Two-Way Kernel Matrix Puncturing: Towards Resource-Efficient PCA and Spectral Clustering

Abstract

The article introduces an elementary cost and storage reduction method for spectral clustering and principal component analysis. The method consists in randomly “puncturing” both the data matrix $X\in\mathbb{C}^{p\times n}$ (or $\mathbb{R}^{p\times n}$) and its corresponding kernel (Gram) matrix $K$ through Bernoulli masks: $S\in\{0,1\}^{p\times n}$ for $X$ and $B\in\{0,1\}^{n\times n}$ for $K$. The resulting “two-way punctured” kernel is thus given by $K=\frac1p[(X\odot S)^\H (X\odot S)]\odot B$. We demonstrate that, for $X$ composed of independent columns drawn from a Gaussian mixture model, as $n,p\to\infty$ with $p/n\to c_0\in(0,\infty)$, the spectral behavior of $K$ – its limiting eigenvalue distribution, as well as its isolated eigenvalues and eigenvectors – is fully tractable and exhibits a series of counter-intuitive phenomena. We notably prove, and empirically confirm on various image databases, that it is possible to drastically puncture the data, thereby providing possibly huge computational and storage gains, for a virtually constant (clustering or PCA) performance. This preliminary study opens as such the path towards rethinking, from a large dimensional standpoint, computational and storage costs in elementary machine learning models.