Scalable Optimal Transport Distance Estimation with Transformer-based Autoencoders

Konsep Inti

Wasserstein Wormhole, a transformer-based autoencoder, embeds empirical distributions into a latent space where Euclidean distances approximate optimal transport (Wasserstein) distances, enabling scalable analysis of large cohorts of distributions.

Abstrak

The paper presents Wasserstein Wormhole, a transformer-based autoencoder that embeds empirical distributions (point clouds) into a latent space where Euclidean distances approximate optimal transport (Wasserstein) distances. This allows for efficient computation of Wasserstein distances, which is otherwise computationally expensive, especially for large cohorts of distributions.
Key highlights:

Wasserstein distance is a powerful tool for comparing distributions, but computing pairwise Wasserstein distances becomes intractable as cohort size grows.
Wasserstein Wormhole learns an embedding space where Euclidean distances between embeddings match Wasserstein distances between the original point clouds.
The encoder network maps point clouds to embeddings, while the decoder network reconstructs point clouds from embeddings, enabling interpretability.
Theoretical analysis provides bounds on the error incurred when embedding non-Euclidean Wasserstein distances.
Empirical results show that Wasserstein Wormhole outperforms existing methods in matching Wasserstein distances and enables scalable analysis of large and high-dimensional datasets, such as spatial transcriptomics data.

Statistik

"Wormhole follows a long line of neural network approaches for Optimal Transport estimation [31]. Sinkhorn iterations, which are quadratic with size of point-clouds, cannot be applied when When point-clouds are massive consisting of thousands of points."
"Wormhole can be easily applied to any point cloud dataset via a pip installable package"

Kutipan

"Optimal transport (OT) and the related Wasserstein metric (W) are powerful and ubiquitous tools for comparing distributions. However, computing pairwise Wasserstein distances rapidly becomes intractable as cohort size grows."
"By lending scalability and interpretability to OT approaches, Wasserstein Wormhole unlocks new avenues for data analysis in the fields of computational geometry and single-cell biology."

Wawasan Utama Disaring Dari

Wasserstein Wormhole: Scalable Optimal Transport Distance with Transformers

by Doron Haviv,... pada arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.09411.pdf

Pertanyaan yang Lebih Dalam

How can the Wasserstein Wormhole framework be extended to learn the optimal transport plan between distributions, in addition to the distance estimation?

In order to extend the Wasserstein Wormhole framework to learn the optimal transport plan between distributions, we can incorporate additional components into the existing architecture. One approach could involve modifying the decoder network to not only reconstruct the input point clouds but also generate the optimal transport plan between them. This would require the decoder to output a mapping that minimizes the cost of transporting mass from one distribution to another, similar to the Sinkhorn algorithm used in optimal transport calculations. By training the decoder to produce this transport plan along with the reconstructed point clouds, the framework can learn to not only estimate distances but also the optimal mapping between distributions.

What are the potential limitations of the transformer-based architecture used in Wasserstein Wormhole, and how could alternative neural network architectures be incorporated to further improve scalability and performance?

While transformers have shown great success in various tasks, they do have limitations that can impact their performance in certain scenarios. One limitation is the quadratic complexity of transformers, which can hinder scalability when dealing with large datasets or high-dimensional input. Additionally, transformers may struggle with capturing long-range dependencies in the data, which can affect the quality of embeddings.
To address these limitations, alternative neural network architectures could be incorporated into the Wasserstein Wormhole framework. For example, graph neural networks (GNNs) could be used to capture relationships between samples in the point clouds, allowing for more efficient processing of the data. GNNs are well-suited for modeling graph-structured data and can handle variable-sized inputs, making them a good alternative for encoding point clouds. Additionally, convolutional neural networks (CNNs) could be used to extract spatial features from the point clouds, which could complement the transformer-based architecture and improve performance in tasks requiring spatial information.

Given the success of Wasserstein Wormhole in high-dimensional settings, how could the insights from this work be applied to other domains that rely on optimal transport, such as in computational chemistry or numerical geometry?

The insights gained from the Wasserstein Wormhole framework can be applied to other domains that rely on optimal transport, such as computational chemistry or numerical geometry, in several ways. One application could be in the analysis of molecular structures in computational chemistry. By representing molecular structures as point clouds and using Wasserstein Wormhole to embed and compare them, researchers can gain insights into the similarities and differences between molecules based on their spatial distribution of atoms.
In numerical geometry, the framework could be used to compare and analyze geometric shapes or surfaces. By converting geometric shapes into point clouds and applying Wasserstein Wormhole, researchers can quantify the differences between shapes and understand the underlying geometry in a more efficient and scalable manner.
Overall, the principles and techniques developed in Wasserstein Wormhole can be adapted and extended to various domains that rely on optimal transport, providing new tools for analyzing and comparing complex data structures in a wide range of fields.