At the core of scientific computing and much of modern machine learning (ML) lies the challenge of estimating the eigenvalues of high-dimensional Hermitian matrices. Such matrices, including kernels, Hessians, and graph representations, encode the intrinsic geometry and connectivity of the data and models built on them, rendering the pursuit of efficient spectral techniques a primary concern for both theory and practice. Studying eigenspectra has become a prominent approach to understanding performance and guiding training in deep learning [10, 20, 36, 53]. In many cases, the spectra of such matrices have non-trivial structure, often containing spikes, multiple multi-modal bulks, and heavy-tails [14, 25]. Conventional algorithms to extract eigenvalue information from these matrices have required that the data are able to be stored in memory, scratch space, or can at least be accessed as an implicit operator (via matrix-vector products). More recently, a new class of algorithms has emerged that is able to provide highly-accurate estimates of the eigenvalues (or summary functionals thereof [2]) of matrices, even without implicit or explicit access to the full matrix, i.e., of so-called impalpable matrices [1]. One such method, termed Free Decompression (FD), shows great promise as a tool for gaining access to the spectral distributions of such impalpable matrices. The central premise is that by appropriately sampling a small sub-matrix from the large impalpable matrix of interest, one can evolve a partial differential equation (PDE) in the Stieltjes transform of a spectral density in the decompression ratio to the desired matrix dimension.

We consider the problem of parameter estimation from a generalized linear model with a random design matrix that is orthogonally invariant in law. Such a model allows the design have an arbitrary distribution of singular values and only assumes that its singular vectors are generic. It is a vast generalization of the i.i.d. Gaussian design typically considered in the theoretical literature, and is motivated by the fact that real data often have a complex correlation structure so that methods relying on i.i.d. assumptions can be highly suboptimal. Building on the paradigm of spectrally-initialized iterative optimization, this paper proposes optimal spectral estimators and combines them with an approximate message passing (AMP) algorithm, establishing rigorous performance guarantees for these two algorithmic steps. Both the spectral initialization and the subsequent AMP meet existing conjectures on the fundamental limits to estimation -- the former on the optimal sample complexity for efficient weak recovery, and the latter on the optimal errors. Numerical experiments suggest the effectiveness of our methods and accuracy of our theory beyond orthogonally invariant data.

Alternatively, the spectral kernels are defined in the spectral domain [2,13,14]. Nonetheless, when modeling graph-structured data, prior kernels face severalchallenges.

Partial Least Squares (PLS) is a widely used method for data integration, designed to extract latent components shared across paired high-dimensional datasets. Despite decades of practical success, a precise theoretical understanding of its behavior in high-dimensional regimes remains limited. In this paper, we study a data integration model in which two high-dimensional data matrices share a low-rank common latent structure while also containing individual-specific components. We analyze the singular vectors of the associated cross-covariance matrix using tools from random matrix theory and derive asymptotic characterizations of the alignment between estimated and true latent directions. These results provide a quantitative explanation of the reconstruction performance of the PLS variant based on Singular Value Decomposition (PLS-SVD) and identify regimes where the method exhibits counter-intuitive or limiting behavior. Building on this analysis, we compare PLS-SVD with principal component analysis applied separately to each dataset and show its asymptotic superiority in detecting the common latent subspace. Overall, our results offer a comprehensive theoretical understanding of high-dimensional PLS-SVD, clarifying both its advantages and fundamental limitations.

Here we study the performance of iterative Hessian sketch for least-squares problems.