Spectral mixture (SM) kernels comprise a powerful class of generalized kernels for Gaussian processes (GPs) to describe complex patterns. This paper introduces model compression and time- and phase (TP) modulated dependency structures to the original (SM) kernel for improved generalization of GPs. Specifically, by adopting Bienaym\'es identity, we generalize the dependency structure through cross-covariance between the SM components. Then, we propose a novel SM kernel with a dependency structure (SMD) by using cross-convolution between the SM components. Furthermore, we ameliorate the expressiveness of the dependency structure by parameterizing it with time and phase delays. The dependency structure has clear interpretations in terms of spectral density, covariance behavior, and sampling path. To enrich the SMD with effective hyperparameter initialization, compressible SM kernel components, and sparse dependency structures, we introduce a novel structure adaptation (SA) algorithm in the end. A thorough comparative analysis of the SMD on both synthetic and real-life applications corroborates its efficacy.

翻译：谱混合（SM）核是一类强大的广义核函数，用于高斯过程（GP）描述复杂模式。本文通过引入模型压缩以及时间-相位（TP）调制依赖结构，对原始SM核进行改进以提升GP的泛化能力。具体地，利用Bienaymé恒等式，我们通过SM分量间的互协方差推广了依赖结构。随后，通过SM分量间的互卷积，提出了一种具有依赖结构的新型SM核（SMD）。此外，通过引入时间延迟和相位延迟参数化该依赖结构，进一步增强了其表达能力。该依赖结构在谱密度、协方差行为及采样路径方面具有清晰的物理解释。为赋予SMD有效的超参数初始化、可压缩SM核分量及稀疏依赖结构，我们最终引入了一种新颖的结构自适应（SA）算法。在合成数据与真实应用场景中的全面对比分析验证了SMD方法的有效性。