Relational Linearity in Theory and Models

• Relational linearity is a concept that imposes linear structures on relations, defining constraints on maps, transformations, and statistical effects across various domains.
• It unifies diverse mathematical formulations from operator theory, linear logic, language models, and graph encodings, illustrating its multifaceted applications.
• Empirical investigations reveal that relational linearity can predict phenomena such as hallucination rates in language models and improve modeling in relational statistics and temporal networks.

Searching arXiv for papers on “relational linearity” and closely related usages across fields. Relational linearity is a cross-disciplinary term rather than a single standardized concept. In the literatures represented here, it denotes linear structure imposed on relations, relational models, or relation-conditioned transformations: linear relations as subspaces of products in operator theory and module theory, relational semantics for linear logic, shared linear structure for subject–relation–object behavior in LLMs, and linear contribution assumptions in statistical relational and relational-event models (Berger et al., 2020, Bennett-Tennenhaus, 2024, Carvalho, 2015, Lu et al., 16 Jan 2026, Valer et al., 21 May 2026, Klimova et al., 2016, Boschi et al., 5 Sep 2025). Taken together, these uses treat linearity not merely as a property of isolated maps, but as a constraint on correspondences, denotations, or relation-indexed effects.

1. Terminological scope and recurring pattern

Across current usage, “relational linearity” names several mathematically distinct constructions.

Domain	Object made linear	Canonical form
Linear relations	A relation itself	$A \subseteq \mathfrak H \times \mathfrak H$, $C \subseteq L \oplus M$
Linear logic	Proof denotation in a relational model	$|!C|=\mathcal{M}_{\mathrm{fin}}(|C|)$
LLMs	Relation-specific subject-to-object transformation	$\hat{\mathbf{o}}_j=\mathbf{s}_j+\bar{\mathbf{d}}_r$
Relational probing	Relation-conditioned output distribution	linear-softmax probe from $\mathbf{f}^{(\ell)}(\mathbf{s})$
Relational statistics	Log-parameter structure	$$\log \boldsymbol{\delta}=\mathbf{A}'\boldsymbol{\theta}$$
Relational hyper-event models	Log-rate contribution of statistics	$$f^{\mathrm{LE}}(\mathbf{x}(t,I,J),t)=\boldsymbol{\theta}^\top \mathbf{x}(t,I,J)$$
Graph encoding	Neighborhoods via multiple orders	$N[x]=\bigcup_{1\le i\le p} I_i$

A recurring synthesis is that the relation, rather than only the underlying carrier, is the locus of linear structure. In some settings the relation is literally a subspace; in others, a fixed relation induces a linear predictor, a shared translation, or a linear softmax readout. This suggests a family resemblance rather than a unified theory.

2. Linear relations as algebraic, operator-theoretic, and categorical objects

In operator theory, a linear relation in a complex linear space $\mathfrak H$ is a linear subspace $A \subseteq \mathfrak H \times \mathfrak H$. Its basic associated subspaces are

$C \subseteq L \oplus M$0

with operator-like behavior characterized by $C \subseteq L \oplus M$1 (Berger et al., 2020). The paper "Linear relations and their singular chains" identifies the singular chain space

$C \subseteq L \oplus M$2

as the key invariant distinguishing linear relations from ordinary operators. Its main identity,

$C \subseteq L \oplus M$3

shows that, unlike operator theory, distinct root spaces for a linear relation may intersect nontrivially, and that the intersection is exactly the singular chain space (Berger et al., 2020).

For monotone analysis, linear relations are set-valued maps with linear graphs. The paper "Five classes of monotone linear relations and operators" proves that monotone linear relations admit a single-valued reduction on a suitable subspace, especially $C \subseteq L \oplus M$4, and that a monotone linear relation with full domain is automatically maximal monotone and single valued (Edwards, 2012). It also shows that for maximal monotone linear relations, $C \subseteq L \oplus M$5-monotonicity implies paramonotonicity, while finite-dimensional equivalences can fail for arbitrary linear relations.

