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 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.

虽然众所周知，非线性逼近方法的性能通常比线性方法要好得多，但如何衡量此类方法的最佳性能仍然存在问题。本文研究了与数值实现兼容的非线性逼近方法，因为它们需要数值稳定。引入了一种最优性能度量，称为稳定流形宽度，用于通过稳定流形方法逼近Banach 空间X 中的模型类K。建立了这些稳定流形宽度和K的熵之间的基本不等式。讨论了在深度学习和压缩感知的设置中要求稳定性的影响。