VRSD: Rethinking Similarity and Diversity for Retrieval in Large Language Models

Vector retrieval algorithms are essential for semantic queries within the rapidly evolving landscape of Large Language Models (LLMs). The ability to retrieve vectors that satisfy both similarity and diversity criteria substantially enhances the performance of LLMs. Although Maximal Marginal Relevance (MMR) is widely employed in retrieval scenarios requiring relevance and diversity, variations in the parameter $λ$ lead to fluctuations that complicate the optimization trajectory in vector spaces. This obscures the direction of improvement and highlights the lack of a robust theoretical analysis regarding similarity and diversity constraints in retrieval processes. To address these challenges, this paper introduces a novel approach that characterizes both constraints through the relationship between the sum vector and the query vector. The proximity of these vectors ensures the similarity constraint, while requiring individual vectors within the sum vector to diverge in their alignment with the query vector satisfies the diversity constraint. We first formulate a new combinatorial optimization problem, selecting k vectors from a candidate set such that their sum vector maximally aligns with the query vector, and demonstrate that this problem is NP-complete. This result underscores the inherent difficulty of simultaneously achieving similarity and diversity in vector retrieval, thereby providing a theoretical foundation for future research. Subsequently, we present the heuristic algorithm Vectors Retrieval with Similarity and Diversity, VRSD, which features a clear optimization objective and eliminates the need for preset parameters. VRSD also achieves a modest reduction in time complexity compared to MMR. Empirical validation confirms that VRSD significantly outperforms MMR across various datasets.