Efficient streaming noise generation for the optimal Toeplitz factorization

Develop an efficient streaming algorithm to generate correlated Gaussian noise for the lower triangular Toeplitz factorization of A(n) proposed by Fichtenberger, Henzinger, and Upadhyay—defined by Bij = Cij = f_{i−j} with f0 = 1 and fk = (1 − 1/(2k)) · fk−1—so that noise can be produced online with polylogarithmic space and time despite the matrices being dense.

Background

The optimal lower triangular Toeplitz factorization of A(n) (within the Toeplitz class) due to Fichtenberger, Henzinger, and Upadhyay achieves near‑optimal MaxErr, but its dense structure impedes efficient streaming noise generation. In contrast, sparse constructions like the binary tree mechanism enable efficient sampling but are suboptimal by a significant multiplicative factor.

The authors’ BLT approach provides efficient streaming algorithms for near‑optimal factorizations via rational approximations, but an efficient sampler for the exact optimal Toeplitz factorization remains unknown.