A Cheeger Inequality for Small Set Expansion

The discrete Cheeger inequality, due to Alon and Milman (J. Comb. Theory Series B 1985), is an indispensable tool for converting the combinatorial condition of graph expansion to an algebraic condition on the eigenvalues of the graph adjacency matrix. We prove a generalization of Cheeger's inequality, giving an algebraic condition equivalent to small set expansion. This algebraic condition is the p-to-q hypercontractivity of the top eigenspace for the graph adjacency matrix. Our result generalizes a theorem of Barak et al (STOC 2012) to the low small set expansion regime, and has a dramatically simpler proof; this answers a question of Barak (2014).