Signed MaxSim: Sign-Sensitive Retrieval

• Signed MaxSim is a late-interaction similarity function that extends MaxSim by incorporating sign information to exactly recover arbitrary signed sparse inner products.
• By attaching a sign to each embedding and applying it after selecting the best document token, the method decouples magnitude from polarity for precise matching.
• Empirical evaluation using the Fallon model shows that Signed MaxSim significantly improves retrieval performance, especially for queries involving negation.

Signed MaxSim is a late-interaction similarity function introduced to extend MaxSim from non-negative matching to sign-sensitive retrieval. In standard MaxSim, a query and a document are represented as sets of embeddings, and each query embedding contributes the inner product of its single best-matching document embedding. The 2026 theoretical treatment shows that this mechanism can exactly reproduce inner products for non-negative sparse vectors, but cannot exactly recover arbitrary signed inner products under a sparsity-preserving encoding regime. Signed MaxSim addresses that limitation by attaching a sign to each embedding and applying the sign only after the maximizing document embedding has been selected. Under this construction, arbitrary sparse real-valued inner products can be recovered exactly while preserving the late-interaction “route-to-best-match” structure (Killingback et al., 7 Jul 2026).

1. Late-interaction setting and the motivation for sign-sensitive scoring

Late-interaction retrieval models such as ColBERT represent a query and a document as sets of token embeddings and score them with MaxSim. In the formulation used in the literature, standard MaxSim is

$$S(\mathcal U,\mathcal V)=\sum_{\mathbf m\in \mathcal U}\max_{\mathbf t\in \mathcal V}\langle \mathbf m,\mathbf t\rangle.$$

This operator is expressive for sparse matching and “match-one-of-these” behavior, because each query token is routed to its best document-side match rather than being compressed into a single global vector. The 2026 capacity analysis shows by construction that MaxSim can exactly replicate the inner product between any two non-negative $k$-sparse vectors with possibly infinite dimension, requiring only $O(k)$ representation space. It also shows that there exist similarities that MaxSim can express while standard vector inner products with the same representation space cannot (Killingback et al., 7 Jul 2026).

The same analysis identifies a limitation that becomes consequential when negation or exclusion is semantically central. The paper’s motivating example is retrieval of documents such as “open-source AI tools that do not mention Google,” where a matched feature should reduce score rather than merely fail to help it. Standard MaxSim is described as unable to natively handle negative contributions in the same exact way that it handles non-negative sparse matching, because the max operator discards sign information too early. Signed MaxSim is proposed as a minimal extension that preserves the late-interaction structure while restoring exact sign-sensitive behavior (Killingback et al., 7 Jul 2026).

2. Formal definition and signed embedding construction

Signed MaxSim is defined on sets of embedding-sign pairs rather than on embeddings alone. For each query pair $(m,s_q)\in U_s$, let the maximizing document pair be

$$(t^*(m), s_d^*(m))=\arg\max_{(t,s_d)\in V_s}\langle m,t\rangle.$$

The Signed MaxSim similarity is then

$$S_{\pm}(U_s,V_s)=\sum_{(m,s_q)\in U_s} s_q\cdot s_d^*(m)\cdot \langle m,t^*(m)\rangle.$$

The defining feature is that the sign of the selected document embedding is multiplied by the query sign only after the maximizing document token has been chosen. This differs from standard MaxSim, where only the magnitude of the best inner product contributes to the final sum (Killingback et al., 7 Jul 2026).

The construction used in the paper separates magnitude from sign. For a real-valued sparse vector $u\in\mathbb R^n$, the query-side set is

$$U_s=\left\{\bigl(|u_i|\phi(i),\operatorname{sgn}(u_i)\bigr)\mid i\in \operatorname{supp}(u)\right\}.$$

For $v\in\mathbb R^n$, the document-side set is

$$V_s=\left\{\bigl(c(i,|v_i|),\operatorname{sgn}(v_i)\bigr)\mid i\in \operatorname{supp}(v)\right\}\cup \{(0,1)\}.$$

The shared baseline pair $k$0 is part of the formal definition. The embedding map is

$k$1

and the polynomial coefficient vector $k$2 is chosen so that a quadratic polynomial evaluates to $k$3 at index $k$4 and to a negative value at all other indices. A concrete construction given in the paper is

$k$5

which yields

$k$6

This arrangement causes the max step to identify the matching coordinate by geometry, after which the stored signs reintroduce the factor $k$7 exactly (Killingback et al., 7 Jul 2026).

