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

We prove a superlinear 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 [D.L. Donoho, P.B. Stark, Uncertainty principles and signal recovery, SIAM Journal of Applied Mathematics 49 (1989) 906-931] given by Tao [T. Tao, An uncertainty principle for cyclic groups of prime order, Mathematical Research Letters 12 (2005) 121-127], and a combinatorial lemma by Raz and Shpilka [R. Raz, A. Shpilka, Lower bounds for matrix product, in arbitrary circuits with bounded gates, SIAM Journal of Computing 32 (2003) 488-513]. 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.