In category theory, linear relations are morphisms $C \subseteq L \oplus M$6 represented by linear subspaces $C \subseteq L \oplus M$7. Weinstein’s "Categories of (co)isotropic linear relations" analyzes when composition is well-behaved. A composable pair $C \subseteq L \oplus M$8 is monic iff

$C \subseteq L \oplus M$9

and transversal iff

$|!C|=\mathcal{M}_{\mathrm{fin}}(|C|)$0

The Wehrheim–Woodward category records the two numerical obstructions to naive composition: excess, measuring failure of monicity, and defect, measuring failure of transversality. In $|!C|=\mathcal{M}_{\mathrm{fin}}(|C|)$1, $|!C|=\mathcal{M}_{\mathrm{fin}}(|C|)$2, and $|!C|=\mathcal{M}_{\mathrm{fin}}(|C|)$3, a Wehrheim–Woodward morphism is determined by its shadow, defect, and excess, so the category is a central extension of the underlying relation category by endomorphisms of the unit object (Weinstein, 2015).

Over arbitrary commutative rings, linear relations are $|!C|=\mathcal{M}_{\mathrm{fin}}(|C|)$4-submodules $|!C|=\mathcal{M}_{\mathrm{fin}}(|C|)$5. "Linear relations over commutative rings" identifies their category with a subcategory of Kronecker representations satisfying

$|!C|=\mathcal{M}_{\mathrm{fin}}(|C|)$6

and proves that this subcategory is definable, faithful, hereditary, and the torsion-free half of a torsion pair (Bennett-Tennenhaus, 2024). The same paper extends the functorial filtrations method from fields to commutative rings, replacing direct splitting by the weaker but functorial reduction framework built from

$|!C|=\mathcal{M}_{\mathrm{fin}}(|C|)$7

3. Relational semantics in linear logic

Within linear logic, relational models are explicitly named among the tools, approaches, and methodologies developed for the study of linear logic’s syntax and semantics (Ehrhard et al., 2019). A concrete and technically decisive instance is de Carvalho’s injectivity theorem for the relational model of MELL proof-nets.

In "The relational model is injective for Multiplicative Exponential Linear Logic", formulas are interpreted by sets with exponentials realized as finite multisets: $|!C|=\mathcal{M}_{\mathrm{fin}}(|C|)$8 The denotation $|!C|=\mathcal{M}_{\mathrm{fin}}(|C|)$9 is the set of results of all experiments observed at the conclusions, and the main theorem states that if two cut-free proof-structures $\hat{\mathbf{o}}_j=\mathbf{s}_j+\bar{\mathbf{d}}_r$0 and $\hat{\mathbf{o}}_j=\mathbf{s}_j+\bar{\mathbf{d}}_r$1 have the same conclusions and

$\hat{\mathbf{o}}_j=\mathbf{s}_j+\bar{\mathbf{d}}_r$2

then

$\hat{\mathbf{o}}_j=\mathbf{s}_j+\bar{\mathbf{d}}_r$3

Equivalently, equality in the relational model is exactly axiomatized by cut-elimination (Carvalho, 2015).

This result matters because exponentials could, in principle, collapse box structure into multiset behavior. The theorem shows that they do not collapse it beyond cut-elimination equivalence. Relational semantics is therefore not merely sound for MELL proof-nets; it is injective on them.

4. Relational linearity in LLMs

Recent language-model work uses the phrase in an explicitly operational sense. "Relational Linearity is a Predictor of Hallucinations" defines relational linearity as the extent to which a relation in hidden-state space can be approximated by a shared translation from subject representations to object representations. For a relation $\hat{\mathbf{o}}_j=\mathbf{s}_j+\bar{\mathbf{d}}_r$4, training pairs $\hat{\mathbf{o}}_j=\mathbf{s}_j+\bar{\mathbf{d}}_r$5 yield

$\hat{\mathbf{o}}_j=\mathbf{s}_j+\bar{\mathbf{d}}_r$6

and held-out linearity is measured by

$\hat{\mathbf{o}}_j=\mathbf{s}_j+\bar{\mathbf{d}}_r$7

Using SyntHal, a dataset of 6000 synthetic entities for six relations, the paper reports a strong correlation $\hat{\mathbf{o}}_j=\mathbf{s}_j+\bar{\mathbf{d}}_r$8 between relational linearity and hallucination rate across four instruction-tuned 7–8B models (Lu et al., 16 Jan 2026). The same study also reports that on natural triples the sign of the association reverses: higher $\hat{\mathbf{o}}_j=\mathbf{s}_j+\bar{\mathbf{d}}_r$9 correlates with lower hallucination rate, which the authors interpret as evidence that linear relations are easier to learn when the fact is known but harder to abstain on when the queried entity is synthetic or unknown.

