In applied mathematics and statistics, basis pursuit denoising (BPDN) refers to a mathematical optimization problem of the form: where is a parameter that controls the trade-off between sparsity and reconstruction fidelity, is an solution vector, is an vector of observations, is an transform matrix and . This is an instance of convex optimization and also of quadratic programming. Some authors refer to basis pursuit denoising as the following closely related problem: which, for any given , is equivalent to the unconstrained formulation for some (usually unknown a priori) value of close to simple in the . As