03:30 PM - 03:55 PM
Parallel Implementations for Solving Matrix Factorization Problems with Optimization
Abstract: We provide a framework for parallelized large-scale matrix factorization problems. One of the most successful and used methods to solve these problems is solving them via optimization techniques. Optimization methods require gradient vectors to update the iterates. The time spent to solve such a problem is mostly spent on calls to gradient and function value evaluations. In this work, we have used a recent method, which has not been used before for matrix factorization. When it comes to parallelization, we present both CPU and GPU implementations. As our experiments show, the proposed parallelization scales quite well. We report our results on MovieLens data set. Our results show that the new method is quite successful in reducing the number of iterations. We obtain very good RMSE values with significant promising scaling figures.
03:55 PM - 04:20 PM
Controlled Cholesky Factorization derived from the Splitting Preconditioner
Interior point methods are very efficient for solving linear programming problems. Iterative methods are used to solve the resulting linear systems when the factorizations are dense. Close to a solution these systems are ill-conditioned and preconditioning is an essential issue. In our work, we used the Cholesky controlled factorization as a preconditioner in the first iterations. A new splitting preconditioner, which has very good performance near a solution, is used in the final iterations. We apply the Cholesky controlled factorization in the relevant split matrix, and use its factor as a preconditioner. Satisfactory preliminary computational results are presented.
04:20 PM - 04:45 PM
A new role for incomplete Cholesky factorizations
In the general context, incomplete Cholesky factorizations are used for building preconditioners. In this work, we have assigned a new role to incomplete Cholesky factorizations, which consists in using them in the direct solution of the normal equations systems that arise in interior points methods. In this way, the accuracy of the computed solution and consequently fast convergence suffer some deterioration, however, one saves the computational effort used to compute it. Numerical experiments show that this approach reduces the solution time of linear systems, leading to a reduction in the total processing time for most of the test problems.