Landscape analysis of an improved power method for tensor decomposition
This provides a provable, efficient algorithm for low-rank symmetric tensor decomposition, which is incremental but improves robustness and theoretical understanding for applications in data analysis and machine learning.
The paper analyzes the optimization landscape of the Subspace Power Method (SPM) for symmetric tensor decomposition, showing that under certain conditions, second-order critical points must equal or approximate tensor components, with guarantees up to rank O(D) or ~o(D^{⌊m/2⌋}) for noisy tensors.
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.