Heitmeier, Chuang & Baayen (2026): linearity and regularity #
The relation between morphological regularity and linearity of the form-meaning mapping (§16.6, Table 16.2), for paradigms with imputed additive semantics — word meaning as stem vector plus exponent vector, the book's regularisation of embeddings for "truly regular" systems.
- Linear maps satisfy proportional analogy by construction
(
isAnalogicallyRegular_comp): an exponent's form shift is stem-independent. Hence a table violating analogy (suppletion) has no linear interpolant over additive semantics (not_exists_linear_of_not_regular) — irregulars survive in a linear system only through word-specific, non-additive semantic vectors, which is how the book's linear models fit English past tense and Maltese plurals. - When stem and exponent vectors are jointly independent the converse holds
(
exists_linear_iff_isAnalogicallyRegular): linear realisability is analogical regularity. - Under ERM training on the paradigm (
paradigmExperience), an irregular table forces positive loss for every linear map (pos_weightedLoss_of_not_regular).
A paradigm's form table satisfies proportional analogy if the form contrast between two cells is the same for every stem (cat : cats :: dog : dogs).
Equations
- HeitmeierChuangBaayen2026.IsAnalogicallyRegular f = ∀ (st st' : Stem) (c c' : Cell), f st c - f st c' = f st' c - f st' c'
Instances For
Under additive semantics, the form shift a linear map assigns to an exponent is stem-independent.
Every linear map's paradigm table over additive semantics is analogically regular.
A table violating proportional analogy has no linear interpolant over additive semantics.
When stem and exponent vectors are jointly linearly independent, a paradigm table is linearly realisable iff it is analogically regular — over imputed additive semantics, linearity and regularity are the same constraint.
Proportional analogy is the vanishing of second differences — the same
equation as the XOR obstruction of linear separability (§16.6's
hub-and-neighbour classification example): a table realising XOR on a
coordinate, f ⊤ ⊤ = f ⊥ ⊥ ≠ f ⊤ ⊥ = f ⊥ ⊤, violates analogy, so no
linear map over additive semantics produces it. Analogy, linear
separability, and absence of interaction terms are one constraint.
The coding interface #
The stem-exponent setting is the two-primitive case of the DLM coding
interface: a paradigm cell's semantic primitives are its stem and cell
tags, and conceptualization over Sum.elim σ ε is the imputed additive
semantics. The fourth proportional above is conceptualize_analogy at the
multiset relation {st, c} + {st', c'} = {st, c'} + {st', c}.
Equations
- HeitmeierChuangBaayen2026.instSemanticPrimitivesProdSum = { primitives := fun (p : Stem × Cell) => {Sum.inl p.1, Sum.inr p.2} }
The paradigm as a training experience: imputed additive semantics as the meanings, the form table as the targets.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Irregularity forces positive training loss: no linear map fits a suppletive paradigm exactly, so every ERM solution carries residual error — Table 16.2's irregular-linear cell runs on word-specific semantics, not on additive imputation.