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Diffstat (limited to 'topology.tex')
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diff --git a/topology.tex b/topology.tex index 6fb6bce..8cd855d 100644 --- a/topology.tex +++ b/topology.tex @@ -122,10 +122,9 @@ solutions in neural networks with ReLu activations and stable equilibrium in the forces between physical objects. Equality constraints naturally appear in the zero-gradient solutions to overparameterized smooth neural networks and in vertex models of tissues. -In such problems, there is great interest in characterizing structure in the +In problems ranging from toy models \cite{Baldassi_2016_Unreasonable, Baldassi_2019_Properties} to real deep neural networks \cite{Goodfellow_2014_Qualitatively, Draxler_2018_Essentially, Frankle_2020_Revisiting, Vlaar_2022_What, Wang_2023_Plateau}, there is great interest in characterizing structure in the set of solutions, which can influence the behavior of algorithms trying -to find them \cite{Baldassi_2016_Unreasonable, Baldassi_2019_Properties, -Beneventano_2023_On}. Here, we show how topological information about +to find them \cite{Beneventano_2023_On}. Here, we show how topological information about the set of solutions can be calculated in a simple problem of satisfying random nonlinear equalities. This allows us to reason about the connectivity and structure of the solution set. The topological properties revealed by this calculation yield |