Rank and computation of the pathlifting Jacobian of a DAG ReLU network.

Abstract

This paper provides a self-contained proof of the rank of the pathlifting Jacobian of a DAG ReLU network by performing an induction on the network's number of hidden nodes. In fact, the induction is elementary, and the key recipe is to consider the skeleton matrix of the network, a sparse matrix encoding the network paths, and transform the representation of one of its hidden neurons into an output node. The proof relies on intermediate propositions which link the pathlifting, its Jacobian, the network parameters, and its skeleton matrix, which, on top of permitting to conclude on the rank of the pathlifting Jacobian, also provide a way to compute it without backpropagation and whose computation cost is super efficient in practice compare to usual backpropagation. The paper is provided with a Python module that implements the different propositions of the paper for feed forward networks and is used to experimentally quantifies the computational gain of computing the pathlifting Jacobian with the proposed theory.

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BibTeX RIS

BibTeX

@misc{verbockhaven2026rank,
  title = {{Rank and computation of the pathlifting Jacobian of a DAG ReLU network.}},
  author = {Manon Verbockhaven},
  howpublished = {HAL},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2609.18682},
  eprint = {2609.18682},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05773803v2},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Rank and computation of the pathlifting Jacobian of a DAG ReLU network.
AU  - Manon Verbockhaven
PY  - 2026
DA  - 2026/08/01
PB  - HAL
DO  - 10.48550/arxiv.2609.18682
UR  - https://hal.science/hal-05773803v2
AB  - This paper provides a self-contained proof of the rank of the pathlifting Jacobian of a DAG ReLU network by performing an induction on the network's number of hidden nodes. In fact, the induction is elementary, and the key recipe is to consider the skeleton matrix of the network, a sparse matrix encoding the network paths, and transform the representation of one of its hidden neurons into an output node. The proof relies on intermediate propositions which link the pathlifting, its Jacobian, the network parameters, and its skeleton matrix, which, on top of permitting to conclude on the rank of the pathlifting Jacobian, also provide a way to compute it without backpropagation and whose computation cost is super efficient in practice compare to usual backpropagation. The paper is provided with a Python module that implements the different propositions of the paper for feed forward networks and is used to experimentally quantifies the computational gain of computing the pathlifting Jacobian with the proposed theory.
ER  -

Abstract

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