  The critical locus of the loss function of a neural network is determined by
the geometry of the functional space and by the parameterization of this space
by the network's weights. We introduce a natural distinction between pure
critical points, which only depend on the functional space, and spurious
critical points, which arise from the parameterization. We apply this
perspective to revisit and extend the literature on the loss function of linear
neural networks. For this type of network, the functional space is either the
set of all linear maps from input to output space, or a determinantal variety,
i.e., a set of linear maps with bounded rank. We use geometric properties of
determinantal varieties to derive new results on the landscape of linear
networks with different loss functions and different parameterizations. Our
analysis clearly illustrates that the absence of "bad" local minima in the loss
landscape of linear networks is due to two distinct phenomena that apply in
different settings: it is true for arbitrary smooth convex losses in the case
of architectures that can express all linear maps ("filling architectures") but
it holds only for the quadratic loss when the functional space is a
determinantal variety ("non-filling architectures"). Without any assumption on
the architecture, smooth convex losses may lead to landscapes with many bad
minima.

Machine Learning;Algebraic Geometry;Machine Learning