Generalized Sliced Wasserstein Distances (1902.00434v1)

Abstract: The Wasserstein distance and its variations, e.g., the sliced-Wasserstein (SW) distance, have recently drawn attention from the machine learning community. The SW distance, specifically, was shown to have similar properties to the Wasserstein distance, while being much simpler to compute, and is therefore used in various applications including generative modeling and general supervised/unsupervised learning. In this paper, we first clarify the mathematical connection between the SW distance and the Radon transform. We then utilize the generalized Radon transform to define a new family of distances for probability measures, which we call generalized sliced-Wasserstein (GSW) distances. We also show that, similar to the SW distance, the GSW distance can be extended to a maximum GSW (max-GSW) distance. We then provide the conditions under which GSW and max-GSW distances are indeed distances. Finally, we compare the numerical performance of the proposed distances on several generative modeling tasks, including SW flows and SW auto-encoders.

• The paper extends the traditional sliced-Wasserstein distance by incorporating nonlinear projections via the generalized Radon transform.
• The authors propose the max-GSW variant, which optimizes computational resources by selecting a single maximal projection.
• The methodology is validated through theoretical proofs and performance tests in generative modeling tasks, demonstrating superior numerical efficiency.

Generalized Sliced Wasserstein Distances: A Novel Approach to Probability Metrics

The paper, "Generalized Sliced Wasserstein Distances," presents an intriguing extension of the sliced-Wasserstein (SW) distance. The work originates from the authors' exploration into the limitations of the conventional Wasserstein distance, particularly its computational inefficiency when applied to high-dimensional data, and its existing variant, the sliced-Wasserstein distance, which attempts to sidestep these limitations by utilizing one-dimensional projections. The authors leverage the Radon transform's mathematics to propose the generalized sliced-Wasserstein (GSW) distance and extend this to maximal GSW (max-GSW) distance, offering a potentially superior computational and representative framework for probability measures.

Core Contributions

The paper lays out several core contributions:

1. Extension of SW to GSW: Utilizing the generalized Radon transform, the authors define a new distance metric, termed the generalized sliced-Wasserstein distance. They construct this by replacing the linear projections in the SW distance with nonlinear ones (e.g., polynomial projections), thus broadening the scope and utility of the slicing concept.
2. Introduction of the max-GSW Distance: The authors propose a maximal version of the GSW distance that requires only a single projection that maximizes the distance in the projected space, thus promising to significantly reduce computational resource requirements traditionally tied to numerous projection operations.
3. Theoretical Validation as Metrics: The paper outlines conditions under which the GSW and max-GSW can be considered legitimate metrics, establishing their theoretical soundness by providing proofs of their metric properties.
4. Computational and Performance Evaluation: Through various generative modeling tasks, such as sliced-Wasserstein flows and auto-encoders, the paper illustrates the numerical superiority of the GSW metrics over both synthetic and real data.

Practical and Theoretical Implications

Practically, the introduction of the GSW and max-GSW distances holds significant promise for emerging machine learning applications, particularly where high-dimensional data is prevalent—fields like computer vision, NLP, and complex data modeling benefit from such computational enhancements. Theoretical implications suggest that by showcasing the utility of nonlinear measures over traditional linear ones, a new class of distances can be applied to optimize specific data model scenarios.

The authors draw attention to computational efficiency improvements brought about by employing fewer projection requirements without sacrificing representation quality—a crucial balancing act for real-world applicability in environments constrained by computational resources.

Speculation on Future Research

The research opens avenues for further investigation into the approximation and optimization strategies used in defining suitable projections for GSW, especially in automated and data-driven settings. Moreover, integrating neural network-driven approaches for dynamically learning appropriate defining functions in the generalized Radon transform could merge effectively with adversarial training techniques. This could advance fields like unsupervised and semi-supervised learning algorithms, potentially rewriting their generative modeling processes.

A deeper integration with neural architectures and real-time applications remains a fascinating aspect of the hypothesis set forth in this work. The potential accessibility and adaptability improvements in processing and analyzing extremely large datasets through refined distance metrics pave the way for exciting developments.

In conclusion, the advancements detailed by the authors substantiate a meaningful contribution to optimal transport metrics, outlining how mathematical complexity and computational efficiency can be harmoniously aligned in statistical measure methodologies. Such efforts support evolutionary leaps in probabilistic modeling and the broader scope of machine learning.