High-dimensional Spectrum Matching: Covariance Inference Based on Universality

We propose a new spectrum matching framework based on universality for high-dimensional covariance inference. The method constructs reference distributions for a broad class of spectral statistics by generating an auxiliary Gaussian matrix whose covariance spectrum matches that of the target covariance, with validity justified by universality. This framework accommodates both known and unknown covariance settings and adapts to different inferential targets, including covariance testing, spike detection, and simultaneous inference for linear functionals of the covariance matrix. Theoretical guarantees are established through anisotropic local laws for general covariance-type random matrices beyond the proportional-growth regime. Numerical studies and a real-data application demonstrate the practical effectiveness of the proposed methodology in high-dimensional settings.