It looks complicated to assess the practical impact of the paper. On the one hand, the thermodynamic limit and the Gaussianity assumption may be hard to check in practice and it is not straightforward to extrapolate what happens in the finite dimensional case. The idea of identifying the problem's phase transitions is conceptually clear but it is not explicitly specified in the paper how this can help the practitioner. The paper only compares the AMP approach to alternate least squares without mention, for example, positive results obtained in the spectral method literature. Finally, it is not easy to understand if the obtained results only regard the AMP method or generalize to any inference method. Questions: - Is the analysis restricted to the AMP inference? In other words, could a tensor that is hard to infer via AMP approach be easily identifiable by other methods (or the other way round)? - Are the easy-hard-impossible phases be related with conditions on the rank of the tensor? - In the introduction the authors mention the fact that tensor decomposition is in general harder in the symmetric than in the non-symmetric case. How is this connected with recent findings about the `nice' landscape of the objective function associated with the decomposition of symmetric (orthogonal) order-4 tensors [1]? - The Gaussian assumption looks crucial for the analysis and seems to be guaranteed in the limit r << N. Is this a typical situation in practice? Is always possible to compute the `effective' variance for non-gaussian outputs? Is there a finite-N expansion that characterize the departure from Gaussianity in the non-ideal case? - For the themodynamic limit to hold, should one require N_alpha / N = O(1) for all alpha? - Given an observed tensor, is it possible to determine the particular phase it belongs to? [1] Rong Ge and Tengyu Ma, 2017, On the Optimization Landscape of Tensor Decompositions
- In the introduction the authors mention the fact that tensor decomposition is in general harder in the symmetric than in the non-symmetric case. How is this connected with recent findings about the `nice' landscape of the objective function associated with the decomposition of symmetric (orthogonal) order-4 tensors [1]?