A zonal function (ZF) network on the $q$ dimensional sphere $mathbb{S}^q$ isa network of the form $mathbf{x}mapsto sum_{k=1}^na_kphi(mathbf{x}cdotmathbf{x}_k)$ where $phi :[-1,1]omathbf{R}$ is theactivation function, $mathbf{x}_kinmathbb{S}^q$ are the centers, and$a_kinmathbb{R}$. While the approximation properties of such networks arewell studied in the context of positive definite activation functions, recentinterest in deep and shallow networks motivate the study of activationfunctions of the form $phi(t)=|t|$, which are not positive definite. In thispaper, we define an appropriate smoothess class and establish approximationproperties of such networks for functions in this class. The centers can bechosen independently of the target function, and the coefficients are linearcombinations of the training data. The constructions preserve rotationalsymmetries.
展开▼
机译：$ q $ diperional sphere $ mathbb {s} ^ $ isa网络上的Zonal函数（zf）网络$ mathbf {x} mapsto sum_ {k = 1} ^ na_k phi（ mathbf {x} cdot mathbf {x} _k）$ where $ phi：[ - 1,1] to mathbf {r} $ the mathbf {r} $ thectivation函数，$ mathbf {x} _k in mathbb {s} ^ $是中心，而$ a_k in mathbb {r} $。虽然这种网络的近似特性在正定的激活功能的上下文中研究，但深度和浅网络中的最新兴趣激励了对 phi（t）= | t |的活化功能的研究，这不是正定的。在此纸纸中，我们定义了适当的平滑类，并在此类中建立这种网络的近似值。该中心可以独立于目标函数强调，系数是训练数据的线性键。结构保持旋转态度。
展开▼
