Optimal Transfer Learning for Missing Not-at-Random Matrix Completion
This addresses a critical data limitation in biological applications where missing data patterns prevent standard matrix completion, though it is incremental in improving estimation methods for specific MNAR scenarios.
The paper tackles the problem of matrix completion with missing not-at-random data, where entire rows and columns are missing, by using transfer learning from a noisy source matrix and achieves minimax optimal estimation error in an active sampling setting.
We study transfer learning for matrix completion in a Missing Not-at-Random (MNAR) setting that is motivated by biological problems. The target matrix $Q$ has entire rows and columns missing, making estimation impossible without side information. To address this, we use a noisy and incomplete source matrix $P$, which relates to $Q$ via a feature shift in latent space. We consider both the active and passive sampling of rows and columns. We establish minimax lower bounds for entrywise estimation error in each setting. Our computationally efficient estimation framework achieves this lower bound for the active setting, which leverages the source data to query the most informative rows and columns of $Q$. This avoids the need for incoherence assumptions required for rate optimality in the passive sampling setting. We demonstrate the effectiveness of our approach through comparisons with existing algorithms on real-world biological datasets.