3. Expressivity relative to standard MaxSim and inner-product retrieval

The first key theoretical result is that standard MaxSim already exactly recovers non-negative sparse inner products. For non-negative vectors $k$8, where $k$9 is $O(k)$0-sparse and $O(k)$1 is $O(k)$2-sparse, there exist sets $O(k)$3 with $O(k)$4 and $O(k)$5 such that

$O(k)$6

The constructions of $O(k)$7 and $O(k)$8 depend only on $O(k)$9 and $(m,s_q)\in U_s$0, respectively. In the paper’s interpretation, this establishes that MaxSim is at least as powerful as inner-product retrieval for non-negative sparse vectors (Killingback et al., 7 Jul 2026).

The second result is a limitation theorem for standard MaxSim under a contextual sparsity-preserving encoding. If $(m,s_q)\in U_s$1 and $(m,s_q)\in U_s$2 satisfy the sparsity-preserving definition and

$(m,s_q)\in U_s$3

for every pair of one-sparse vectors $(m,s_q)\in U_s$4, then $(m,s_q)\in U_s$5. Consequently, when $(m,s_q)\in U_s$6, some pair of one-sparse vectors cannot have its inner product exactly recovered. The proof is summarized in the paper as a rank argument: exact recovery of signed one-hot inner products would require a diagonal matrix of pairwise similarities of rank $(m,s_q)\in U_s$7, whereas any factorization through $(m,s_q)\in U_s$8 has rank at most $(m,s_q)\in U_s$9. This is the formal reason standard MaxSim cannot exactly simulate arbitrary signed inner products under the stated sparsity-preserving constraints (Killingback et al., 7 Jul 2026).

The central theorem of Signed MaxSim removes that restriction. For arbitrary real-valued sparse vectors $$(t^*(m), s_d^*(m))=\arg\max_{(t,s_d)\in V_s}\langle m,t\rangle.$$0, where $$(t^*(m), s_d^*(m))=\arg\max_{(t,s_d)\in V_s}\langle m,t\rangle.$$1 is $$(t^*(m), s_d^*(m))=\arg\max_{(t,s_d)\in V_s}\langle m,t\rangle.$$2-sparse and $$(t^*(m), s_d^*(m))=\arg\max_{(t,s_d)\in V_s}\langle m,t\rangle.$$3 is $$(t^*(m), s_d^*(m))=\arg\max_{(t,s_d)\in V_s}\langle m,t\rangle.$$4-sparse, there exist sets of pairs $$(t^*(m), s_d^*(m))=\arg\max_{(t,s_d)\in V_s}\langle m,t\rangle.$$5, with $$(t^*(m), s_d^*(m))=\arg\max_{(t,s_d)\in V_s}\langle m,t\rangle.$$6 and $$(t^*(m), s_d^*(m))=\arg\max_{(t,s_d)\in V_s}\langle m,t\rangle.$$7, such that

$$(t^*(m), s_d^*(m))=\arg\max_{(t,s_d)\in V_s}\langle m,t\rangle.$$8

Because the max operator acts on magnitudes while the sign is reintroduced after maximization, the selected coordinate contributes exactly

