Tensor sparse representation via Einstein product

Sparse representation has garnered significant attention across multiple fields, including signal processing, statistics, and machine learning. The fundamental concept of this technique is that we can express the signal as a linear combination of only a few elements from a known basis. Compressed sensing (CS) is an interesting application of this technique. It is valued for its potential to improve data collection and ensure efficient acquisition and recovery from just a few measurements. In this paper, we propose a novel approach for the high-order CS problem based on the Einstein product, utilizing a tensor dictionary instead of the commonly used matrix-based dictionaries in the Tucker model. Our approach provides a more general framework for compressed sensing. We present two novel models to address the CS problem in the multidimensional case. The first model represents a natural generalization of CS to higher-dimensional signals; we extend the traditional CS framework to effectively capture the sparsity of multidimensional signals and enable efficient recovery. In the second model, we introduce a complexity reduction technique by utilizing a low-rank representation of the signal. We extend the OMP and the homotopy algorithms to solve the high-order CS problem. Through various simulations, we validate the effectiveness of our proposed method, including its application to solving the completion tensor problem in 2D and 3D colored and hyperspectral images.