The \emph{ sum-of-squares (SoS) complexity} of a $d$-multiquadratic polynomial $f$ (quadratic in each of $d$ blocks of $n$ variables) is the minimum $s$ such that $f = \sum_{ i=1}^s g_i^2$ with each $g_i$ $d$-multilinear. In the case $d=2$, Hrube\v{ s}, Wigderson and Yehudayoff (2011) showed that an $n^{ 1+\Omega(1)}$ lower bound on the SoS complexity of explicit biquadratic polynomials implies an exponential lower bound for non-commutative arithmetic circuits. In this paper, we establish an analogous connection between general \emph{ multiquadratic sum-of-squares} and \emph{ commutative arithmetic formulas}. Specifically, we show that an $n^{ d-o(\log d)}$ lower bound on the SoS complexity of explicit $d$-multiquadratic polynomials, for any $d = d(n)$ with $\omega(1) \le d(n) \le O(\frac{ \log n}{ \log\log n})$, would separate the algebraic complexity classes VNC$^1$ and VNP.
Categories: cs.CC