Optimal Exact Matrix Completion Under new Parametrization
This work addresses matrix completion problems in data science and machine learning, offering incremental improvements in efficiency for specific scenarios.
The paper tackles exact matrix completion for rank-r matrices by introducing a sparsity-number relation and adaptive sampling algorithms, achieving exact recovery with high probability using significantly fewer observations than the state-of-the-art, especially for coherent matrices.
We study the problem of exact completion for $m \times n$ sized matrix of rank $r$ with the adaptive sampling method. We introduce a relation of the exact completion problem with the sparsest vector of column and row spaces (which we call \textit{sparsity-number} here). Using this relation, we propose matrix completion algorithms that exactly recovers the target matrix. These algorithms are superior to previous works in two important ways. First, our algorithms exactly recovers $μ_0$-coherent column space matrices by probability at least $1 - ε$ using much smaller observations complexity than $\mathcal{O}(μ_0 rn \mathrm{log}\frac{r}ε)$ the state of art. Specifically, many of the previous adaptive sampling methods require to observe the entire matrix when the column space is highly coherent. However, we show that our method is still able to recover this type of matrices by observing a small fraction of entries under many scenarios. Second, we propose an exact completion algorithm, which requires minimal pre-information as either row or column space is not being highly coherent. At the end of the paper, we provide experimental results that illustrate the strength of the algorithms proposed here.