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.

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