HiPPO Functional Memory: Coordinate-Queryable Attention for Irregular Time Series

Irregular time series are observed at nonuniform coordinates, yet predictions may be required at different times. We introduce HiPPO Functional Memory (HiPPO-FM), a functional-memory attention operator that compiles irregular observations into a fixed-rank, coordinate-addressable state for direct evaluation at arbitrary query times. HiPPO-derived functional coordinates and matched kernel geometry couple memory write and read, while content interaction can precede compression without changing the functional query space. We characterize the resulting finite-dimensional query space and its compression limits, and demonstrate strong continuous-coordinate reconstruction, competitive irregular representation learning, and real-data extrapolation. Because the encoded state is reusable across query sets, HiPPO-FM replaces observation-indexed retrieval with direct queries to a compact function-valued memory.