Fast matrix completion without the condition number
This addresses a key bottleneck in matrix completion for applications like recommendation systems by reducing computational costs, though it is incremental as it builds on Alternating Minimization.
The paper tackles the problem of matrix completion by introducing the first algorithm with polynomial time and sample complexity in rank, linear in dimension, and logarithmic in condition number, improving over prior methods that had quadratic dependencies.
We give the first algorithm for Matrix Completion whose running time and sample complexity is polynomial in the rank of the unknown target matrix, linear in the dimension of the matrix, and logarithmic in the condition number of the matrix. To the best of our knowledge, all previous algorithms either incurred a quadratic dependence on the condition number of the unknown matrix or a quadratic dependence on the dimension of the matrix in the running time. Our algorithm is based on a novel extension of Alternating Minimization which we show has theoretical guarantees under standard assumptions even in the presence of noise.