Faithful Embeddings of Irregular and Asynchronous Data for Online Log-NCDEs

Continuous-time models are a natural choice for irregular and asynchronous data. A central design choice is how to embed discrete observations into continuous time. Interpolation- and imputation-based embeddings reconstruct a continuous observation path, making the model sensitive to the choice of reconstruction. We show that this reconstruction step is unnecessary. On compact sets, universality for continuous functionals of paths transfers to universality for continuous functionals of discrete observation streams under any continuous and injective embedding. Guided by this result, and building on the rectilinear control path for Neural Controlled Differential Equations (NCDEs), we introduce a continuous and injective embedding for Log-NCDEs, a universal class of continuous-time models. This embedding records observations as increments and composes them over arbitrary query intervals to form log-signatures, giving interval-level summaries of a faithful embedding of the observed data. This avoids interpolation of observed variables, supports online computation, and allows prediction on output grids independent of the input sampling times. Experiments on synthetic controlled dynamics and real-world time-series datasets show that the representation is accurate, efficient, and robust to irregular, asynchronous, and sparse observations.