  We consider rank-one symmetric tensor estimation when the tensor is corrupted
by Gaussian noise and the spike forming the tensor is a structured signal
coming from a generalized linear model. The latter is a mathematically
tractable model of a non-trivial hidden lower-dimensional latent structure in a
signal. We work in a large dimensional regime with fixed ratio of
signal-to-latent space dimensions. Remarkably, in this asymptotic regime, the
mutual information between the spike and the observations can be expressed as a
finite-dimensional variational problem, and it is possible to deduce the
minimum-mean-square-error from its solution. We discuss, on examples,
properties of the phase transitions as a function of the signal-to-noise ratio.
Typically, the critical signal-to-noise ratio decreases with increasing
signal-to-latent space dimensions. We discuss the limit of vanishing ratio of
signal-to-latent space dimensions and determine the limiting tensor estimation
problem. We also point out similarities and differences with the case of
matrices.
