A Robust Partial p-Wasserstein-Based Metric for Comparing Distributions

Core Concepts

A new family of distances called (p, k)-RPW that is robust to outlier noise and sampling discrepancies while retaining the sensitivity of p-Wasserstein distance to minor geometric differences between distributions.

Abstract

The content introduces a new family of distances called (p, k)-RPW that combines the total variation distance and the p-Wasserstein distance to achieve robustness to outlier noise and sampling discrepancies while retaining the sensitivity of p-Wasserstein distance to minor geometric differences between distributions. Key highlights: The (p, k)-RPW distance satisfies the metric properties, including the triangle inequality, unlike the (1-δ)-partial p-Wasserstein distance. The (p, k)-RPW distance is robust to outlier noise - an outlier mass of δ can only change the distance by at most ±δ, unlike the p-Wasserstein distance which can be significantly impacted. The empirical (p, 1)-RPW distance converges to the true distance at a rate of Õ(n^(-p/(4p-2))), which is significantly faster than the convergence rate of Õ(n^(-1/2p)) for the p-Wasserstein distance. The (p, k)-RPW distance interpolates between the total variation distance and the p-Wasserstein distance, and can be reduced to other well-known distances like the Lévy-Prokhorov distance. Experiments on image retrieval tasks show that the (p, k)-RPW distance outperforms the 1-Wasserstein, 2-Wasserstein, and total variation distances on noisy real-world datasets.

Stats

The mass of a distribution µ is denoted by M(µ). The p-Wasserstein distance between distributions µ and ν is denoted by Wp(µ, ν). The (1-ε)-partial p-Wasserstein distance between µ and ν is denoted by Wp,1-ε(µ, ν).

Quotes

"The 2-Wasserstein distance is sensitive to minor geometric differences between distributions, making it a very powerful dissimilarity metric. However, due to this sensitivity, a small outlier mass can also cause a significant increase in the 2-Wasserstein distance between two similar distributions." "For p ≥ 2, outliers can disproportionately increase the distance between distributions." "In 2d, the convergence rate of the empirical p-Wasserstein distance to the true distance drops to n^(-1/2p) and for p = ∞, the empirical distance does not even converge to the real one."

Key Insights Distilled From

A New Robust Partial $p$-Wasserstein-Based Metric for Comparing Distributions

by Sharath Ragh... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.03664.pdf

Deeper Inquiries

How can the (p, k)-RPW distance be incorporated as a loss function in training generative models to make them more robust to noise and outliers

To incorporate the (p, k)-RPW distance as a loss function in training generative models for increased robustness to noise and outliers, we can follow these steps: Loss Function Formulation: Modify the loss function of the generative model to include the (p, k)-RPW distance between the generated samples and the true data distribution. This can be achieved by adding a regularization term based on the (p, k)-RPW distance to the existing loss function. Regularization Strength: Adjust the regularization strength parameter (k) to control the trade-off between the sensitivity to geometric differences and robustness to noise and outliers. A higher value of k will prioritize robustness, while a lower value will emphasize sensitivity. Training Procedure: During training, optimize the generative model parameters to minimize the combined loss function, which now includes the (p, k)-RPW distance term. This will encourage the model to generate samples that are not only close to the true distribution but also resilient to noise and outliers. Evaluation and Fine-Tuning: Evaluate the performance of the generative model using metrics that reflect its robustness to noise and outliers. Fine-tune the model and the regularization parameter based on the evaluation results to achieve the desired balance between accuracy and robustness. By incorporating the (p, k)-RPW distance as a loss function, generative models can learn to generate samples that are not only faithful to the true distribution but also more resilient to perturbations, making them more suitable for real-world applications where data may contain noise and outliers.

What are the theoretical guarantees and practical implications of using the (p, k)-RPW distance for clustering and barycenter computation tasks compared to the p-Wasserstein distance

Theoretical guarantees and practical implications of using the (p, k)-RPW distance for clustering and barycenter computation tasks compared to the p-Wasserstein distance are as follows: Theoretical Guarantees: Metric Properties: The (p, k)-RPW distance satisfies metric properties, ensuring a valid distance metric for clustering tasks. Robustness: The (p, k)-RPW distance is robust to small outlier masses and sampling discrepancies, providing more reliable clustering results in the presence of noise. Convergence Rate: The empirical (p, k)-RPW distance converges faster to the true distance compared to the p-Wasserstein distance, leading to more accurate clustering results with fewer samples. Practical Implications: Improved Robustness: The (p, k)-RPW distance can handle noisy data and outliers better than the p-Wasserstein distance, resulting in more robust clustering outcomes. Efficient Computation: The faster convergence rate of the (p, k)-RPW distance allows for quicker and more accurate computation of barycenters, enhancing the efficiency of clustering algorithms. Balanced Sensitivity: By adjusting the parameters p and k, the (p, k)-RPW distance can strike a balance between sensitivity to geometric differences and robustness to noise, providing more flexibility in clustering tasks. Overall, the (p, k)-RPW distance offers both theoretical guarantees and practical benefits that make it a promising metric for clustering and barycenter computation tasks, outperforming the traditional p-Wasserstein distance in scenarios with noisy data and outliers.

Can the ideas behind the (p, k)-RPW distance be extended to other optimal transport-based metrics beyond the Wasserstein distance to achieve similar robustness properties

The ideas behind the (p, k)-RPW distance can be extended to other optimal transport-based metrics beyond the Wasserstein distance to achieve similar robustness properties in various applications. Here are some ways to extend these concepts: Generalized Optimal Transport Metrics: Develop generalized versions of the (p, k)-RPW distance for different optimal transport metrics, such as Gromov-Wasserstein distance or Sinkhorn divergence. This extension can enhance the robustness of these metrics to noise and outliers. Domain-Specific Applications: Apply the principles of the (p, k)-RPW distance to domain-specific optimal transport problems, such as image registration, shape matching, or time series analysis. By customizing the parameters p and k, tailored robustness can be achieved for specific applications. Hybrid Metrics: Explore hybrid metrics that combine the (p, k)-RPW distance with other dissimilarity measures, such as Mahalanobis distance or Jensen-Shannon divergence. This fusion can lead to novel metrics with enhanced robustness properties in diverse scenarios. Scalability and Efficiency: Develop scalable algorithms and optimization techniques to compute the extended optimal transport metrics efficiently, especially for large-scale datasets. This will enable the practical implementation of these metrics in real-world applications. By extending the concepts of the (p, k)-RPW distance to other optimal transport-based metrics, researchers can create a versatile framework for robust and reliable distance computations in various fields, opening up new possibilities for applications in machine learning, computer vision, and beyond.