Linear Independence of Polynomial Compositions with Bounded Input Degree

Xiaotian Chang; Yacoub Hendi

math.AGOct 14, 2026

Linear Independence of Polynomial Compositions with Bounded Input Degree

Authors: Xiaotian Chang, Yacoub Hendi

Abstract

We prove the weak composition-independence conjectures of Kohn, Marchetti, Massarenti and Mella. The proof uses deformations of rational curves and a single kernel of the linearized equation. Riemann-Roch gives a contradiction from tangent fields with controlled poles. By settling these conjectures, we prove that a neural network with a general polynomial activation function of sufficiently large degree is identifiable.

Representation Identifiability

Representation identifiability studies when learned internal codes can be uniquely recovered from observations or behavior, and which ambiguities remain.

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Linear Independence of Polynomial Compositions and Identifiability of Deep Neural Networks
Aug 27, 2026Kathlén Kohn, Giovanni Luca Marchetti, Alex Massarenti +1Parameter Identifiability Polynomial Neural Networks
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Oct 24, 2026Isabel Dahlgren, Giovanni Luca Marchetti, Kathlén KohnParameter Identifiability Polynomial Neural Networks

Aug 27, 2026J/K move · Enter open · S save

Linear Independence of Polynomial Compositions and Identifiability of Deep Neural Networks

Kathlén Kohn, Giovanni Luca Marchetti, Alex Massarenti, Massimiliano Mella

KTH Stockholm · AI4S AB Stockholm · University of Ferrara

Parameter Identifiability Polynomial Neural Networks Multilayer Perceptrons

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Motivated by theoretical problems in deep learning, we conjecture that post-composing a fixed number of pairwise distinct nonconstant polynomials with a generic polynomial of sufficiently large degree yields linearly independent polynomials. This generalizes Newman--Slater's theorem on powers of polynomials. We establish several cases of this conjecture and its origin-passing variant: We prove the result for two polynomials, and for an arbitrary number of polynomials when their degrees are bounded. Furthermore, we show how the conjecture implies a complete understanding of the identifiability (i.e., parameter symmetries) of deep fully connected neural network architectures with generic polynomial activation functions. In particular, for network architectures with layer-specific activations of increasing degree, our established versions of the conjecture fully characterize the set of parameters yielding the same end-to-end network function. As a special case, we fully resolve the identifiability of shallow polynomial networks.

Abstract

Representation Identifiability

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