Over-parametrization is an important technique in training neural networks. In both theory and practice, training a larger network allows the optimization algorithm to avoid bad local optimal solutions. In this paper we study a closely related tensor decomposition problem: given an l-th order tensor in (R^d)^{?l} of rank r (where r«d), can variants of gradient descent find a rank m decomposition where m>r? We show that in a lazy training regime (similar to the NTK regime for neural networks) one needs at least m=O(d^{l-1}), while a variant of gradient descent can find an approximate tensor when m=O(r^{2.5l}logd). Our results show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data.
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