2014/11/04 by Prateek Jain, Praneeth Netrapalli, Jain, Prateek +1 · 3 citations
Computer Science · Engineering · Mathematics · #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Numerical Analysis (math.NA) #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1411.1087
openalex publication_date 2014/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Matrix completion is the problem of recovering a low rank matrix by observing a small fraction of its entries. A series of recent works [KOM12,JNS13,HW14] have proposed fast non-convex optimization based iterative algorithms to solve this problem. However, the sample complexity in all these results is sub-optimal in its dependence on the rank, condition number and the desired accuracy. In this paper, we present a fast iterative algorithm that solves the matrix completion problem by observing O(nr5 log3 n) entries, which is independent of the condition number and the desired accuracy. The run time of our algorithm is O(nr7log3 nlog 1/ε) which is near linear in the dimension of the matrix. To the best of our knowledge, this is the first near linear time algorithm for exact matrix completion with finite sample complexity (i.e. independent of ε). Our algorithm is based on a well known projected gradient descent method, where the projection is onto the (non-convex) set of low rank matrices. There are two key ideas in our result: 1) our argument is based on a ℓ∞ norm potential function (as opposed to the spectral norm) and provides a novel way to obtain perturbation bounds for it. 2) we prove and use a natural extension of the Davis-Kahan theorem to obtain perturbation bounds on the best low rank approximation of matrices with good eigen-gap. Both of these ideas may be of independent interest.