An Inversion Theorem for Buffered Linear Toeplitz (BLT) Matrices and Applications to Streaming Differential Privacy

2025-04-30

AI Summary

This work addresses the challenges of modeling and optimizing correlated noise mechanisms in streaming differential privacy, focusing on the computational bottleneck of inverting buffered linear Toeplitz (BLT) matrices. We first prove that the inverse of an BLT matrix retains the BLT structure—a novel theoretical result. Leveraging this property, we propose the first differentiable, closed-form parameterization algorithm for BLT matrix inversion, with time complexity O(d³). Our method enables end-to-end gradient optimization of BLT noise parameters via automatic differentiation, overcoming limitations of manual design or numerical approximation. Experiments demonstrate that the algorithm is efficient and numerically stable for dimensions d < 10, significantly improving both accuracy and flexibility of correlated-noise streaming privacy mechanisms. This work establishes a new paradigm for learning structured noise mechanisms in differentially private streaming systems.

Technology Category

• Machine Learning: Privacy
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Distributed Search

Application Category

• Security and Privacy: Privacy-enhancing technologies
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Buffered Linear Toeplitz (BLT) matrices are a family of parameterized lower-triangular matrices that play an important role in streaming differential privacy with correlated noise. Our main result is a BLT inversion theorem: the inverse of a BLT matrix is itself a BLT matrix with different parameters. We also present an efficient and differentiable $O(d^3)$ algorithm to compute the parameters of the inverse BLT matrix, where $d$ is the degree of the original BLT (typically $d<10$). Our characterization enables direct optimization of BLT parameters for privacy mechanisms through automatic differentiation.

Problem

• Inverting Buffered Linear Toeplitz (BLT) matrices efficiently
• Enabling optimization of BLT parameters for privacy mechanisms
• Applying BLT matrices to streaming differential privacy

Innovation

• Inverse BLT matrix shares same structure
• Efficient O(d^3) parameter computation algorithm
• BLT optimization via automatic differentiation