A General Theory for Kernel Packets: from state space model to compactly supported basis

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.