The traditional method of computing singular value decomposition (SVD) of a data matrix is based on a least squares principle, thus, is very sensitive to the presence of outliers. Hence the resulting inferences across different applications using the classical SVD are extremely degraded in the presence of data contamination (e.g., video surveillance background modelling tasks, etc.). A robust singular value decomposition method using the minimum density power divergence estimator (rSVDdpd) has been found to provide a satisfactory solution to this problem and works well in applications. For example, it provides a neat solution to the background modelling problem of video surveillance data in the presence of camera tampering. In this paper, we investigate the theoretical properties of the rSVDdpd estimator such as convergence, equivariance and consistency under reasonable assumptions. Since the dimension of the parameters, i.e., the number of singular values and the dimension of singular vectors can grow linearly with the size of the data, the usual M-estimation theory has to be suitably modified with concentration bounds to establish the asymptotic properties. We believe that we have been able to accomplish this satisfactorily in the present work. We also demonstrate the efficiency of rSVDdpd through extensive simulations.

翻译：传统的数据矩阵奇异值分解（SVD）计算方法基于最小二乘原则，因此对异常值的存在极为敏感。当数据受到污染时（例如视频监控背景建模任务等），基于经典SVD的各类应用推断结果会严重退化。基于最小密度幂散度估计量的稳健奇异值分解方法（rSVDdpd）已被证明能为该问题提供令人满意的解决方案，并在实际应用中表现良好。例如，该方法能在摄像机篡改场景下为视频监控数据的背景建模问题提供简洁的解决方案。本文在合理假设条件下，研究了rSVDdpd估计量的理论性质，包括收敛性、等变性及一致性。由于参数维度（即奇异值数量及奇异向量维度）可能随数据规模线性增长，标准M估计理论需结合集中不等式进行适当修正，方能建立渐近性质。我们相信本工作已圆满完成此项任务。此外，我们通过大量模拟实验验证了rSVDdpd的有效性。