  In this work, we consider the optimization formulation for symmetric tensor
decomposition recently introduced in the Subspace Power Method (SPM) of Kileel
and Pereira. Unlike popular alternative functionals for tensor decomposition,
the SPM objective function has the desirable properties that its maximal value
is known in advance, and its global optima are exactly the rank-1 components of
the tensor when the input is sufficiently low-rank. We analyze the non-convex
optimization landscape associated with the SPM objective. Our analysis accounts
for working with noisy tensors. We derive quantitative bounds such that any
second-order critical point with SPM objective value exceeding the bound must
equal a tensor component in the noiseless case, and must approximate a tensor
component in the noisy case. For decomposing tensors of size $D^{\times m}$, we
obtain a near-global guarantee up to rank $\widetilde{o}(D^{\lfloor m/2
\rfloor})$ under a random tensor model, and a global guarantee up to rank
$\mathcal{O}(D)$ assuming deterministic frame conditions. This implies that SPM
with suitable initialization is a provable, efficient, robust algorithm for
low-rank symmetric tensor decomposition. We conclude with numerics that show a
practical preferability for using the SPM functional over a more established
counterpart.
