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

Partial Least Squares (PLS) spectral analysis for multimodal learning is characterized by sharp phase transitions when both data views are subject to independent entry-wise missing-completely-at-random masking. This study investigates how dual missingness attenuates the cross-view spike in a high-dimensional spiked model, revealing that the effective signal strength is reduced by a factor of the joint retention probability, leading to a critical threshold for recovery and providing closed-form asymptotic overlap formulas.

The gist

Dual missingness acts on the PLS-SVD phase diagram as a multiplicative attenuation of the cross-view spike: masking induces an effective spike strength θeff = √ρ θ, shifting the recoverability boundary by 1/√ρ. The resulting model is a spiked rectangular singular-vector problem with a Baik–Ben Arous–Péché (BBP) transition.

Model and Setup

The analysis is conducted in a proportional high-dimensional spiked model where the complete design satisfies the whitened condition, X⊤⋆ X⋆ = NIDx. The response Y⋆ is generated as Y⋆ = θ(X⋆u0)v⊤0 + Z, where Z are i.i.d. Gaussian noise entries with variance 1. Independent entry-wise missing-completely-at-random (MCAR) masks Sx and Sy are applied to X and Y, respectively, with retention probabilities ρx = 1 − mx and ρy = 1 − my, resulting in a joint retention probability ρ = (1 - mx)(1 - my). The observed cross-covariance is analyzed via the rescaled matrix C = (N√ρ)−1X⊤Y.

Phase Transition Prediction

The replica-symmetric analysis predicts a sharp transition at the critical threshold θcrit = 1/((αxαy)1/4√ρ), where αx and αy are the aspect ratios (N/Dx and N/Dy). Below this value, the leading singular vectors have zero asymptotic overlap with the planted directions. Above it, they achieve nontrivial alignment with the latent shared directions. The replica-symmetric prediction yields closed-form asymptotic overlaps for both views:

r2x =





0, αxαy ρ2θ4 ≤ 1,

(8) r2x = αxαyρ2θ4 − 1 / (αy√ρ θ2) (αx√ρ θ2) + 1, αxαy ρ2θ4 > 1.

r2y =





0, αxαy ρ2θ4 ≤ 1,

(9) r2y = αxαyρ2θ4 − 1 / (αx√ρ θ2) (√αy√ρ θ2) + 1, αxαy ρ2θ4 > 1.

Theoretical Derivation

The derivation proceeds by reducing the high-dimensional singular-vector problem to a scalar optimization over overlaps with planted directions using a Gibbs formulation. The effective spike strength is established as θeff = √ρ θ. The replica method is employed to compute the free energy density, which is then analyzed in the zero-temperature limit by introducing susceptibilities χu and χv. Optimizing these susceptibilities leads to the two-parameter objective function Ψ(ru, rv), whose stationarity conditions yield the threshold and the closed-form overlaps (21).

Experimental Validation

The theoretical predictions are validated through extensive Monte Carlo simulations across multiple experimental designs. Synthetic experiments confirm the predicted phase boundary and overlap curves, with a correlation of r = 0.994 between theory and empirical results across various aspect ratios, signal strengths, and missingness levels. Semi-synthetic experiments on real biological datasets like TCGA BRCA and PBMC Multiome show that the transition is robust to the specific geometry of (u0, v0), confirming that only the signal-to-noise ratio θ/θcrit matters. Furthermore, split-half stability serves as a practical diagnostic to identify trustworthy recovery without ground-truth directions.

Practical Implications

The analysis establishes a direct signal-strength penalty imposed by missingness: PLS-SVD requires a factor 1/√ρ more signal than the fully observed case. For instance, 30% dropout per view results in a 1.43× penalty, while 50% dropout yields a 2× penalty. This threshold is crucial for model-based calibration in biomedical settings where missingness may be informative, and it suggests that iterative re-imputation does not improve the observed recovery curve under MCAR conditions. The finite-rank extension conjecture suggests that the single-spike threshold applies componentwise when latent spikes are separated.

Limitations and Robustness

The analysis relies on assumptions such as MCAR masking, a rank-1 latent signal, i.i.d.

As a fastidious researcher, I have thoroughly analyzed this paper on Missing-Data-Induced Phase Transitions in Spectral PLS for Multimodal Learning. The core contribution is establishing a theoretically grounded framework for understanding how missing data in two views of paired high-dimensional data (like multi-omics) affects the performance and reliability of Partial Least Squares (PLS) spectral estimation.

Here are the specific improvements to AI systems that can be made, based directly on the findings and theoretical results presented in this paper:

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1. 】Missing Data Robustness Calibration for Spectral Learning

2. The system can now automatically calibrate its reliance on multi-view inputs by estimating a dynamic signal strength penalty. By calculating the joint retention probability, it can predict how much its performance will degrade due to dropout before even running the full estimation, allowing for proactive resource allocation or data acquisition decisions in real-time (e.g., deciding if a missing data point is likely noise or signal).

3. 】Threshold-Guided Feature Selection for High-Dimensional Data

4. The system can implement a principled signal strength threshold to filter latent variables derived from PLS-SVD. If the estimated signal strength falls below the theoretically predicted critical threshold (derived from aspect ratios and missingness), the system can flag those specific components as unreliable or uninformative, preventing erroneous predictions based on corrupted shared structure.

5. 】Robust Representation Learning in Multi-View Deep Learning

6. The system can be adapted for multimodal deep learning architectures (like contrastive learning or joint embedding networks). By treating the latent representations as the views and the PLS-SVD as a spectral method, it can quantify how missing data affects the alignment of shared features. This allows for training representations that are inherently robust to expected dropout patterns in downstream inference tasks.

7. 】Adaptive Noise Modeling for Uncertainty Quantification

8. The system can move beyond simple Gaussian noise assumptions by incorporating learned noise characteristics (e.g., via Student-t distributions or factor models, as tested in Appendix B). This enables more accurate uncertainty quantification; when the noise structure deviates from i.i.d. Gaussian, the system provides a more honest assessment of its prediction reliability than a standard PLS implementation would allow.

9. 】Inference under Missingness and MAR Mechanisms

10. The system can be designed to explicitly model and mitigate common missing data mechanisms (MCAR, MAR). If the input data is suspected of having signal-dependent missingness (MAR), the system can apply a learned correction factor derived from the phase diagram analysis to restore performance closer to the optimal MCAR recovery boundary.

10.】Optimized Imputation Strategies

10. Instead of relying on simple mean imputation, the system can employ iterative or oracle-based imputation methods guided by PLS-SVD principles. This means it can fill in missing data points in a way that maximizes the resulting cross-view covariance structure, effectively using the spectral estimator itself as an intelligent imputation engine to improve final performance.