  Vector retrieval algorithms are vital for semantic queries in the evolving
landscape of Large Language Models (LLMs). Retrieving vectors that
simultaneously meet criteria for both similarity and diversity significantly
enhances the capabilities of LLM-based agents. Despite the widespread use of
the Maximal Marginal Relevance (MMR) in retrieval scenarios with relevance and
diversity requirements, fluctuations caused by variations in the parameter $
\lambda $ within the MMR complicate the determination of the optimization
trajectory in vector spaces, thus obscuring the direction of enhancement.
Moreover, there is a lack of a robust theoretical analysis for the constraints
of similarity and diversity in retrieval processes. This paper introduces a
novel approach to characterizing both constraints through the relationship
between the sum vector and the query vector. The proximity of these vectors
addresses the similarity constraint, while necessitating that individual
vectors within the sum vector divergently align with the query vector to
satisfy the diversity constraint. We also formulate a new combinatorial
optimization challenge, taking a selection of $k$ vectors from a set of
candidates such that their sum vector maximally aligns with the query vector, a
problem we demonstrate to be NP-complete. This establishes the profound
difficulty of pursuing similarity and diversity simultaneously in vector
retrieval and lays a theoretical groundwork for further research. Additionally,
we present the heuristic algorithm Vectors Retrieval with Similarity and
Diversity (VRSD) which not only has a definitive optimization goal and eschews
the need for preset parameters but also offers a modest reduction in time
complexity compared to MMR. Empirical validation further confirm that VRSD
significantly surpasses MMR across various datasets.
