AI 中文总结
本文引入函数在某点的余弦测度新概念，围绕其展开数学理论研究，给出计算该测度的算法，相关结果为无穷集合与非凸锥的余弦测度提供了新视角。
AI 中文摘要
$\boldsymbol{\text{R}}^n$中一组向量的余弦测度用于衡量该组向量对$\boldsymbol{\text{R}}^n$中所有方向的覆盖程度，它能识别出与该组向量夹角最远的方向，既应用于多种优化算法的收敛理论，也能凸显集合的有趣几何性质。例如，集合$S$的余弦测度大于零当且仅当：对任意在点$\boldsymbol{x}$处梯度非零的$\boldsymbol{\text{C}}^1$函数$f$，$S$必包含$f$在$\boldsymbol{x}$处的一个下降方向。本文研究函数$f$不可微或梯度为零向量时的相关问题，为此引入函数在某点的余弦测度这一新概念，该值给出了一组向量需满足的余弦测度的下确界，以保证其包含函数在目标点处的下降方向。本文围绕该概念展开数学理论研究，包括示例表明光滑函数的余弦测度可取值于$[-1,1]$，还给出计算函数余弦测度的算法，并展示该算法在光滑与非光滑函数上的应用示例，这些结果也为无穷集合与非凸锥的余弦测度提供了新的视角。
英文摘要
The cosine measure of a set of vectors in $\mathbb{R}^n$ measures how well the set covers all directions in $\mathbb{R}^n$. It identifies the direction furthest, in angle, from the set. It is used in the convergence theory of various optimization algorithms, but also highlights interesting geometric properties of sets. For example, the cosine measure of a set $S$ is greater than zero if and only if given any $\mathcal{C}^1$ function $f$ at a point $\mathbf{x}$ where the gradient is nonzero, $S$ must contain a descent direction of $f$ at $\mathbf{x}$. In this paper, we examine the question of what can be said when the function $f$ is non-differentiable or if it has a gradient equal to the zero vector. To examine these cases, we introduce the novel concept of the {\em cosine measure of a function} at a point. This value provides an infimum on the value of the cosine measure that a set of vectors requires to guarantee it contains a descent direction of the function at the point of interest. We present mathematical theory around this concept, including examples showing that the cosine measure of a smooth function can have any value in $[-1,1]$. We further present algorithms to compute the cosine measure of a function, and examples demonstrating the algorithm on smooth and nonsmooth functions. These results also shed light on the the cosine measure of infinite sets and nonconvex cones.
