A robust 3d-tensor completion method with entry-wise bounds

Abstract

The technological advances in recent decades, including computing power and data collection techniques, have highlighted the need for advanced data analyzing tools. The proposed tools are required to reveal the data’s underlying patterns while being robust to significant data issues such as noise contamination, presence of anomalies, and missing data. Such research has diverse applications ranging from entertainment such as music/video processing to medicine such as medical image/ultra sound processing. Principal Component Analysis is a famous fundamental tool for linear data analysis, which suffers from high sensitivity to noise and anomalies. Ro bust Principal Component Analysis (RPCA) was capable of decomposing a given 2D data-matrix into both a sparse component and a low-rank component where it is capable of excluding noise and anomalies from the analysis. ERPCA was further improved to handle incomplete data by allowing to impose bounds on the data. Even though the aforesaid evolution granted improved tools to the data analyzing literature, they all possess limited applicability only to 2D data matrices. While it is possible to apply 2D methods to 3D matrices, resizing causes a suboptimal loss of information. Thus, analysis of 3D data matrices (3D tensors) such as color images with noise and missing values have been a problem. Here, we propose a robust principal component analysis method for 3D tensors via the aid of a new Fourier based tensor product that incorporates entry-wise bounds. These entry-wise constraints serve to make our method more adaptive to various types of noise and anomaly scenarios.