Improved Accuracy for Private Continual Cardinality Estimation in Fully Dynamic Streams via Matrix Factorization

We study differentially-private statistics in the fully dynamic continual observation model, where many updates can arrive at each time step and updates to a stream can involve both insertions and deletions of an item. Earlier work (e.g., Jain et al., NeurIPS 2023 for counting distinct elements; Raskhodnikova & Steiner, PODS 2025 for triangle counting with edge updates) reduced the respective cardinality estimation problem to continual counting on the difference stream associated with the true function values on the input stream. In such reductions, a change in the original stream can induce many changes in the difference stream. This poses a challenge for applying private continual counting algorithms to obtain optimal error bounds. We improve the accuracy of several such reductions by studying the associated ℓ p sensitivity vectors of the resulting difference streams and isolating their properties. We demonstrate that our framework gives improved bounds for counting distinct elements, estimating degree histograms, and estimating triangle counts (under a slightly relaxed privacy model), thus offering a general approach to private continual cardinality estimation in streaming settings. Our improved accuracy stems from tight analysis of known factorization mechanisms for the counting matrix in this setting; the key technical challenge is arguing that one can use state-of-the-art factorizations for sensitivity vector sets with the properties we isolate. In particular, we show that for approximate DP, the celebrated square-root factorization of the counting matrix (Henzinger et al., SODA 2023; Fichtenberger et al., ICML 2023) can be employed to give concrete improvements in accuracy over past work based on the binary tree mechanism. We also give a tight analysis of b -ary tree mechanisms with subtraction under these sensitivity patterns, including a polytime routine for computing the ℓ p sensitivity, yielding time and space efficient algorithms for both pure and approximate DP. Empirically and analytically, we demonstrate that our improved error bounds offer a substantial improvement in accuracy for cardinality estimation problems over a large range of parameters.