  We develop new tools to study landscapes in nonconvex optimization. Given one
optimization problem, we pair it with another by smoothly parametrizing the
domain. This is either for practical purposes (e.g., to use smooth optimization
algorithms with good guarantees) or for theoretical purposes (e.g., to reveal
that the landscape satisfies a strict saddle property). In both cases, the
central question is: how do the landscapes of the two problems relate? More
precisely: how do desirable points such as local minima and critical points in
one problem relate to those in the other problem? A key finding in this paper
is that these relations are often determined by the parametrization itself, and
are almost entirely independent of the cost function. Accordingly, we introduce
a general framework to study parametrizations by their effect on landscapes.
The framework enables us to obtain new guarantees for an array of problems,
some of which were previously treated on a case-by-case basis in the
literature. Applications include: optimizing low-rank matrices and tensors
through factorizations; solving semidefinite programs via the Burer-Monteiro
approach; training neural networks by optimizing their weights and biases; and
quotienting out symmetries.
