Beyond Imputation: Rethinking Matrix Multiplication under Missing Data

Matrix multiplication is a fundamental operation behind machine learning, yet the usual operator becomes undefined when either operand contains missing entries. Existing solutions typically rely on task-specific adaptations or imputation-based pipelines, both of which can introduce additional complexity, computational costs, and task-specific development efforts. To address these limitations, we introduce Missing-aware Matrix Multiplication, denoted M${}^3$, which estimates unavailable multiplicative contributions directly through conditional moments. Under a pair-local Gaussian working model, four observation patterns admit closed-form contributions, while marginal moments exploit all available observations and estimate dependence only from overlapping entries. We prove pair-local oracle optimality and show that exploiting cross-factor dependence strictly reduces mean-squared error relative to mean imputation. Across direct product recovery, Gram and PCA recovery, cross-view spectral estimation, and recommendation tasks, our method generally improves effectiveness while remaining computationally efficient. These findings suggest that M${}^3$ provides a promising and efficient approach to matrix multiplication in the presence of missing data.