Let us consider the deconvolution problem, that is, to recover a latent source $x(\cdot)$ from the observations $\mathbf{y} = [y_1,\ldots,y_N]$ of a convolution process $y = x\star h + \eta$, where $\eta$ is an additive noise, the observations in $\mathbf{y}$ might have missing parts with respect to $y$, and the filter $h$ could be unknown. We propose a novel strategy to address this task when $x$ is a continuous-time signal: we adopt a Gaussian process (GP) prior on the source $x$, which allows for closed-form Bayesian nonparametric deconvolution. We first analyse the direct model to establish the conditions under which the model is well defined. Then, we turn to the inverse problem, where we study i) some necessary conditions under which Bayesian deconvolution is feasible, and ii) to which extent the filter $h$ can be learnt from data or approximated for the blind deconvolution case. The proposed approach, termed Gaussian process deconvolution (GPDC) is compared to other deconvolution methods conceptually, via illustrative examples, and using real-world datasets.

翻译：考虑反卷积问题，即从卷积过程$y = x\star h + \eta$的观测值$\mathbf{y} = [y_1,\ldots,y_N]$中恢复潜在源信号$x(\cdot)$，其中$\eta$为加性噪声，$\mathbf{y}$中的观测值相对$y$可能存在缺失部分，且滤波器$h$可能未知。我们提出一种处理该问题的新策略（适用于$x$为连续时间信号的情形）：对源信号$x$采用高斯过程先验，从而实现闭式贝叶斯非参数反卷积。首先，我们分析正向模型以建立模型良定义的条件；随后转向逆问题，研究：(i)贝叶斯反卷积可行的必要条件；(ii)在盲反卷积情形下，滤波器$h$可从数据中学习或被近似的程度。我们将所提出的高斯过程反卷积方法与其他反卷积方法进行概念比较，通过示例说明及真实数据集验证其性能。