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PreprintCC BY-NC-SA 4.0arXiv

On the identifiability and semi-algebraicity of ReLU neuromanifolds

  • Axel Flinth
  • Stefano Mereta
  • Michele Pernice
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Abstract

We study several aspects of the semi-algebraicity of the neuromanifold ℳd of a feedforward ReLU neural network. We prove that ℳd in general is not a semi-algebraic quotient of the space of parameters of the network, but it can be approximated (in a precise sense) by semi-algebraic spaces. We then consider the notion of identifiable subsets of the space of parameters, where the network does not have any hidden symmetries. We prove that the maximal identifiable subset is semi-algebraic for any neural network with semi-algebraic activation functions. Lastly, we compute explicitly the identifiable subset of a shallow ReLU network and prove that for input dimensions ≥ 3 and width ≥ 6 it is not euclidean open.

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Cite

BibTeXRIS
BibTeX
@misc{flinth2026identifiability,
  title = {{On the identifiability and semi-algebraicity of ReLU neuromanifolds}},
  author = {Axel Flinth and Stefano Mereta and Michele Pernice},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.02826},
  eprint = {2606.02826},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.02826},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - On the identifiability and semi-algebraicity of ReLU neuromanifolds
AU  - Axel Flinth
AU  - Stefano Mereta
AU  - Michele Pernice
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2606.02826
UR  - https://arxiv.org/abs/2606.02826
AB  - We study several aspects of the semi-algebraicity of the neuromanifold ℳ_d of a feedforward ReLU neural network. We prove that ℳ_d in general is not a semi-algebraic quotient of the space of parameters of the network, but it can be approximated (in a precise sense) by semi-algebraic spaces. We then consider the notion of identifiable subsets of the space of parameters, where the network does not have any hidden symmetries. We prove that the maximal identifiable subset is semi-algebraic for any neural network with semi-algebraic activation functions. Lastly, we compute explicitly the identifiable subset of a shallow ReLU network and prove that for input dimensions ≥ 3 and width ≥ 6 it is not euclidean open.
ER  -