For the pure-DP Laplace matrix mechanism on prefix sums, the paper claims optimized maximum and mean squared errors of order Θ(log^3 n / ε^2) for arbitrary real factorizations, matching the best-known upper bound.
Pietsch,Approximation Spaces, Journal of Approximation Theory 32 (1981), 115–134
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Costs of Arbitrary Real Matrix Factorizations for Pure-DP Continual Counting
For the pure-DP Laplace matrix mechanism on prefix sums, the paper claims optimized maximum and mean squared errors of order Θ(log^3 n / ε^2) for arbitrary real factorizations, matching the best-known upper bound.