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The article introduces an elementary cost and storage reduction method forspectral clustering and principal component analysis. The method consists inrandomly "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 isthus given by $K=\frac{1}{p}[(X \odot S)^{\sf H} (X \odot S)] \odot B$. Wedemonstrate that, for $X$ composed of independent columns drawn from a Gaussianmixture model, as $n,p\to\infty$ with $p/n\to c_0\in(0,\infty)$, the spectralbehavior of $K$ -- its limiting eigenvalue distribution, as well as itsisolated eigenvalues and eigenvectors -- is fully tractable and exhibits aseries of counter-intuitive phenomena. We notably prove, and empiricallyconfirm on GAN-generated image databases, that it is possible to drasticallypuncture the data, thereby providing possibly huge computational and storagegains, for a virtually constant (clustering of PCA) performance. Thispreliminary study opens as such the path towards rethinking, from a largedimensional standpoint, computational and storage costs in elementary machinelearning models.

Two-way kernel matrix puncturing: towards resource-efficient PCA and spectral clustering