  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.
