Approximate Real Symmetric Tensor Rank

We investigate the effect of an ε -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 ε> 0 are given. What is the smallest symmetric tensor rank in the ε -neighborhood of f? In other words, what is the symmetric tensor rank of f after a clever ε -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. © 2023, Institute for Mathematical Sciences (IMS), Stony Brook University, NY.