  We present a faster interior-point method for optimizing sum-of-squares (SOS)
polynomials, which are a central tool in polynomial optimization and capture
convex programming in the Lasserre hierarchy. Let $p = \sum_i q^2_i$ be an
$n$-variate SOS polynomial of degree $2d$. Denoting by $L := \binom{n+d}{d}$
and $U := \binom{n+2d}{2d}$ the dimensions of the vector spaces in which
$q_i$'s and $p$ live respectively, our algorithm runs in time
$\tilde{O}(LU^{1.87})$. This is polynomially faster than state-of-art SOS and
semidefinite programming solvers, which achieve runtime
$\tilde{O}(L^{0.5}\min\{U^{2.37}, L^{4.24}\})$.
  The centerpiece of our algorithm is a dynamic data structure for maintaining
the inverse of the Hessian of the SOS barrier function under the polynomial
interpolant basis, which efficiently extends to multivariate SOS optimization,
and requires maintaining spectral approximations to low-rank perturbations of
elementwise (Hadamard) products. This is the main challenge and departure from
recent IPM breakthroughs using inverse-maintenance, where low-rank updates to
the slack matrix readily imply the same for the Hessian matrix.
