Spectral Thresholds in Correlated Spiked Models and Fundamental Limits of Partial Least Squares

We study a fundamental problem in machine learning: partial recovery of a planted signal $\boldsymbol{v} \in \mathbb{S}^{n-1}$ from a spiked tensor $\boldsymbol{\mathsf{X}} = λ\, \boldsymbol{v}^{\otimes p} + \boldsymbol{\mathsf{Z}}$, where $\boldsymbol{\mathsf{Z}}$ has symmetric Gaussian entries of variance $\asymp n^{-1}$. While previous works replace $\boldsymbol{\mathsf{Z}}$ by an i.i.d. standard Gaussian tensor to circumvent technical difficulties caused by dependencies, this paper directly analyzes the symmetric model via random matrix theory. We show that the square unfolding algorithm is subject to a sharp phase transition for partial recovery at $λ\asymp n^{(p-2) / 4}$, validating an open conjecture from Montanari and Richard (2014). We get precise expressions for the limiting spectral distribution, deterministic equivalent, phase transition threshold and overlap. The main message is that unfolded symmetric tensors spectrally behave like standard Wigner matrices, up to simple transformations accounting for high-order symmetries. We consider extensions to multiple orthogonal spikes, recursive unfolding and non-square unfolding via disproportionate random matrix theory, leading to an interesting conjectured statistical-computational tradeoff regarding the choice of unfolding: square versus long.

We study distinguishability and recovery of a sparse rank-one signal in a spiked Wigner model with independently sparse noise through a variety of algorithms and parameter regimes. The signal and noise have separate sparsity parameters, while the signal-to-noise ratio (SNR) remains of constant order as the matrix size grows. For sub-Gaussian noise and spikes whose nonzero coordinates are bounded away from zero, a leave-one-out quadratic thresholding procedure (a generalization of Ding et al., 2024) identifies the support. Sign synchronization yields exact recovery for a Rademacher spike, and spectral refinement yields strong recovery. The quadratic thresholding search parameter gives an explicit tradeoff between signal sparsity, noise sparsity, and computational cost. We also study extensions to sub-Gaussian spike priors through diagonal and quadratic thresholding. For Rademacher spikes with continuous nonzero noise, we develop an identical search algorithm that exploits unblurred signal entries, the existence of which permits distinguishability; connectivity of the graph they reveal provides a sufficient condition for exact recovery. These guarantees do not require moment assumptions on the continuous noise distribution. Together with our previous PCA results (Dumitriu et al., 2026), our analysis covers almost the entire space of parameters and regimes for distinguishing and recovering in our doubly sparse spiked Wigner model.

Principal component analysis (PCA) is a fundamental tool to reduce the dimensionality of the data in many applications. PCA finds a few signal directions that contain most of the variability of the data by computing the eigenvectors of the sample covariance matrix. In this work, we focus on the spiked covariance model, in which the data vectors are defined by a few orthogonal signals plus an isotropic Gaussian noise, and our goal is to estimate one or more of the leading signals. Our main theoretical finding is that the subspace spanned by several leading eigenvectors of the sample covariance matrix contains significant information about the desired signals long before the individual eigenvectors converge to the population principal components. To prove this, we derive a posteriori bounds for the angle between the subspace spanned by the desired population signals and the subspace obtained from the sample using perturbation theory for singular vectors. This leads to a new algorithm, SuperPCA (SUbsPace subsamplER PCA), which capitalizes on an approximate eigenspace of the sample covariance matrix to find the leading signals far more efficiently and accurately than classical PCA in the high-dimensional, multi-signal setting. SuperPCA exploits only a small number of subsampled coordinates of the data, which can lead to tremendous savings in data acquisition cost, especially when the signals are approximately sparse. For the same number of measurements, SuperPCA can offer a factor $10$ improvement in accuracy compared to the classical PCA method.