Sum Estimation via Vector Similarity Search

2026-01-16

AI Summary

This work addresses the challenge of efficiently estimating aggregate quantities—such as kernel density estimates or softmax normalization constants—over large-scale datasets. The authors propose a novel algorithm grounded in vector similarity search, which leverages a hierarchical random assignment scheme with exponentially decaying probabilities, combined with multi-layer approximate nearest neighbor structures and top-k retrieval. This approach achieves unbiased estimation by accessing only $O(\log n)$ most similar vectors and provides high-probability relative error guarantees. Compared to existing methods requiring $O(\sqrt{n})$ operations, the proposed technique substantially reduces computational overhead. Experimental results on OpenImages and Amazon Reviews datasets demonstrate consistently lower estimation errors and superior runtime efficiency across tasks including density estimation, softmax denominator computation, and spherical range counting.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Semantic embeddings to represent objects such as image, text and audio are widely used in machine learning and have spurred the development of vector similarity search methods for retrieving semantically related objects. In this work, we study the sibling task of estimating a sum over all objects in a set, such as the kernel density estimate (KDE) and the normalizing constant for softmax distributions. While existing solutions provably reduce the sum estimation task to acquiring $\mathcal{O}(\sqrt{n})$ most similar vectors, where $n$ is the number of objects, we introduce a novel algorithm that only requires $\mathcal{O}(\log(n))$ most similar vectors. Our approach randomly assigns objects to levels with exponentially-decaying probabilities and constructs a vector similarity search data structure for each level. With the top-$k$ objects from each level, we propose an unbiased estimate of the sum and prove a high-probability relative error bound. We run experiments on OpenImages and Amazon Reviews with a vector similar search implementation to show that our method can achieve lower error using less computational time than existing reductions. We show results on applications in estimating densities, computing softmax denominators, and counting the number of vectors within a ball.

Problem

• sum estimation
• vector similarity search
• kernel density estimation
• softmax normalization
• semantic embeddings

Innovation

• vector similarity search
• sum estimation
• kernel density estimation
• softmax normalization
• logarithmic complexity