LSV-Based Tail Inequalities for Sums of Random Matrices

Area of Science:

• Machine Learning
• Probability Theory
• Linear Algebra

Background:

• Random matrices are integral to various machine learning algorithms.
• Existing methods for tail inequalities in random matrices have limitations.
• Understanding the behavior of sums of random matrices is crucial for theoretical guarantees.

Purpose of the Study:

• To introduce a novel method for analyzing tail inequalities of random matrix sums.
• To develop dimension-independent tail bounds.
• To leverage the largest singular value (LSV) for improved analysis.

Main Methods:

• Introduced a diagonalization method to handle non-commutative LSV operations.
• Converted the LSV operation into a trace operation on an infinite-dimensional diagonal matrix.
• Derived new Laplace-transform bounds.

Main Results:

• Achieved LSV-based tail inequalities for sums of random matrices.
• The derived tail results are independent of matrix dimensions.
• The new method provides an alternative to existing approaches.

Conclusions:

• The proposed method offers a powerful tool for studying random matrix theory in machine learning.
• Dimension-independent tail inequalities enhance the applicability of these results.
• This work advances the theoretical understanding of random matrix sums.