Improved Bounds for Star Discrepancy in High Dimensions
Analysis
This paper significantly improves upon existing bounds for the star discrepancy of double-infinite random matrices, a crucial concept in high-dimensional sampling and integration. The use of optimal covering numbers and the dyadic chaining framework allows for tighter, explicitly computable constants. The improvements, particularly in the constants for dimensions 2 and 3, are substantial and directly translate to better error guarantees in applications like quasi-Monte Carlo integration. The paper's focus on the trade-off between dimensional dependence and logarithmic factors provides valuable insights.
Key Takeaways
- •Provides sharper non-asymptotic probabilistic bounds for the star discrepancy of double-infinite random matrices.
- •Utilizes optimal covering numbers to achieve explicitly computable constants.
- •Demonstrates significant improvements in constants, particularly for dimensions 2 and 3.
- •Offers improved error guarantees for quasi-Monte Carlo integration and related applications.
- •Highlights a precise trade-off between dimensional dependence and logarithmic factors.
“The paper achieves explicitly computable constants that improve upon all previously known bounds, with a 14% improvement over the previous best constant for dimension 3.”