. ...and 33 more sections

Figures (26)

  • Figure 1: Our GPU-parallelised schemes to differentiate the Exponential map and improve learning and optimisation on meshes: the Extrinsic Proxy (EP) and Geodesic Finite Differences (GFD).
  • Figure 2: Left $\theta_l$ and right $\theta_r$ angles on a vertex, edge, and face.
  • Figure 3: Median runtime comparisons with different batch sizes (left) and face counts (right). Medians are computed over 5 consecutive executions (as well as 5 different meshes for the left). While variance is negligible with face counts, $25\%$ and $75\%$ intervals are reported for batch sizes. The time axis was split in right to highlight the massive performance gap with PI.
  • Figure 4: Differentiation correctness against closed-form.
  • Figure 5: AGC (ours) on body parts segmentation.
  • ...and 21 more figures