Principal Component Analysis (PCA) is a widely used technique in machine learning, data analysis and signal processing. With the increase in the size and complexity of datasets, it has become important to develop low-space usage algorithms for PCA. Streaming PCA has gained significant attention in recent years, as it can handle large datasets efficiently. The kernel method, which is commonly used in learning algorithms such as Support Vector Machines (SVMs), has also been applied in PCA algorithms. We propose a streaming algorithm for Kernel PCA problems based on the traditional scheme by Oja. Our algorithm addresses the challenge of reducing the memory usage of PCA while maintaining its accuracy. We analyze the performance of our algorithm by studying the conditions under which it succeeds. Specifically, we show that, when the spectral ratio $R := \lambda_1/\lambda_2$ of the target covariance matrix is lower bounded by $C \cdot \log n\cdot \log d$, the streaming PCA can be solved with $O(d)$ space cost. Our proposed algorithm has several advantages over existing methods. First, it is a streaming algorithm that can handle large datasets efficiently. Second, it employs the kernel method, which allows it to capture complex nonlinear relationships among data points. Third, it has a low-space usage, making it suitable for applications where memory is limited.

翻译：主成分分析（PCA）是机器学习、数据分析和信号处理中广泛使用的技术。随着数据集规模和复杂度的增加，开发低空间占用的PCA算法变得日益重要。近年来，流式PCA因能高效处理大规模数据集而备受关注。核方法作为支持向量机（SVM）等学习算法中的常用技术，也已应用于PCA算法。本文基于Oja的传统方案，提出了一种面向核PCA问题的流式算法。该算法在保持PCA精度的同时，着力解决内存占用降低的挑战。我们通过分析算法的成功条件来评估其性能，具体表明：当目标协方差矩阵的谱比$R := \lambda_1/\lambda_2$满足下界条件$C \cdot \log n\cdot \log d$时，流式PCA可在$O(d)$空间复杂度下求解。与现有方法相比，所提算法具有多重优势：首先，作为流式算法，能够高效处理大规模数据；其次，采用核方法，可捕捉数据点间的复杂非线性关系；此外，其低空间占用特性使其特别适用于内存受限的应用场景。