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In this work, we consider the optimization formulation for symmetric tensordecomposition recently introduced in the Subspace Power Method (SPM) of Kileeland Pereira. Unlike popular alternative functionals for tensor decomposition,the SPM objective function has the desirable properties that its maximal valueis known in advance, and its global optima are exactly the rank-1 components ofthe tensor when the input is sufficiently low-rank. We analyze the non-convexoptimization landscape associated with the SPM objective. Our analysis accountsfor working with noisy tensors. We derive quantitative bounds such that anysecond-order critical point with SPM objective value exceeding the bound mustequal a tensor component in the noiseless case, and must approximate a tensorcomponent in the noisy case. For decomposing tensors of size $D^{\times m}$, weobtain 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 SPMwith suitable initialization is a provable, efficient, robust algorithm forlow-rank symmetric tensor decomposition. We conclude with numerics that show apractical preferability for using the SPM functional over a more establishedcounterpart.

Landscape analysis of an improved power method for tensor decomposition