Spectral Phase Transition and Optimal PCA in Block-Structured Spiked models

The statistical challenge of inferring a low-dimensional signal from a noisy, high-dimensional observation is ubiquitous across statistics, probability, and machine learning. Spiked random matrix models have recently gained extensive interest, serving as a valuable platform for exploring this issue [30, 51, 42]. A prominent example is the spiked Wigner model, where a rank one matrix is observed through a component-wise homogeneous noise, that has been studied extensively in random matrix theory [10]. Most models, with the spiked Wigner model at the forefront, have focused however on scenarios where the noise is "homogeneous", aiming to understand how the performance of the inference depends on the noise level. Yet in practice, datasets are inherently structured and the exploration of inhomogeneity plays a pivotal role in unraveling their complexities. A prototypical model to study this phenomenon is to improve the aforementioned spiked Wigner model by introducing a block structure in the noise, a model which has been recently introduced in a series of papers [17, 5, 7, 34] and that arises in many different learning contexts such as community detection [17, 34], deep Boltzmann machines [6], or the dense limit of the celebrated degree-corrected stochastic block model [34, 39]. Our goal in this paper is to apply rigorous random matrix theory to such "inhomogenous" spiked models, and to provide an optimal reconstruction method from a spectral algorithm, to generalize the seminal work of [10] (BBP) to inhomogenous matrices.