"Relational Linear Properties in LLMs: An Empirical Investigation" studies a related but not identical formulation. For a fixed query $\mathbf{f}^{(\ell)}(\mathbf{s})$0, it tests whether the answer distribution induced by a prompted input can be recovered linearly from the context-only hidden state. Its experimental criterion is a linear-softmax probe

$\mathbf{f}^{(\ell)}(\mathbf{s})$1

trained by a KL-based objective. Across four sentence-level datasets, it finds that relational linearity varies across models, follows layer-wise patterns, and is affected differently by paraphrased relational queries. The strongest evidence appears for Tense and Truth, especially in middle layers, and the proposed KL-RP method is reported as more efficient than the LRE baseline: about 30 minutes per relation for one layer for LRE versus about 40 minutes per relation across all layers for KL-RP (Valer et al., 21 May 2026).

A common misconception is that these two language-model notions are interchangeable. They are not. One uses a translation-only probe on subject/object representations and $\mathbf{f}^{(\ell)}(\mathbf{s})$2; the other uses a relation-specific linear-softmax probe on subject/context states and KL divergence. Both are relation-indexed linearity tests, but they target different objects.

5. Linearity assumptions in relational statistics, networks, and graph encodings

In discrete statistical modeling, "Testing the fit of relational models" treats relational models as generalizations of log-linear models on arbitrary discrete sample spaces. Their defining form is

$\mathbf{f}^{(\ell)}(\mathbf{s})$3

or dually $\mathbf{f}^{(\ell)}(\mathbf{s})$4 (Klimova et al., 2016). Here linearity lies in the canonical coordinates $\mathbf{f}^{(\ell)}(\mathbf{s})$5, not in $\mathbf{f}^{(\ell)}(\mathbf{s})$6 itself. The paper emphasizes the role of the overall effect, formally $\mathbf{f}^{(\ell)}(\mathbf{s})$7: with it, Poisson and multinomial MLEs behave analogously to ordinary log-linear models; without it, scale invariance fails, Poisson and multinomial MLEs are not equivalent, and the likelihood-ratio statistic must be interpreted via Bregman divergence rather than as twice a Kullback–Leibler divergence.

In temporal network modeling, "Beyond Linearity and Time-homogeneity: Relational Hyper Event Models with Time-Varying Non-Linear Effects" defines the standard RHEM linearity assumption by

$\mathbf{f}^{(\ell)}(\mathbf{s})$8

which implies linearity in covariate magnitude and time-homogeneity of effects (Boschi et al., 5 Sep 2025). The paper retains additivity on the log-hazard scale but replaces fixed linear effects by smooths: $\mathbf{f}^{(\ell)}(\mathbf{s})$9

$$\log \boldsymbol{\delta}=\mathbf{A}'\boldsymbol{\theta}$$0

$$\log \boldsymbol{\delta}=\mathbf{A}'\boldsymbol{\theta}$$1

The paper’s central claim is that standard relational hyper-event models are often too restrictive because many relational mechanisms are time-varying, non-linear, or jointly both.

