Computing sparse representations of multidimensional signals using Kronecker bases.

Area of Science:

• Signal Processing
• Numerical Analysis
• Data Science

Background:

• Sparse representations are crucial for efficiently encoding signals using few dictionary elements.
• Existing methods primarily focus on one-dimensional signals (vectors).
• Multidimensional signals (tensors) require advanced techniques for sparse representation.

Purpose of the Study:

• To generalize sparse representation theory to multiway arrays (tensors).
• To develop efficient algorithms for sparse tensor representation and compressed sensing.
• To address the challenges posed by large-scale multidimensional data.

Main Methods:

• Generalization of the orthogonal matching pursuit (OMP) algorithm to tensors using the Tucker model, resulting in Kronecker-OMP.
• Introduction of multiway block-sparse representation and development of the N-way block OMP (N-BOMP) algorithm.
• Theoretical analysis of algorithm complexity and precision under block-sparsity assumptions.

Main Results:

• Kronecker-OMP and N-BOMP algorithms effectively handle sparse representations of tensors.
• N-BOMP demonstrates lower complexity and higher precision than classical OMP under block-sparsity.
• Algorithms are suitable for very large-scale problems intractable for standard methods.

Conclusions:

• The proposed Kronecker-OMP and N-BOMP algorithms offer significant advancements in sparse tensor representation.
• These methods provide efficient solutions for compressed sensing in large-scale multidimensional data, including 2D and 3D imaging.
• The algorithms pave the way for processing complex, real-world multidimensional signals.