Spectral Estimation with Deformed Decompression

Sample covariance matrices are fundamental objects in machine learning and statistics, with valuable information encoded into their eigenvalues. When treating modern problems, the scale of these matrices can become prohibitive for tractable computation. This creates a critical need for "spectral decompression": a method for inferring the eigenspectrum of a very large-scale model using only information from its smaller realizations. Existing methods fall into two main categories: those relying on spectral inversion; and those that rely on Free Decompression techniques. On one hand, spectral inversion is a fundamentally ill-posed problem that is notoriously unstable, often failing to yield reliable results in finite-size settings. On the other hand, while Free Decompression was recently proposed to address this scaling, its reliance on the Nica-Speicher framework restricts its use to unitarily invariant ensembles, a symmetry condition that subsampled covariance matrices fail to satisfy. In this work, we introduce Deformed Decompression (DD), a novel framework for the stable, forward-extrapolation of spectral densities. By leveraging properties of the companion matrix, DD recasts the spectral decompression problem as a directed evolution of the Stieltjes transform of the companion. This approach bypasses the instabilities of traditional inversion and provides a robust methodology for scaling covariance matrices and even their generalized variants (including Hessian matrices) that lack unitary invariance. We demonstrate the utility of our framework through experiments on a variety of random matrix examples, showing that DD successfully captures the complex spectral signatures of large-scale systems.