Open Problem: How much overparametrization is needed for ALS in tensor decomposition?

We ask how much overparameterization is needed for simple iterative methods such as alternating least squares (ALS) and gradient descent to decompose a third-order tensor. This question can be viewed as a basic setting to study feature learning: when a rank-$r$ tensor in ambient dimension $n$ has $r\ll n$, the latent rank-one components are the features, and $k$ is the amount of overparameterization used by the algorithm. For rank $r$ tensors, recent work shows that overparametrized rank $k=O(r^2)$ suffices for the popular ALS heuristic (with random initialization) to converge to a global optima. Is the quadratic dependence on $r$ an inherent barrier for ALS-like methods? We pose the open problem of proving convergence to the global optimum for $k=o(r^2)$, or proving that a lower bound on the overparametrized rank of $k=\Omega(r^{1+c})$ for some absolute constant $c>0$ is necessary.