Online Tensor Dictionary Learning under the Einstein Product with an Application to Completion

Dictionary learning is a powerful technique widely used in signal processing to construct adaptive data representations by learning overcomplete dictionaries that promote sparsity. In this work, we present an online tensor dictionary learning framework formulated using the Einstein product. Within this framework, we propose two distinct online learning methods: a stochastic online variant of the Tensor K-SVD (TKSVD) algorithm and an online dictionary learning approach based on projected stochastic gradient descent (PSGD). The proposed methods are designed to efficiently handle multidimensional data in streaming and large-scale settings. In addition, we incorporate extrapolation techniques to accelerate the Iterative Shrinkage–Thresholding Algorithm (ISTA) during the sparse coding stage. Extensive numerical experiments demonstrate that the proposed online methods achieve competitive and, in several cases, superior performance compared to state-of-the-art tensor completion techniques. MSC Classification: 15A69 , 15A72 , 15A83