The Implicit Bias of Batch Normalization in Linear Models and Two-layer Linear Convolutional Neural Networks
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Abstract A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in ann-vertex graph not containingC2k, the cycle of length 2k, isO(n1+1/k). Simonovits established a corresponding supersaturation result forC2k’s, showing that there exist positive constantsC,cdepending only onksuch that everyn-vertex graphGwithe(G)⩾Cn1+1/kcontains at leastc(e(G)/v(G))2kcopies ofC2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant). In this paper we extend Simonovits' result to a supersaturation result ofr-uniform linear cycles of even length inr-uniform linear hypergraphs. Our proof is self-contained and includes ther= 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.more » « less
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