论文标题
最佳稳定非线性近似
Optimal Stable Nonlinear Approximation
论文作者
论文摘要
众所周知，近似的非线性方法通常可以比线性方法更好地执行，但仍然存在有关如何测量此类方法可能的最佳性能的问题。本文研究了与数值实现兼容的非线性近似方法，因为它们必须在数值上稳定。引入了一种最佳性能，称为{\ em稳定的流形宽度}，用于通过稳定的歧管方法近似Banach Space $ x $中的模型类$ k $。这些稳定的歧管宽度与$ K $的熵之间的根本不平等现象。讨论了在深度学习和压缩感测的设置中需要稳定性的影响。
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.
