On the Power of Border of Depth-3 Arithmetic Circuits

We show that over the field of complex numbers, every homogeneous polynomial of degree d can be approximated (in the border complexity sense) by a depth-3 arithmetic circuit of top fan-in at most 2. This is quite surprising, since there exist homogeneous polynomials P on n variables of degree 2, such that any depth-3 arithmetic circuit computing P must have top fan-in at least Ω (n). As an application, we get a new tradeoff between the top fan-in and formal degree in an approximate analog of the celebrated depth reduction result of Gupta, Kamath, Kayal, and Saptharishi [7, 10]. Formally, we show that if a degree d homogeneous polynomial P can be computed by an arithmetic circuit of size s ≥ d, then for every t ≤ d, P is in the border of a depth-3 circuit of top fan-in sO(t) and formal degree sO(d/t). To the best of our knowledge, the upper bound on the top fan-in in the original proof of Reference [7] is always at least sΩ (√d), regardless of the formal degree.