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Non-Euclidean Sliced Optimal Transport Sampling
Graphics · Submitted 26 Feb 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim These applications collectively underscore the efficacy of our methodology.
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The abstract, sentence by sentence
In machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well-dispersed collection of samples. Providing a formal metric for measuring the distance between probability measures on general spaces, Optimal Transport (OT) emerges as a pivotal theoretical framework within this context. However, the associated computational burden is prohibitive in most real-world scenarios. Leveraging the simple structure of OT in 1D, Sliced Optimal Transport (SOT) has appeared as an efficient alternative to generate samples in Euclidean spaces. This paper pushes the boundaries of SOT utilization in computational geometry problems by extending its application to sample densities residing on more diverse mathematical domains, including the spherical space Sd , the hyperbolic plane Hd , and the real projective plane Pd . Moreover, it ensures the quality of these samples by achieving a blue noise characteristic, regardless of the dimensionality involved. The robustness of our approach is highlighted through its application to various geometry processing tasks, such as the intrinsic blue noise sampling of meshes, as well as the sampling of directions and rotations. These applications collectively underscore the efficacy of our methodology.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well-dispersed collection of samples. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s1 | Providing a formal metric for measuring the distance between probability measures on general spaces, Optimal Transport (OT) emerges as a pivotal theoretical framework within this context. | background 1.00 | 0.04 | 0.03 | asserted 1.00 |
| s2 | However, the associated computational burden is prohibitive in most real-world scenarios. | background 0.60 | 0.06 | 0.02 | asserted 1.00 |
| s3 | Leveraging the simple structure of OT in 1D, Sliced Optimal Transport (SOT) has appeared as an efficient alternative to generate samples in Euclidean spaces. | background 1.00 | 0.05 | 0.04 | asserted 0.99 |
| s4 | This paper pushes the boundaries of SOT utilization in computational geometry problems by extending its application to sample densities residing on more diverse mathematical domains, including the spherical space Sd , the hyperbolic plane Hd , and the real projective plane Pd . | contribution 1.00 | 0.87 | 0.16 | asserted 1.00 |
| s5 | Moreover, it ensures the quality of these samples by achieving a blue noise characteristic, regardless of the dimensionality involved. | result 0.71 | 0.84 | 0.24 | asserted 1.00 |
| s6 | The robustness of our approach is highlighted through its application to various geometry processing tasks, such as the intrinsic blue noise sampling of meshes, as well as the sampling of directions and rotations. | result 0.49 | 0.87 | 0.17 | asserted 1.00 |
| s7 | These applications collectively underscore the efficacy of our methodology. | interpretation 0.96 | 0.60 | 0.44 | asserted 0.97 |
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The paper
Original abstract
In machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well-dispersed collection of samples. Providing a formal metric for measuring the distance between probability measures on general spaces, Optimal Transport (OT) emerges as a pivotal theoretical framework within this context. However, the associated computational burden is prohibitive in most real-world scenarios. Leveraging the simple structure of OT in 1D, Sliced Optimal Transport (SOT) has appeared as an efficient alternative to generate samples in Euclidean spaces. This paper pushes the boundaries of SOT utilization in computational geometry problems by extending its application to sample densities residing on more diverse mathematical domains, including the spherical space Sd , the hyperbolic plane Hd , and the real projective plane Pd . Moreover, it ensures the quality of these samples by achieving a blue noise characteristic, regardless of the dimensionality involved. The robustness of our approach is highlighted through its application to various geometry processing tasks, such as the intrinsic blue noise sampling of meshes, as well as the sampling of directions and rotations. These applications collectively underscore the efficacy of our methodology.
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How the literature uses it
1 arXiv paper cite this in 1 sentence
What it cites · 11 sentences without a confident label
Uses the method or tool of 2 papers
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These key ingredients are mostly borrowed from Bonet et al. [ref] dedicated to the computation of $SW$ on $^d$ and $^d$.
Spherical Sliced-Wasserstein · Sliced optimal transport sampling on constant curvature manifolds · 0.97
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These key ingredients are mostly borrowed from Bonet et al. [ref] dedicated to the computation of $SW$ on $^d$ and $^d$.
Hyperbolic Sliced-Wasserstein via Geodesic and Horospherical Projections · Sliced optimal transport sampling on constant curvature manifolds · 0.97
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