Two-way kernel matrix puncturing: towards resource-efficient PCA and spectral clustering (2102.12293v3)

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 $K=\frac{1}{p}[(X \odot S)^{\sf H} (X \odot S)] \odot B$. 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 GAN-generated image databases, that it is possible to drastically puncture the data, thereby providing possibly huge computational and storage gains, for a virtually constant (clustering of 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.