Half-quadratic alternating direction method of multipliers for robust orthogonal tensor approximation

Higher-order tensor canonical polyadic decomposition (CPD) with one or more of the latent factor matrices being columnwisely orthonormal has been well studied in recent years. However, most existing models penalize the noises, if occurring, by employing the least squares loss, which may be sensitive to non-Gaussian noise or outliers, leading to bias estimates of the latent factors. In this paper, we derive a robust orthogonal tensor CPD model with Cauchy loss, which is resistant to heavy-tailed noise such as the Cauchy noise, or outliers. By exploring the half-quadratic property of the model, we develop the so-called half-quadratic alternating direction method of multipliers (HQ-ADMM) to solve the model. Each subproblem involved in HQ-ADMM admits a closed-form solution. Thanks to some nice properties of the Cauchy loss, we show that the whole sequence generated by the algorithm globally converges to a stationary point of the problem under consideration. Numerical experiments on synthetic and real data demonstrate the effectiveness of the proposed model and algorithm.