Kernel principal component analysis (kPCA) is a widely studied method to construct a low-dimensional data representation after a nonlinear transformation. The prevailing method to reconstruct the original input signal from kPCA -- an important task for denoising -- requires us to solve a supervised learning problem. In this paper, we present an alternative method where the reconstruction follows naturally from the compression step. We first approximate the kernel with random Fourier features. Then, we exploit the fact that the nonlinear transformation is invertible in a certain subdomain. Hence, the name \emph{invertible kernel PCA (ikPCA)}. We experiment with different data modalities and show that ikPCA performs similarly to kPCA with supervised reconstruction on denoising tasks, making it a strong alternative.

翻译：核主成分分析（kPCA）是一种广泛研究的方法，用于在非线性变换后构建低维数据表示。从kPCA中重建原始输入信号（一项对去噪至关重要的任务）的主流方法需要解决监督学习问题。本文提出了一种替代方法，其中重建过程自然地从压缩步骤中导出。我们首先用随机傅里叶特征近似核函数。然后，我们利用非线性变换在某个子域内可逆这一事实，因此得名“可逆核主成分分析（ikPCA）”。我们在不同数据模态上进行实验，结果表明，在去噪任务上，ikPCA的性能与结合监督重建的kPCA相当，使其成为一种强有力的替代方案。