  Ranking is a fundamental and popular problem in search. However, existing
ranking algorithms usually restrict the granularity of ranking to full passages
or require a specific dense index for each desired level of granularity. Such
lack of flexibility in granularity negatively affects many applications that
can benefit from more granular ranking, such as sentence-level ranking for
open-domain question-answering, or proposition-level ranking for attribution.
In this work, we introduce the idea of any-granularity ranking, which leverages
multi-vector embeddings to rank at varying levels of granularity while
maintaining encoding at a single (coarser) level of granularity. We propose a
multi-granular contrastive loss for training multi-vector approaches, and
validate its utility with both sentences and propositions as ranking units.
Finally, we demonstrate the application of proposition-level ranking to
post-hoc citation addition in retrieval-augmented generation, surpassing the
performance of prompt-driven citation generation.
