  While it is well known that nonlinear methods of approximation can often
perform dramatically better than linear methods, there are still questions on
how to measure the optimal performance possible for such methods. This paper
studies nonlinear methods of approximation that are compatible with numerical
implementation in that they are required to be numerically stable. A measure of
optimal performance, called {\em stable manifold widths}, for approximating a
model class $K$ in a Banach space $X$ by stable manifold methods is introduced.
Fundamental inequalities between these stable manifold widths and the entropy
of $K$ are established. The effects of requiring stability in the settings of
deep learning and compressed sensing are discussed.
