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

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.

@article{mcmahan2025_2504.21413,
  title={ An Inversion Theorem for Buffered Linear Toeplitz (BLT) Matrices and Applications to Streaming Differential Privacy },
  author={ H. Brendan McMahan and Krishna Pillutla },
  journal={arXiv preprint arXiv:2504.21413},
  year={ 2025 }
}