具有复谱结构的非光滑信号建模
Modelling Non-Smooth Signals with Complex Spectral Structure
摘要：
高斯过程卷积模型（GPCM; Tobar等人，2015 a）是具有复杂频谱结构的信号的模型。GPCM的一个重要限制是它假设了一个快速衰减的频谱：它只能模拟平滑信号。此外，GPCM中的推断目前需要（1）平均场假设，导致校准不良的不确定性，以及（2）对大的协方差阵进行繁琐的变分优化。我们重新设计的GPCM模型，以诱导一个更丰富的分布在频谱上与宽松的假设平滑度：因果高斯过程卷积模型（CGPCM）引入了因果关系的假设GPCM，和粗糙高斯过程卷积模型（RGPCM）可以被解释为贝叶斯非参数概括的分数奥恩斯坦-乌伦贝克过程。我们还提出了一个更有效的变分推断计划，超越了平均场假设：我们设计了一个吉布斯采样器，直接从最佳变分解的样本，完全规避任何变分优化。在合成和真实世界的数据实验中，GPCM的建议的变化进行了验证，显示出良好的效果。
英文摘要：
The Gaussian Process Convolution Model (GPCM; Tobar et al., 2015a) is a model for signals with complex spectral structure. A significant limitation of the GPCM is that it assumes a rapidly decaying spectrum: it can only model smooth signals. Moreover, inference in the GPCM currently requires (1) a mean-field assumption, resulting in poorly calibrated uncertainties, and (2) a tedious variational optimisation of large covariance matrices. We redesign the GPCM model to induce a richer distribution over the spectrum with relaxed assumptions about smoothness: the Causal Gaussian Process Convolution Model (CGPCM) introduces a causality assumption into the GPCM, and the Rough Gaussian Process Convolution Model (RGPCM) can be interpreted as a Bayesian nonparametric generalisation of the fractional Ornstein-Uhlenbeck process. We also propose a more effective variational inference scheme, going beyond the mean-field assumption: we design a Gibbs sampler which directly samples from the optimal variational solution, circumventing any variational optimisation entirely. The proposed variations of the GPCM are validated in experiments on synthetic and real-world data, showing promising results.
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