# Approximate Hausdorff Distance for Multi-Vector Databases

arXiv: https://arxiv.org/abs/2503.06833 · HTML: https://mlsanity.com/paper/2503.06833/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Mar 2025.

## Built on by (most cited first, 1 of 1)

- [ProHD: Projection-Based Hausdorff Distance Approximation](https://mlsanity.com/paper/2511.18207/) (2025)

## Abstract (arXiv)

The Hausdorff distance is a fundamental measure for comparing sets of vectors, widely used in database theory and geometric algorithms. However, its exact computation is computationally expensive, often making it impractical for large-scale applications such as multi-vector databases. In this paper, we introduce an approximation framework that efficiently estimates the Hausdorff distance while maintaining rigorous error bounds. Our approach leverages approximate nearest-neighbor (ANN) search to construct a surrogate function that preserves essential geometric properties while significantly reducing computational complexity. We provide a formal analysis of approximation accuracy, deriving both worst-case and expected error bounds. Additionally, we establish theoretical guarantees on the stability of our method under transformations, including translation, rotation, and scaling, and quantify the impact of non-uniform scaling on approximation quality. This work provides a principled foundation for integrating Hausdorff distance approximations into large-scale data retrieval and similarity search applications, ensuring both computational efficiency and theoretical correctness.