Approximate Algorithms for k-Sparse Wasserstein Barycenter with Outliers

Core Concepts

The authors propose approximate algorithms for the k-sparse Wasserstein barycenter problem in the presence of outliers, which is a more practical setting as real-world data often contains noise.

Abstract

The authors study the k-sparse Wasserstein barycenter (WB) problem with outliers, which is a more practical setting than the vanilla k-sparse WB problem as real-world data often contains noise. They first investigate the relation between k-sparse WB with outliers and the clustering (with outliers) problems. They propose a clustering-based LP method that yields a constant approximation factor for the k-sparse WB with outliers problem. They further utilize the coreset technique to achieve a (1 + ε)-approximation factor for any ε > 0 if the dimensionality is not high. The authors conduct experiments to illustrate the efficiency of their proposed algorithms in practice.
The key contributions are:

Establishing the relation between k-sparse WB with outliers and clustering (with outliers) problems, and developing new insights to handle the influence of outliers.
Proposing a clustering-based LP algorithm that achieves a constant approximation factor for k-sparse WB with outliers.
Developing a more sophisticated algorithm using coresets that achieves a (1 + ε)-approximation factor for any ε > 0 in low-dimensional spaces.
Conducting experiments on synthetic and real-world datasets to demonstrate the practical performance of the proposed algorithms.

Stats

The authors use the following key metrics in their analysis:

The total weight of the input point sets P1, P2, ..., Pm, which is n.
The number of outliers z, which is a fraction of the total weight n.
The support size k of the k-sparse Wasserstein barycenter.
The dimensionality d of the input point sets.
The number of input point sets m.

Quotes

"Wasserstein Barycenter (WB) is one of the most fundamental optimization problems in optimal transportation. Given a set of distributions, the goal of WB is to find a new distribution that minimizes the average Wasserstein distance to them."
"The problem becomes even harder if we restrict the solution to be 'k-sparse'. In this paper, we study the k-sparse WB problem in the presence of outliers, which is a more practical setting since real-world data often contains noise."

Key Insights Distilled From

Approximate Algorithms For $k$-Sparse Wasserstein Barycenter With Outliers

by Qingyuan Yan... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2404.13401.pdf

Deeper Inquiries

How can the proposed algorithms be extended to handle the case where both the input distributions and the barycenter can have outliers

To extend the proposed algorithms to handle cases where both the input distributions and the barycenter can have outliers, we can modify the existing algorithms to incorporate the presence of outliers in both the input distributions and the barycenter. This extension would involve adjusting the distance calculations and optimization criteria to account for outliers in both sets of points. Specifically, we would need to consider the outliers in the computation of the Wasserstein distance and the determination of the optimal barycenter. By including outliers in both the input distributions and the barycenter, the algorithms would need to adapt to find a solution that minimizes the average Wasserstein distance while accounting for the presence of outliers in both sets of points.

What are the potential applications of the k-sparse Wasserstein barycenter with outliers in real-world problems, and how can the algorithms be further tailored to those applications

The k-sparse Wasserstein barycenter with outliers has several potential applications in real-world problems. One application could be in image processing, where the algorithm could be used to find a representative image that captures the essential features of a set of images, even in the presence of outliers or noise. This could be valuable in tasks such as image retrieval, object recognition, and image clustering. Additionally, the algorithms could be applied in data analysis tasks where finding a sparse representation of the data is important, such as in medical imaging, financial data analysis, or natural language processing. To tailor the algorithms to these applications, we could incorporate domain-specific constraints or features into the optimization process, ensuring that the resulting barycenter is meaningful and useful for the specific problem at hand.

Is it possible to develop efficient algorithms for the k-sparse Wasserstein barycenter with outliers problem in high-dimensional spaces, where the coreset-based approach may not be as effective

Developing efficient algorithms for the k-sparse Wasserstein barycenter with outliers problem in high-dimensional spaces presents a significant challenge due to the increased complexity and computational requirements in higher dimensions. In high-dimensional spaces, the coreset-based approach may not be as effective due to the curse of dimensionality and the increased computational cost of processing high-dimensional data. To address this challenge, alternative techniques such as dimensionality reduction, sparse sampling, or advanced optimization methods tailored for high-dimensional spaces could be explored. Additionally, leveraging parallel computing, distributed algorithms, or approximation methods specifically designed for high-dimensional data could help improve the efficiency of the algorithms in handling the k-sparse Wasserstein barycenter with outliers problem in high-dimensional spaces.