Sparse solution of nonnegative least squares problems with applications in the construction of probabilistic Boolean networks

Summary In this paper, we consider finding a sparse solution of nonnegative least squares problems with a linear equality constraint. We propose a projection-based gradient descent method for solving huge size constrained least squares problems. Traditional Newton-based methods require solving a linear system. However, when the matrix is huge, it is neither practical to store it nor possible to solve it in a reasonable time. We therefore propose a matrix-free iterative scheme for solving the minimizer of the captured problem. This iterative scheme can be explained as a projection-based gradient descent method. In each iteration, a projection operation is performed to ensure the solution is feasible. The projection operation is equivalent to a shrinkage operator, which can guarantee the sparseness of the solution obtained. We show that the objective function is decreasing. We then apply the proposed method to the inverse problem of constructing a probabilistic Boolean network. Numerical examples are then given to illustrate both the efficiency and effectiveness of our proposed method. Copyright © 2015 John Wiley & Sons, Ltd.