It is well known that the state space (SS) model formulation of a Gaussian process (GP) can lower its training and prediction time both to O(n) for n data points. We prove that an $m$-dimensional SS model formulation of GP is equivalent to a concept we introduce as the general right Kernel Packet (KP): a transformation for the GP covariance function $K$ such that $\sum_{i=0}^{m}a_iD_t^{(j)}K(t,t_i)=0$ holds for any $t \leq t_1$, 0 $\leq j \leq m-1$, and $m+1$ consecutive points $t_i$, where ${D}_t^{(j)}f(t) $ denotes $j$-th order derivative acting on $t$. We extend this idea to the backward SS model formulation of the GP, leading to the concept of the left KP for next $m$ consecutive points: $\sum_{i=0}^{m}b_i{D}_t^{(j)}K(t,t_{m+i})=0$ for any $t\geq t_{2m}$. By combining both left and right KPs, we can prove that a suitable linear combination of these covariance functions yields $m$ compactly supported KP functions: $\phi^{(j)}(t)=0$ for any $t\not\in(t_0,t_{2m})$ and $j=0,\cdots,m-1$. KPs further reduces the prediction time of GP to O(log n) or even O(1) and can be applied to more general problems involving the derivative of GPs.

翻译：众所周知，高斯过程（GP）的状态空间（SS）模型公式可将其训练和预测时间均降低至$O(n)$（其中$n$为数据点数）。我们证明，GP的$m$维SS模型公式等价于我们引入的广义右核包（KP）概念：对GP协方差函数$K$的变换，使得对任意$t \leq t_1$、$0 \leq j \leq m-1$以及$m+1$个连续点$t_i$，均满足$\sum_{i=0}^{m}a_iD_t^{(j)}K(t,t_i)=0$，其中${D}_t^{(j)}f(t)$表示对$t$求$j$阶导数。我们将此思想拓展至GP的后向SS模型公式，从而引出左KP概念（针对后续$m$个连续点）：对任意$t\geq t_{2m}$，有$\sum_{i=0}^{m}b_i{D}_t^{(j)}K(t,t_{m+i})=0$。通过结合左、右KP，可证明这些协方差函数的适当线性组合能生成$m个$紧支撑KP函数：对任意$t\not\in(t_0,t_{2m})$和$j=0,\cdots,m-1$，有$\phi^{(j)}(t)=0$。KP进一步将GP的预测时间降低至$O(\log n)$甚至$O(1)$，并可应用于涉及GP导数的更广泛问题。