Provable Model for Tensor Ring Completion

Tensor is a natural way to represent the high-dimensional data, thus it preserves more intrinsic information than matrix when dealing with high-order data [1, 2, 3]. In practice, parts of the tensor entries are missing during data acquisition and transformation, tensor completion estimates the missing entries based on the assumption that most elements are correlated [4]. This correlation can be modeled as low-rank data structures which can be used in a series of applications, including signal processing [2], machine learning [5], remote sensing [6], computer vision [7], etc. There are two main frameworks for tensor completion, namely, variational energy minimization as well as tensor rank minimization [8, 9], where the energy is usually a recovery error in the context of tensor completion and the definition of rank varies with diverse tensor decompositions. The first method is realized by means of the alternating least square (ALS), in which each core tensor is updated one by one while others are fixed [8]. The ALSbased method requires a predefined tensor rank, while the rank minimization does not. Common forms of tensor decompositions are summarized as follows.