# Approximate Kernel PCA Using Random Features: Computational vs. Statistical Trade-off

arXiv: https://arxiv.org/abs/1706.06296 · HTML: https://mlsanity.com/paper/1706.06296/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jun 2017.

## Built on by (most cited first, 8 of 8)

- [Contrastive Learning Can Find An Optimal Basis For Approximately View-Invariant Functions](https://mlsanity.com/paper/2210.01883/) (2022)
- [HyperFast: Instant Classification for Tabular Data](https://mlsanity.com/paper/2402.14335/) (2024)
- [Spectral Regularized Kernel Two-Sample Tests](https://mlsanity.com/paper/2212.09201/) (2022)
- [Statistical Optimality and Computational Efficiency of Nystr"om Kernel PCA](https://mlsanity.com/paper/2105.08875/) (2021)
- [Random Fourier Signature Features](https://mlsanity.com/paper/2311.12214/) (2023)
- [Statistical and Computational Trade-offs in Variational Inference: A Case Study in Inferential Model Selection](https://mlsanity.com/paper/2207.11208/) (2022)
- [Spectral Regularized Kernel Goodness-of-Fit Tests](https://mlsanity.com/paper/2308.04561/) (2023)
- [Kernel PCA with the Nystr"om method](https://mlsanity.com/paper/2109.05578/) (2021)

## Abstract (arXiv)

Kernel methods are powerful learning methodologies that allow to perform non-linear data analysis. Despite their popularity, they suffer from poor scalability in big data scenarios. Various approximation methods, including random feature approximation, have been proposed to alleviate the problem. However, the statistical consistency of most of these approximate kernel methods is not well understood except for kernel ridge regression wherein it has been shown that the random feature approximation is not only computationally efficient but also statistically consistent with a minimax optimal rate of convergence. In this paper, we investigate the efficacy of random feature approximation in the context of kernel principal component analysis (KPCA) by studying the trade-off between computational and statistical behaviors of approximate KPCA. We show that the approximate KPCA is both computationally and statistically efficient compared to KPCA in terms of the error associated with reconstructing a kernel function based on its projection onto the corresponding eigenspaces. The analysis hinges on Bernstein-type inequalities for the operator and Hilbert-Schmidt norms of a self-adjoint Hilbert-Schmidt operator-valued U-statistics, which are of independent interest.