  Tensors are often studied by introducing preorders such as restriction and
degeneration: the former describes transformations of the tensors by local
linear maps on its tensor factors; the latter describes transformations where
the local linear maps may vary along a curve, and the resulting tensor is
expressed as a limit along this curve. In this work we introduce and study
partial degeneration, a special version of degeneration where one of the local
linear maps is constant whereas the others vary along a curve. Motivated by
algebraic complexity, quantum entanglement and tensor networks, we present
constructions based on matrix multiplication tensors and find examples by
making a connection to the theory of prehomogeneous tensor spaces. We highlight
the subtleties of this new notion by showing obstruction and classification
results for the unit tensor. To this end, we study the notion of aided rank, a
natural generalization of tensor rank. The existence of partial degenerations
gives strong upper bounds on the aided rank of a tensor, which allows one to
turn degenerations into restrictions. In particular, we present several
examples, based on the W-tensor and the Coppersmith-Winograd tensors, where
lower bounds on aided rank provide obstructions to the existence of certain
partial degenerations.
