Active subspace: toward scalable low-rank learning.

Area of Science:

• Machine Learning
• Optimization
• Numerical Analysis

Background:

• Low-rank matrix learning problems often involve computationally intensive nuclear norm regularized optimization problems (NNROPs).
• Existing solvers for NNROPs face significant scalability challenges in large-scale applications.
• The optimal solutions to NNROPs are frequently characterized by a low-rank structure.

Purpose of the Study:

• To address the computational scalability issues inherent in large-scale low-rank matrix learning.
• To develop an efficient algorithm for solving nuclear norm regularized optimization problems (NNROPs).
• To reduce the complexity of NNROPs by transforming them into smaller-scale problems.

Main Methods:

• Revisiting low-rank matrix factorization to exploit the low-rank property of NNROP solutions.
• Developing an active subspace algorithm that factorizes the large solution matrix into a small orthonormal matrix (active subspace) and another small matrix.
• Employing the augmented Lagrange alternating direction method to find suboptimal solutions for the resulting nonconvex problems.

Main Results:

• The proposed active subspace algorithm transforms large-scale NNROPs into manageable, small-scale problems.
• Theoretical analysis for Robust PCA (RPCA) confirms the suboptimality of the algorithm's solution.
• Empirical results demonstrate significant reductions in computational complexity for general NNROPs, with no loss in optimality.

Conclusions:

• The active subspace algorithm provides an efficient and scalable approach to solving large-scale low-rank matrix learning problems.
• The method effectively reduces computational demands for NNROPs, maintaining solution quality.
• This approach offers a practical alternative for handling complex matrix learning tasks in computationally constrained environments.