Improve space complexity of near‑optimal streaming factorizations

Determine whether it is possible to construct either (i) a lower triangular Toeplitz matrix factorization B, C with BC = A(n) whose associated streaming algorithm uses O(log n) space, or (ii) any matrix factorization B, C with BC = A(n) satisfying MaxErr(B, C) ≤ Opt(n) + O(1) while improving upon the O(log^2 n) space complexity, thereby achieving near‑optimal utility with reduced memory requirements.

Background

The paper’s main result (Theorem 1.1) achieves near‑optimal utility for differentially private continual counting using lower triangular Toeplitz factorizations and provides a streaming algorithm with O(log^2 n) space. A secondary result (Theorem 1.2) attains O(log n) space via a recursive approach but does not preserve the Toeplitz structure and only guarantees multiplicative near‑optimality.

Bridging this gap—obtaining O(log n) space while maintaining Toeplitz structure or achieving additive constant near‑optimality over all factorizations—would substantially improve the practicality and performance of streaming differential privacy mechanisms.