Logarithmic Norm Regularized Low-Rank Factorization for Matrix and Tensor Completion.

Area of Science:

• Data Science
• Machine Learning
• Numerical Analysis

Background:

• Matrix and tensor completion are crucial for reconstructing incomplete data.
• Conventional methods using nuclear norm surrogates for rank often yield suboptimal low-rank recovery.
• Existing techniques struggle with large-scale datasets and computational efficiency.

Purpose of the Study:

• To introduce a novel logarithmic norm definition for matrices and tensors.
• To develop new algorithms for accurate and efficient low-rank matrix and tensor completion.
• To address the limitations of convex surrogate norms in rank minimization.

Main Methods:

• Definition of a new matrix/tensor logarithmic norm to create a sparsity-driven rank surrogate.
• Derivation of factor matrix/tensor norm surrogate theorems for efficient computation.
• Development of Logarithmic norm Regularized Matrix Factorization (LRMF) and Tensor Factorization (LRTF) algorithms.
• Application of alternating minimization for solving optimization problems with convergence guarantees.

Main Results:

• The proposed logarithmic norm effectively serves as a sparsity-driven surrogate for rank.
• LRMF and LRTF algorithms demonstrate enhanced accuracy in low-rank approximation.
• The new algorithms achieve significant improvements in computational efficiency compared to state-of-the-art methods.
• Simulation results confirm superior performance on both synthetic and real-world data.

Conclusions:

• The novel logarithmic norm provides a more effective approach to low-rank recovery than traditional methods.
• LRMF and LRTF offer a powerful and efficient solution for matrix and tensor completion tasks.
• This work advances the field of low-rank approximation with practical implications for data recovery.