Binary Measure Sketches for Persistent Homology: All-Scale Interleavings and Memory Lower Bounds

Persistent homology summarizes geometry across scales, but computing it from a compressed representation requires knowing which geometric information must survive compression. For distribution-sensitive filtrations, this question concerns both distances and the masses they organize, rather than the number of distinct distance values alone. We introduce binary measure sketches that aggregate a sample into weighted cells and replace representative coordinates by calibrated binary geometry. For finite distance-to-a-measure weighted Vietoris–Rips persistence, we show that mass-preserving aggregation and uniform distance distortion compose into an all-scale interleaving with error \(2\rpart+\eta\), separating partition loss from coding loss without an additional penalty for the scale axis or homological degree. A complementary packing of degree-zero persistence diagrams gives a topology-specific memory lower bound, showing that the information required by the topological output cannot in general be eliminated by quantizing distances. Experiments examine the resulting error decomposition and the utility of the retained representation. Together, these results connect the geometry a sketch preserves to the persistence it can recover, providing a principled basis for allocating representative and coding budgets.