Generalization Bounds for Inductive Matrix Completion in Low-noise Settings
This work provides theoretical guarantees for matrix completion with side information, addressing a foundational problem in machine learning, though it is incremental as it builds on existing literature.
The paper tackles inductive matrix completion in low-noise settings, deriving generalization bounds that scale with noise standard deviation, converge to zero with infinite samples, and have logarithmic dependence on matrix size for fixed side information dimensions.
We study inductive matrix completion (matrix completion with side information) under an i.i.d. subgaussian noise assumption at a low noise regime, with uniform sampling of the entries. We obtain for the first time generalization bounds with the following three properties: (1) they scale like the standard deviation of the noise and in particular approach zero in the exact recovery case; (2) even in the presence of noise, they converge to zero when the sample size approaches infinity; and (3) for a fixed dimension of the side information, they only have a logarithmic dependence on the size of the matrix. Differently from many works in approximate recovery, we present results both for bounded Lipschitz losses and for the absolute loss, with the latter relying on Talagrand-type inequalities. The proofs create a bridge between two approaches to the theoretical analysis of matrix completion, since they consist in a combination of techniques from both the exact recovery literature and the approximate recovery literature.