Low-Rank Tensor Learning by Generalized Nonconvex Regularization

Area of Science:

• Machine Learning
• Tensor Analysis
• Optimization

Background:

• Low-rank tensor learning is crucial for analyzing high-dimensional data with limited observations.
• Existing methods using sum of nuclear norms may not fully exploit tensor low-rank structure.
• Developing more effective low-rank tensor learning techniques is an active research area.

Purpose of the Study:

• To propose a novel nonconvex model for low-rank tensor learning.
• To effectively characterize the low-rankness of underlying tensors using a transformed tensor nuclear norm.
• To establish theoretical guarantees and demonstrate practical effectiveness.

Main Methods:

• A nonconvex model utilizing a transformed tensor nuclear norm is proposed.
• Error bounds are established under restricted strong convexity and regularity conditions.
• A proximal majorization-minimization (PMM) algorithm is developed to solve the nonconvex model.
• Global convergence and convergence rates of the PMM algorithm are analyzed.

Main Results:

• The proposed nonconvex model effectively captures tensor low-rankness.
• Theoretical error bounds are derived for the stationary point of the model.
• The PMM algorithm demonstrates global convergence and established convergence rates.
• Numerical experiments show superior performance compared to state-of-the-art methods in tensor completion and binary classification.

Conclusions:

• The proposed transformed tensor nuclear norm offers a more effective approach to low-rank tensor learning.
• The PMM algorithm provides a robust and efficient method for solving the nonconvex model.
• The method shows significant promise for applications in tensor completion and classification.