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

arXiv: https://arxiv.org/abs/2605.30213 · HTML: https://mlsanity.com/paper/2605.30213/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 47 of its 82 references that are in the map.
- First posted on arXiv 28 May 2026.

## Builds on (most cited first, 30 of 47)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
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- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Recurrent Neural Networks for Multivariate Time Series with Missing Values](https://mlsanity.com/paper/1606.01865/) (2016)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [The UEA multivariate time series classification archive, 2018](https://mlsanity.com/paper/1811.00075/) (2018)
- [Multi-Time Attention Networks for Irregularly Sampled Time Series](https://mlsanity.com/paper/2101.10318/) (2021)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [N-HiTS: Neural Hierarchical Interpolation for Time Series Forecasting](https://mlsanity.com/paper/2201.12886/) (2022)
- [Graph Neural Controlled Differential Equations for Traffic Forecasting](https://mlsanity.com/paper/2112.03558/) (2021)
- [A Primer on the Signature Method in Machine Learning](https://mlsanity.com/paper/1603.03788/) (2016)
- [Interpolation-Prediction Networks for Irregularly Sampled Time Series](https://mlsanity.com/paper/1909.07782/) (2019)
- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [Theoretical Foundations of Deep Selective State-Space Models](https://mlsanity.com/paper/2402.19047/) (2024)
- [ContiFormer: Continuous-Time Transformer for Irregular Time Series Modeling](https://mlsanity.com/paper/2402.10635/) (2024)
- [Continuous-Time Modeling of Counterfactual Outcomes Using Neural Controlled Differential Equations](https://mlsanity.com/paper/2206.08311/) (2022)
- [The Signature Kernel is the solution of a Goursat PDE](https://mlsanity.com/paper/2006.14794/) (2020)
- [Deep Learning for Multivariate Time Series Imputation: A Survey](https://mlsanity.com/paper/2402.04059/) (2024)
- [The iisignature library: efficient calculation of iterated-integral signatures and log signatures](https://mlsanity.com/paper/1802.08252/) (2018)
- [Oscillatory State-Space Models](https://mlsanity.com/paper/2410.03943/) (2024)
- [Log Neural Controlled Differential Equations: The Lie Brackets Make a Difference](https://mlsanity.com/paper/2402.18512/) (2024)
- [Signature Methods in Machine Learning](https://mlsanity.com/paper/2206.14674/) (2022)
- [Distribution Regression for Sequential Data](https://mlsanity.com/paper/2006.05805/) (2020)
- [SigGPDE: Scaling Sparse Gaussian Processes on Sequential Data](https://mlsanity.com/paper/2105.04211/) (2021)
- [EXIT: Extrapolation and Interpolation-based Neural Controlled Differential Equations for Time-series Classification and Forecasting](https://mlsanity.com/paper/2204.08771/) (2022)
- [Signature Kernel Conditional Independence Tests in Causal Discovery for Stochastic Processes](https://mlsanity.com/paper/2402.18477/) (2024)
- [Structured Linear CDEs: Maximally Expressive and Parallel-in-Time Sequence Models](https://mlsanity.com/paper/2505.17761/) (2025)

## Abstract (arXiv)

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; under mild conditions, compact-set universality on the model input space transfers to the data space whenever the embedding from data to input is continuous and injective. 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. Our approach records observations as increments and composes them over arbitrary query intervals to directly form log-signatures. This provides interval-level summaries without first interpolating the observed variables, while supporting online computation. 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.