  In the realm of tensor optimization, the low-rank Tucker decomposition is
crucial for reducing the number of parameters and for saving storage. We
explore the geometry of Tucker tensor varieties -- the set of tensors with
bounded Tucker rank -- which is notably more intricate than the well-explored
matrix varieties. We give an explicit parametrization of the tangent cone of
Tucker tensor varieties and leverage its geometry to develop provable
gradient-related line-search methods for optimization on Tucker tensor
varieties. To the best of our knowledge, this is the first work concerning
geometry and optimization on Tucker tensor varieties. In practice, low-rank
tensor optimization suffers from the difficulty of choosing a reliable rank
parameter. To this end, we incorporate the established geometry and propose a
Tucker rank-adaptive method that aims to identify an appropriate rank with
guaranteed convergence. Numerical experiments on tensor completion reveal that
the proposed methods are in favor of recovering performance over other
state-of-the-art methods. The rank-adaptive method performs the best across
various rank parameter selections and is indeed able to find an appropriate
rank.
