Lp Quasi-norm Minimization

Abstract

The ℓp (0 < p < 1) quasi-norm is used as a sparsity-inducing function, and has applications in diverse areas, e.g., statistics, machine learning, and signal processing. This paper proposes a heuristic based on a two-block ADMM algorithm for tackling ℓp quasi-norm minimization problems. For p = s/q < 1, s, q ∈ ℤ +, the proposed algorithm requires solving for the roots of a scalar degree 2q polynomial as opposed to applying a soft thresholding operator in the case of ℓ1. We show numerical results for two example applications, sparse signal reconstruction from few noisy measurements and spam email classification using support vector machines. Our method obtains significantly sparser solutions than those obtained by ℓ1 minimization while achieving similar level of measurement fitting in signal reconstruction, and training and test set accuracy in classification.