Missing-Data-Induced Phase Transitions in Spectral Partial Least Squares

Abstract

Spectral partial least squares (PLS-SVD) estimates the directions shared by two views of the same samples. We analyze it using a rank-one regression model, with entries of both views missing completely at random and filled with zeros. Missing response entries weaken the signal, whereas missing design entries also tilt the recovered direction. The setting is the proportional regime, with Gaussian response noise and a design that is column-orthogonal before masking. For incoherent designs and planted directions, we prove that as the signal diverges, the squared overlap with the truth approaches $1$ on the response side but only a ceiling on the design side. This ceiling is strictly below $1$ whenever design entries are missing, unless samples of vanishing leverage carry the signal. For this estimator, a stronger signal therefore cannot remove the design-side error. Under equal leverage and an explicit spectral condition on the masked design, which we prove for designs taken from nested Hadamard matrices, we establish a phase transition and derive the recovery curves. Under the same conditions, the critical signal strength is exactly proportional to $\rho_y^{-1/2}$, with $\rho_y$ the fraction of response entries retained. A heuristic replica derivation reproduces the equations behind these curves, and simulations follow them, including on random orthogonal designs outside the verified class. In semi-synthetic experiments on real biological designs with unequal leverage, the predicted ceiling varies across directions, and high-signal recovery tracks it. These results bring the theory of PLS-SVD closer to what is known for PCA under homogeneous missingness and show that the loss of signal-to-noise ratio observed there does not, by itself, describe missing design entries.