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DOI: 10.4230/LIPIcs.SoCG.2026.91
The Hierarchy of Manifolds in a Stratification of the Set of Equivalent Linear Neural Networks

Authors: Jonathan Richard Shewchuk and Sagnik Bhattacharya

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
A linear neural network computes a linear transformation of its input vector. Given a fully-connected linear network, the set of all weight vectors for which the network computes the same linear transformation is an algebraic variety in weight space, called a fiber under the matrix multiplication map. Sometimes this variety is a manifold, but usually not. The rank stratification of a fiber is a natural partition of the fiber into manifolds of various dimensions called strata. We characterize how these strata are connected to each other. They satisfy the frontier condition: if a stratum intersects the closure of another stratum, then the former stratum is a subset of the closure of the latter stratum. This subset relationship can be expressed as a partial order with a single minimal element. Our main result describes the relationship between this partial order and the ranks of certain matrices in the network. Each stratum represents a different pattern of information flow through the network, expressed as a barcode. Connections among the strata are best understood through simple transformations of the barcodes called barcode moves.
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Jonathan Richard Shewchuk and Sagnik Bhattacharya. The Hierarchy of Manifolds in a Stratification of the Set of Equivalent Linear Neural Networks. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 91:1-91:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{shewchuk_et_al:LIPIcs.SoCG.2026.91,
  author =	{Shewchuk, Jonathan Richard and Bhattacharya, Sagnik},
  title =	{{The Hierarchy of Manifolds in a Stratification of the Set of Equivalent Linear Neural Networks}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{91:1--91:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.91},
  URN =		{urn:nbn:de:0030-drops-258971},
  doi =		{10.4230/LIPIcs.SoCG.2026.91},
  annote =	{Keywords: Linear neural network, real algebraic variety, stratification, multilinear algebra, product of matrices, persistence barcode, real algebraic geometry, discrete geometry}
}
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