Linear Independence of Polynomial Compositions and Identifiability of Deep Neural Networks

Abstract: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.

Commutative Algebra (math.AC); Machine Learning (cs.LG); Algebraic Geometry (math.AG)

arXiv:2608.27113 [math.AC]

(or arXiv:2608.27113v1 [math.AC] for this version)

https://doi.org/10.48550/arXiv.2608.27113 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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