论文标题
下降模量和应用
Descent modulus and applications
论文作者
论文摘要
梯度$ \ nabla $ f（x）的标准测量了x时实现的平滑函数f的最大下降。对于（非滑动）凸函数，这是由距离距离的距离dist（0，$ \ partial $ f（x））表达的，而对于原始空间定义的一般真实价值函数，这是由公制斜率| $ \ nabla $ f |（x）所定义的。在这项工作中，我们提出了在每个点x下的下降模量t [f]（x）的公理定义，该函数f在一般（不一定是度量）空间上定义。该定义包括上述所有实例，以及在概率空间上定义的函数的平均下降。我们表明，大量的功能完全取决于其下降模量和相应的临界值。在平滑的情况下，这个结果已经令人惊讶：一维信息（梯度的规范）几乎与对完整梯度映射的知识一样强大。在非平滑情况下，确定结果的关键要素是凭着公制斜率的定义，由下坡方向引起的对称性的破裂。最后一节研究了在有限空间上定义的功能的特定情况。在这种情况下，我们获得了典型的下降算子的明确分类。
The norm of the gradient $\nabla$f (x) measures the maximum descent of a real-valued smooth function f at x. For (nonsmooth) convex functions, this is expressed by the distance dist(0, $\partial$f (x)) of the subdifferential to the origin, while for general real-valued functions defined on metric spaces by the notion of metric slope |$\nabla$f |(x). In this work we propose an axiomatic definition of descent modulus T [f ](x) of a real-valued function f at every point x, defined on a general (not necessarily metric) space. The definition encompasses all above instances as well as average descents for functions defined on probability spaces. We show that a large class of functions are completely determined by their descent modulus and corresponding critical values. This result is already surprising in the smooth case: a one-dimensional information (norm of the gradient) turns out to be almost as powerful as the knowledge of the full gradient mapping. In the nonsmooth case, the key element for this determination result is the break of symmetry induced by a downhill orientation, in the spirit of the definition of the metric slope. The particular case of functions defined on finite spaces is studied in the last section. In this case, we obtain an explicit classification of descent operators that are, in some sense, typical.
