Khái niệm cốt lõi
Kernel packets (KPs) provide a general framework to construct compactly supported basis functions for Gaussian processes (GPs) driven by stochastic differential equations (SDEs), enabling efficient training and prediction of GP models.
Tóm tắt
The paper presents a general theory for constructing kernel packets (KPs) - a set of compactly supported basis functions - for Gaussian processes (GPs) driven by stochastic differential equations (SDEs).
Key highlights:
- The authors prove that KPs generally exist for GPs defined by SDEs and provide a framework to obtain them.
- KPs are derived from the forward and backward Markov properties of state-space models, in contrast to previous work that used harmonic analysis.
- The minimum number of equations required to construct a minimal KP system is shown to be 2m+1, where m is the order of the SDE.
- The KP basis functions are proven to be linearly independent and can be used to achieve O(n) training time and O(log n) or O(1) prediction time for GP regression.
- The KP framework is extended to handle combined kernels formed by addition and multiplication of individual kernels.
- Examples are provided for the Matérn-3/2 and integrated Brownian motion kernels to illustrate the KP construction.
The proposed KP theory provides a general and efficient approach for GP modeling and inference, with applications in various domains.