2026-09-05T11:58:36Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:25897 2026-06-23T13:00:25Z ddc:004 open_access

The Hierarchy of Manifolds in a Stratification of the Set of Equivalent Linear Neural Networks Shewchuk, Jonathan Richard Bhattacharya, Sagnik Linear neural network real algebraic variety stratification multilinear algebra product of matrices persistence barcode real algebraic geometry discrete geometry 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jonathan Richard Shewchuk and Sagnik Bhattacharya 2026 Is Part Of LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2026.91 urn:nbn:de:0030-drops-258971 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.91 eng https://creativecommons.org/licenses/by/4.0/legalcode