Basis pursuit denoising

In applied mathematics and statistics, basis pursuit denoising (BPDN) refers to a mathematical optimization problem of the form:

where ${\displaystyle \lambda }$ is a parameter that controls the trade-off between sparsity and reconstruction fidelity, ${\displaystyle x}$ is an ${\displaystyle N\times 1}$ solution vector, ${\displaystyle y}$ is an ${\displaystyle M\times 1}$ vector of observations, ${\displaystyle A}$ is an ${\displaystyle M\times N}$ transform matrix and ${\displaystyle M<N}$. This is an instance of convex optimization and also of quadratic programming.

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