Fast matrix completion without the condition number
نویسندگان
چکیده
We give the first algorithm for Matrix Completion that achieves running time and sample complexity that 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. 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.
منابع مشابه
Fast Exact Matrix Completion with Finite Samples
Matrix completion is the problem of recovering a low rank matrix by observing a small fraction of its entries. A series of recent works (Keshavan, 2012; Jain et al., 2013; Hardt, 2014) 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 ...
متن کاملMinimizing the Condition Number of a Positive Deenite Matrix by Completion
We consider the problem of minimizing the spectral condition number of a positive deenite matrix by completion: minfcond(positive deeniteg; where A is an n n Hermitian positive deenite matrix, B a p n matrix and X is a free p p Hermitian matrix. We reduce this problem to an optimization problem for a convex function in one variable. Using the minimal solution of this problem we characterize the...
متن کاملThe Leave-one-out Approach for Matrix Completion: Primal and Dual Analysis
In this paper, we introduce a powerful technique, Leave-One-Out, to the analysis of lowrank matrix completion problems. Using this technique, we develop a general approach for obtaining fine-grained, entry-wise bounds on iterative stochastic procedures. We demonstrate the power of this approach in analyzing two of the most important algorithms for matrix completion: the non-convex approach base...
متن کاملMinimizing the condition number of a positive definite matrix by completion
We consider the problem of minimizing the spectral condition number of a positive definite matrix by completion: min{cond( [ A BH B X ] ) : [ A BH B X ] positive definite}, where A is an n × n Hermitian positive definite matrix, B a p × n matrix and X is a free p× p Hermitian matrix. We reduce this problem to an optimization problem for a convex function in one variable. Using the minimal solut...
متن کاملConvergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent
We address the rectangular matrix completion problem by lifting the unknown matrix to a positive semidefinite matrix in higher dimension, and optimizing a nonconvex objective over the semidefinite factor using a simple gradient descent scheme. WithO(μr2κ2nmax(μ, log n)) random observations of a n1×n2 μ-incoherent matrix of rank r and condition number κ, where n = max(n1, n2), the algorithm line...
متن کامل