$$(t^*(m), s_d^*(m))=\arg\max_{(t,s_d)\in V_s}\langle m,t\rangle.$$9

The same paper also proves an “Impossibility of Dimensionality Compression” theorem for standard inner products: for the set of $$S_{\pm}(U_s,V_s)=\sum_{(m,s_q)\in U_s} s_q\cdot s_d^*(m)\cdot \langle m,t^*(m)\rangle.$$0 standard basis vectors in $$S_{\pm}(U_s,V_s)=\sum_{(m,s_q)\in U_s} s_q\cdot s_d^*(m)\cdot \langle m,t^*(m)\rangle.$$1, no mappings into $$S_{\pm}(U_s,V_s)=\sum_{(m,s_q)\in U_s} s_q\cdot s_d^*(m)\cdot \langle m,t^*(m)\rangle.$$2 can preserve all pairwise inner products exactly. Within the paper’s framework, this separates three regimes: finite-dimensional single-vector inner products are bottlenecked by dimension, standard MaxSim exactly covers non-negative sparse inner products and more, and Signed MaxSim extends exact recovery to arbitrary signed sparse vectors (Killingback et al., 7 Jul 2026).

4. Logical interpretation: Max-OR, positive CNF, and the role of negation

The same theoretical framework interprets MaxSim as an aggregation of soft logical operations. For a query group $$S_{\pm}(U_s,V_s)=\sum_{(m,s_q)\in U_s} s_q\cdot s_d^*(m)\cdot \langle m,t^*(m)\rangle.$$3 and a document set $$S_{\pm}(U_s,V_s)=\sum_{(m,s_q)\in U_s} s_q\cdot s_d^*(m)\cdot \langle m,t^*(m)\rangle.$$4, the paper defines a weighted OR:

$$S_{\pm}(U_s,V_s)=\sum_{(m,s_q)\in U_s} s_q\cdot s_d^*(m)\cdot \langle m,t^*(m)\rangle.$$5

With a higher-dimensional polynomial map

$$S_{\pm}(U_s,V_s)=\sum_{(m,s_q)\in U_s} s_q\cdot s_d^*(m)\cdot \langle m,t^*(m)\rangle.$$6

MaxSim computes the sum of these groupwise MaxOR values. This gives a formal account of the operator’s “soft OR” behavior: within each group, the best matching key dominates, and across groups those maxima are added (Killingback et al., 7 Jul 2026).

The paper then proves that MaxSim can evaluate positive CNF queries rank-equivalently. For a Boolean query

$$S_{\pm}(U_s,V_s)=\sum_{(m,s_q)\in U_s} s_q\cdot s_d^*(m)\cdot \langle m,t^*(m)\rangle.$$7

where each clause $$S_{\pm}(U_s,V_s)=\sum_{(m,s_q)\in U_s} s_q\cdot s_d^*(m)\cdot \langle m,t^*(m)\rangle.$$8 is a disjunction of positive keys, the resulting MaxSim score is rank-equivalent to exact CNF satisfaction. Each clause contributes $$S_{\pm}(U_s,V_s)=\sum_{(m,s_q)\in U_s} s_q\cdot s_d^*(m)\cdot \langle m,t^*(m)\rangle.$$9 if satisfied and $u\in\mathbb R^n$0 otherwise, so the final score equals the number of satisfied clauses. This logical interpretation is used in the paper to explain why MaxSim is naturally effective for synonym grouping and positive disjunctions (Killingback et al., 7 Jul 2026).

The same analysis also clarifies why negation is difficult for standard MaxSim. A positive clause can be handled by “choose the best witness” semantics, but exclusion requires a matched concept to penalize score. Signed MaxSim alters only this point: it preserves the route-to-best-match mechanism while allowing the selected match to carry a negative contribution. In the paper’s framing, this closes a specific expressivity gap rather than replacing the late-interaction paradigm (Killingback et al., 7 Jul 2026).

5. Empirical evaluation and the Fallon model

The empirical model used for Signed MaxSim is called Fallon. The training setup reported in the paper uses 100k queries and 200k documents built from synthetic LIMIT-style documents with attributes such as hobbies and foods. Each query has 1–4 inclusion terms and exactly one negated term, and each training query includes 32 hard negatives consisting of documents that match the inclusion terms but contain the negated term. The backbone is ModernBERT with 149M parameters and hidden size 768. Token embeddings are projected to 128 dimensions via a five-layer MLP, and Fallon adds an additional five-layer MLP that predicts a real-valued weight/sign-like scalar. Training uses contrastive cross-entropy with in-batch negatives and hard negatives; retrieval is exhaustive over 100k documents; the reported metrics are nDCG@10, P@10, and AP (Killingback et al., 7 Jul 2026).

