A topological description of loss surfaces based on Betti Numbers

Maria Sofia Bucarelli, Giuseppe Alessio D'Inverno, Monica Bianchini, Franco Scarselli, Fabrizio Silvestri

Abstract: Highlights•Establishment of bounds on the topological complexity of loss functions in terms of Betti numbers’s characterization.•Bounds are provided for Mean Squared Error and Binary Cross Entropy loss functions and for networks with Pfaffian activation functions.•Analysis of the effect of skip connections and ℓ2<math><msub is="true"><mrow is="true"><mi is="true">ℓ</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub></math> regularization on the topological complexity.