Approximate Real Symmetric Tensor Rank
This work addresses a theoretical problem in tensor decomposition for researchers in computational mathematics, but it appears incremental as it builds on known results and focuses on expository insights.
The paper tackles the problem of determining the smallest symmetric tensor rank achievable within an ε-neighborhood of a given real symmetric tensor, providing constructive upper bounds through two theorems and three algorithms. The results include theoretical proofs and algorithmic approaches, though no concrete numerical improvements are specified.
We investigate the effect of an $\varepsilon$-room of perturbation tolerance on symmetric tensor decomposition. To be more precise, suppose a real symmetric $d$-tensor $f$, a norm $||.||$ on the space of symmetric $d$-tensors, and $\varepsilon >0$ are given. What is the smallest symmetric tensor rank in the $\varepsilon$-neighborhood of $f$? In other words, what is the symmetric tensor rank of $f$ after a clever $\varepsilon$-perturbation? We prove two theorems and develop three corresponding algorithms that give constructive upper bounds for this question. With expository goals in mind; we present probabilistic and convex geometric ideas behind our results, reproduce some known results, and point out open problems.