Evaluation is reported on three 2,000-query benchmarks over 100k documents each. The In-Domain benchmark uses the same vocabulary and query format as training. Different Vocabulary preserves the same structure but uses new vocabulary terms generated by Gemini. Negation Only contains queries consisting only of negations, such as “People who do not like Cherries,” sometimes with 1–2 negations. On nDCG@10, ColBERT scores 0.982 / 0.597 / 0.008 across these three settings, whereas Fallon scores 0.997 / 1.000 / 0.788. The abstract summarizes the out-of-domain improvement as nDCG@10 increasing from 0.597 to 1.000 under a vocabulary shift and from 0.008 to 0.788 on negation-only queries (Killingback et al., 7 Jul 2026).

The corresponding P@10 scores are 0.875 / 0.587 / 0.009 for ColBERT and 0.893 / 0.955 / 0.789 for Fallon. The AP scores are 0.943 / 0.511 / 0.002 for ColBERT and 0.997 / 1.000 / 0.039 for Fallon. All improvements are reported as statistically significant at $u\in\mathbb R^n$1. The paper emphasizes the Negation Only condition as the strongest empirical signal: the standard ColBERT/MaxSim baseline essentially fails, whereas Signed MaxSim performs much better, though not perfectly. The stated interpretation is that standard MaxSim tends to seek a positive match for each query token and therefore cannot easily punish documents containing excluded concepts, while Signed MaxSim directly allows negative contributions (Killingback et al., 7 Jul 2026).

6. Relation to other MaxSim variants and common terminological confusions

Signed MaxSim is distinct from BiMax, despite the superficial similarity of names. “BiMax: Bidirectional MaxSim Score for Document-Level Alignment” defines a cross-lingual document alignment score that computes MaxSim once in each direction, source-to-target and target-to-source, and averages the two values:

$u\in\mathbb R^n$2

That method is explicitly described as not a signed variant of MaxSim. Its “directional” aspect is bidirectionality in segment matching, not positive or negative polarity. It is a document-level late-interaction alignment score used in hierarchical mining and web-scale parallel corpus construction, and the paper stresses its efficiency relative to optimal transport (Wang et al., 17 Oct 2025).

Signed MaxSim is also distinct from FLASH-MAXSIM. “FLASH-MAXSIM: IO-Aware Fused Kernels for Late-Interaction Scoring” optimizes the computation of the standard MaxSim score

$u\in\mathbb R^n$3

without materializing the full similarity tensor. That work discusses standard MaxSim over ordinary real-valued dot products, which may be negative, and notes that padded positions are masked by setting them to $u\in\mathbb R^n$4 before the row reduction. It explicitly does not introduce a special “signed MaxSim” operator, absolute-value MaxSim, positive-only MaxSim, or separate handling of positive and negative contributions. Negative similarities are handled simply as ordinary real numbers under the max reduction (Pony et al., 28 May 2026).

A further terminological confusion arises from unrelated graph-theoretic usage. In “Signed graphs with maximal index,” a signed graph is a graph with edge labels in $u\in\mathbb R^n$5, and the index is the largest eigenvalue of its adjacency matrix. There, “signed” refers to edge signs and “maximal index” refers to a spectral extremal problem over complete signed graphs with a fixed number of negative edges. This is unrelated to MaxSim or late-interaction retrieval (Ghorbani et al., 2021).

Within the retrieval literature represented here, the precise distinction is therefore threefold. Standard MaxSim is a sum of maxima over inner products; BiMax is a bidirectional document-alignment average of MaxSim scores; Signed MaxSim is the sign-augmented late-interaction similarity that restores exact recovery of arbitrary sparse real-valued inner products and improves negation-sensitive retrieval (Killingback et al., 7 Jul 2026).