Abstract
In this dissertation, we focus on tensor completion, which is closely related to the ubiquitous missing data problem in real-world applications. Given a tensor with incomplete entries, existing methods assume the desired tensor exhibits low-rank structure. Predicting missing entries then boils down to recovering a low-rank tensor from given entries. Factorization schemes and completion schemes are two popular methodologies. As the number of missing entries increases, factorization schemes overfit the model structure due to their incorrectly predefined tensor’s rank, while completion schemes fail to interpret the model factors because they solely rely on rank minimization. Therefore, we introduce a novel concept to break the current limitations: complete the missing entries and simultaneously capture the underlying model structure. We propose a method called Simultaneous Tensor Decomposition and Completion (STDC). The major contributions are three-fold. First, we leverage rank minimization with Tucker model decomposition; i.e., we automate rank estimation while carefully maintain the latent tensor structure. Second, considering the informative semantics (named factor priors in our work) of real-world tensor objects, we discover the latent manifold with a new presented methodology, called Multilinear Graph Embedding (MGE), and study its significance in tensor completion. Finally, because factor priors are task-dependent and can be unavailable, we further propose a prior-free extension with a new presented methodology, called Permutation on Manifolds (PoM), to automate joint-manifold learning. We conducted experiments to empirically verify the convergence of our algorithm on synthetic data, and evaluate its effectiveness on various kinds of real-world data. The results demonstrate the superiority of our method and its potential usage in tensor-based applications.