Missing-Data-Induced Phase Transitions in Spectral PLS for Multimodal Learning

Partial Least Squares (PLS) learns shared structure from paired data via the top singular vectors of the empirical cross-covariance (PLS-SVD), but multimodal datasets often have missing entries in both views. We study PLS-SVD under independent entry-wise missing-completely-at-random masking in a proportional high-dimensional spiked model. After appropriate normalization, the masked cross-covariance behaves like a spiked rectangular random matrix whose effective signal strength is attenuated by $\sqrtρ$, where $ρ$ is the joint entry retention probability. As a result, PLS-SVD exhibits a sharp BBP-type phase transition: below a critical signal-to-noise threshold the leading singular vectors are asymptotically uninformative, while above it they achieve nontrivial alignment with the latent shared directions, with closed-form asymptotic overlap formulas. Simulations and semi-synthetic multimodal experiments corroborate the predicted phase diagram and recovery curves across aspect ratios, signal strengths, and missingness levels.

💡 Research Summary

This paper investigates the behavior of Partial Least Squares (PLS) when both views of a multimodal dataset are subject to independent entry‑wise missing‑completely‑at‑random (MCAR) masking. The authors consider a proportional high‑dimensional spiked model: a whitened design matrix X*∈ℝ^{N×D_x} with XᵀX = N I_{D_x}, a latent shared direction (u₀, v₀) of unit norm, and a response Y* = θ (X* u₀) v₀ᵀ + Z where Z is i.i.d. Gaussian noise. Independent Bernoulli masks S_x and S_y with retention probabilities ρ_x = 1−m_x and ρ_y = 1−m_y are applied, yielding observed matrices X = S_x⊙X* and Y = S_y⊙Y*. The joint retention probability is ρ = ρ_x ρ_y.

PLS‑SVD computes the leading singular vectors of the empirical cross‑covariance Σ̂_{XY}=N^{-1}XᵀY. To compensate for the average loss of entries, the authors normalize the matrix by 1/√ρ, defining C = (1/√ρ) Σ̂_{XY}. They prove (Lemma 3.1) that, in the proportional limit (N, D_x, D_y → ∞ with fixed aspect ratios α_x = N/D_x and α_y = N/D_y), C converges in distribution to a spiked rectangular random matrix C = θ_eff u₀ v₀ᵀ + N^{-1/2}W, where W has i.i.d. N(0,1) entries and the effective spike strength is attenuated by the joint retention probability: θ_eff = √ρ θ.

This reduction shows that missingness does not merely reduce the effective sample size; it scales down the signal‑to‑noise ratio by √ρ. Consequently, the problem becomes identical to the classic spiked rectangular model, for which Baik–Ben Arous–Péché (BBP) phase transition results apply. The authors derive the critical signal strength required for the top singular value to separate from the bulk: θ_crit = 1 /