A nonlinear lower bound for constant depth arithmetical circuits via the discrete uncertainty principle

We prove a super-linear lower bound on the size of a bounded depth bilinear arithmetical circuit computing cyclic convolution. Our proof uses the strengthening of the Donoho-Stark uncertainty principle [DS89] given by Tao [Tao05], and a combinatorial lemma by Raz and Shpilka [RS03]. This combination and an observation on ranks of circulant matrices, which we use to give a much shorter proof of the Donoho-Stark principle, may have other applications.