A different discrete use appears in graph compression. "Linearity is Strictly More Powerful than Contiguity for Encoding Graphs" defines a closed $$\log \boldsymbol{\delta}=\mathbf{A}'\boldsymbol{\theta}$$2-line-model of a graph $$\log \boldsymbol{\delta}=\mathbf{A}'\boldsymbol{\theta}$$3 as a tuple of linear orders $$\log \boldsymbol{\delta}=\mathbf{A}'\boldsymbol{\theta}$$4 such that for every vertex $$\log \boldsymbol{\delta}=\mathbf{A}'\boldsymbol{\theta}$$5, there exist intervals $$\log \boldsymbol{\delta}=\mathbf{A}'\boldsymbol{\theta}$$6 with

$$\log \boldsymbol{\delta}=\mathbf{A}'\boldsymbol{\theta}$$7

The minimum $$\log \boldsymbol{\delta}=\mathbf{A}'\boldsymbol{\theta}$$8 is $$\log \boldsymbol{\delta}=\mathbf{A}'\boldsymbol{\theta}$$9, and the paper proves that linearity is strictly more powerful than contiguity as a graph-encoding parameter. For a family of connected cographs with complete-binary cotrees,

$$f^{\mathrm{LE}}(\mathbf{x}(t,I,J),t)=\boldsymbol{\theta}^\top \mathbf{x}(t,I,J)$$0

(Crespelle et al., 2018). Here linearity means multiple coordinated orders rather than a single-order interval union.

6. Related extension-based notions and recurrent distinctions

A related but distinct functional-analytic line of work treats linearity relationally through extension properties among subspaces. "General criteria for a strong notion of lineability" defines $$f^{\mathrm{LE}}(\mathbf{x}(t,I,J),t)=\boldsymbol{\theta}^\top \mathbf{x}(t,I,J)$$1-lineability by requiring that every $$f^{\mathrm{LE}}(\mathbf{x}(t,I,J),t)=\boldsymbol{\theta}^\top \mathbf{x}(t,I,J)$$2-dimensional subspace $$f^{\mathrm{LE}}(\mathbf{x}(t,I,J),t)=\boldsymbol{\theta}^\top \mathbf{x}(t,I,J)$$3 extend to a $$f^{\mathrm{LE}}(\mathbf{x}(t,I,J),t)=\boldsymbol{\theta}^\top \mathbf{x}(t,I,J)$$4-dimensional subspace $$f^{\mathrm{LE}}(\mathbf{x}(t,I,J),t)=\boldsymbol{\theta}^\top \mathbf{x}(t,I,J)$$5 with

$$f^{\mathrm{LE}}(\mathbf{x}(t,I,J),t)=\boldsymbol{\theta}^\top \mathbf{x}(t,I,J)$$6

and defines $$f^{\mathrm{LE}}(\mathbf{x}(t,I,J),t)=\boldsymbol{\theta}^\top \mathbf{x}(t,I,J)$$7-spaceability and $$f^{\mathrm{LE}}(\mathbf{x}(t,I,J),t)=\boldsymbol{\theta}^\top \mathbf{x}(t,I,J)$$8-dense lineability analogously (Fávaro et al., 2023). Earlier general criteria show how dense-lineability and spaceability can be transferred from stability under addition, complement-of-subspaces structures, and support separation (Bernal-González et al., 2013). This does not use the phrase “relational linearity” in the same way as operator theory or language-model probing, but it exhibits the same structural move: linearity is studied through extension relations among already-present linear pieces.

Several distinctions recur across the surveyed fields. First, “relational linearity” is not a single invariant. A linear relation $$f^{\mathrm{LE}}(\mathbf{x}(t,I,J),t)=\boldsymbol{\theta}^\top \mathbf{x}(t,I,J)$$9, a relational proof semantics, a translation probe in hidden-state space, and a linear log-rate term in a hyper-event model are formally different objects. Second, moving beyond linearity does not necessarily abandon additivity: the RHEM generalization keeps an additive predictor while replacing linear terms by smooths (Boschi et al., 5 Sep 2025). Third, higher relational linearity is not uniformly beneficial or harmful in LLMs: one study finds that it predicts more hallucinations on synthetic unknown entities, yet lower hallucination on natural known triples (Lu et al., 16 Jan 2026). Fourth, equivalences that hold in finite-dimensional settings may fail for arbitrary linear relations or infinite-dimensional Hilbert spaces; the monotonicity results for linear relations make this especially explicit (Edwards, 2012).

The expression therefore functions best as an umbrella description for several research programs in which linear structure is assigned to relations, relation-indexed transformations, or relationally defined families. Its exact content depends on the ambient category, semantics, model class, or representation space.