Quantifying and Expanding the Theoretical Capacity of Late-Interaction Retrieval Models

Abstract: Late-interaction retrieval models that use the MaxSim similarity function have shown strong empirical performance, often outperforming single-vector dense and sparse retrieval models. Despite these empirical findings, little is known about the theoretical representation power of MaxSim and how it compares to other retrieval approaches. This paper shows by construction that MaxSim similarity can exactly replicate the inner product between any two non-negative k-sparse vectors with possibly infinite dimension, requiring only O(k) representation space. Moreover, there exist similarities that MaxSim can express while standard vector inner products with the same representation space cannot. Leveraging our theoretical framework, we introduce Signed MaxSim which allows late-interaction models to exactly replicate any real-valued inner product, something we prove standard MaxSim is not capable of. We also show that MaxSim can act as an aggregation of soft-OR operations and as an evaluator of logical expressions in positive Conjunctive Normal Form. Our findings show that MaxSim is at least as capable as standard vector inner products for any non-negative vectors and our extension, Signed MaxSim, is as capable for any vectors. Both similarities possess additional capabilities that inner product cannot replicate, marking one of the first theoretical justifications and quantifications of late-interaction methods. Our theoretical findings are supported empirically: on a retrieval task featuring queries with negations, Signed MaxSim improves out-of-domain performance significantly over a standard ColBERT/MaxSim baseline with nDCG@10 increasing from 0.597 to 1.000 under a vocabulary shift and from 0.008 to 0.788 on negation-only queries.

• The paper establishes that MaxSim exactly replicates inner products for non-negative k-sparse vectors using O(k) storage with modular independent query and document representations.
• It introduces Signed MaxSim to overcome standard limitations by incorporating sign information, achieving full replication of real-valued vector inner products with significant performance improvements.
• Empirical validations demonstrate that the enhanced model outperforms single-vector methods, particularly in handling rare terms, vocabulary shifts, and complex negation queries.

Quantifying and Expanding the Theoretical Capacity of Late-Interaction Retrieval Models

Introduction and Motivation

The investigation into the theoretical underpinnings of late-interaction neural retrieval models—particularly those employing the MaxSim similarity function—addresses a key gap in understanding why these approaches frequently outperform both dense and learned sparse retrievers that rely purely on single-vector inner products. Previous empirical work has highlighted the inability of single-vector approaches to match late-interaction performance, despite attempts to close the gap via increased embedding dimensionality. This work formalizes the hypothesis that the primary differentiator for late-interaction models is not merely increased representational capacity, but fundamentally greater expressivity of the similarity function itself.

Theoretical Analysis of MaxSim Expressivity

A principal contribution of the paper is the proof that MaxSim similarity can exactly replicate the inner product between any two non-negative $k$-sparse vectors, potentially of infinite ambient dimension, using only $O(k)$ storage. This is achieved via a low-dimensional ($\mathbb{R}^3$) vector construction for each nonzero index, with the aggregation over set elements enabling preservation of the original inner product through a sum of maximized pairwise similarities. Importantly, this construction is strictly modular: the query and document representations can be constructed independently, as required for dual-encoder retrieval architectures.

The authors further establish a theoretical separation between MaxSim-based late-interaction models and single-vector inner-product retrieval paradigms. Specifically, they show that it is impossible to preserve inner products among more than $d$ orthogonal sparse patterns using any $d$-dimensional embedding with a standard inner product. In contrast, MaxSim can represent arbitrary numbers of sparse features by simply scaling the set size, not the embedding dimension. This result formalizes the intuition that late-interaction models can robustly capture rare terms and out-of-vocabulary concepts without compression losses endemic to low-dimensional dense encoding.

Extension to Real-Valued Vector Spaces: Signed MaxSim

While MaxSim subsumes the modeling capacity of non-negative inner product retrieval, a fundamental limitation is exposed: it cannot, in its standard form, compute inner products between general real-valued vectors with positive and negative entries under the same sparsity-preserving constraint. The impossibility is formally proved even with aggressively flexible encoding schemes. To address this, the paper introduces Signed MaxSim, an extension in which each embedding encodes both magnitude and sign, with the sign applied after maximization. This innovation enables exact replication of real-valued vector inner products, thus elevating the expressiveness of late-interaction models to full parity with any inner-product-based retriever.

Logical Expressivity and Connections to Boolean IR

Beyond vector similarity, the analysis connects MaxSim's behavior to soft logical aggregation. The authors formalize how MaxSim operates as a differentiable, aggregation function analogous to a "Soft-OR" across term groups, and they prove that—in the binary-weighted case—it is rank-equivalent to the evaluation of positive Conjunctive Normal Form (CNF) Boolean expressions. Through polynomial embedding techniques, they demonstrate that queries comprising ANDs of ORs (without negation) can be perfectly evaluated in terms of relevance ranking via MaxSim, without manually constructed logical forms. This bridges classic Boolean IR with neural approaches, providing a theoretical foundation for MaxSim’s practical superiority at capturing complex query intent.

Empirical Validation on Synthetic Retrieval Tasks

Experimental results support the theoretical developments. Rigorous evaluation on controlled synthetic datasets designed to test negations and distribution shifts demonstrates substantial performance gains for Signed MaxSim over standard MaxSim (ColBERT baseline), especially for queries requiring exclusion (negation) and for out-of-domain scenarios. For example, under a vocabulary shift, nDCG@10 increases from 0.597 to 1.000, and for negation-only queries, from 0.008 to 0.788. These results confirm that the enhanced expressivity of Signed MaxSim is not merely theoretical, but translates to improved robustness and capability in realistic retrieval settings.

Implications, Limitations, and Future Directions

This work clarifies the representational and functional advantages of late-interaction models, formally separating them from single-vector architectures and elucidating the mathematical justification for their empirical superiority. The results underscore the potential for late-interaction retrieval to support richer forms of logical and semantic reasoning directly within neural similarity computation, pushing beyond what standard inner products can realize—even as embedding dimensionality grows arbitrarily.

Practically, these insights call for greater attention to the inductive biases encoded in similarity functions and set-based representations. In particular, the Signed MaxSim framework is positioned as a superior default for neural IR systems in domains demanding expressive handling of semantic inclusion and exclusion as well as robust generalization to rare and emergent patterns.

Future research should explore the geometric and algorithmic properties of these constructions under end-to-end learning, the feasibility of efficient approximate retrieval under these richer similarity functions, and further connections to logical IR formalisms. Another open direction concerns the limits of MaxSim under relaxed encoding constraints, and the characterization of the full closure of its set-expressive power relative to classical vector spaces.

Conclusion

The theoretical and empirical findings demonstrate that MaxSim-based late-interaction retrieval models encapsulate and exceed the representational power of single-vector inner-product models for non-negative vectors, and with Signed MaxSim, can exactly replicate any real-valued inner product. The logical interpretation of MaxSim establishes strong connections to Boolean retrieval, bolstering explainability and providing formal justification for observed empirical strengths, particularly in out-of-domain and negation-intensive tasks. This work advances the mathematical foundations of neural information retrieval, laying groundwork for further unification of logic-based and neural similarity methods.