# Distribution Regression for Sequential Data

arXiv: https://arxiv.org/abs/2006.05805 · HTML: https://mlsanity.com/paper/2006.05805/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2020.

## Built on by (most cited first, 30 of 31)

- [Theoretical Foundations of Deep Selective State-Space Models](https://mlsanity.com/paper/2402.19047/) (2024)
- [Non-adversarial training of Neural SDEs with signature kernel scores](https://mlsanity.com/paper/2305.16274/) (2023)
- [Signature Methods in Machine Learning](https://mlsanity.com/paper/2206.14674/) (2022)
- [Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes](https://mlsanity.com/paper/2109.03582/) (2021)
- [Neural signature kernels as infinite-width-depth-limits of controlled ResNets](https://mlsanity.com/paper/2303.17671/) (2023)
- [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)
- [New directions in the applications of rough path theory](https://mlsanity.com/paper/2302.04586/) (2023)
- [Exact Gradients for Stochastic Spiking Neural Networks Driven by Rough Signals](https://mlsanity.com/paper/2405.13587/) (2024)
- [Sparse Signature Coefficient Recovery via Kernels](https://mlsanity.com/paper/2412.08579/) (2024)
- [pySigLib -- Fast Signature-Based Computations on CPU and GPU](https://mlsanity.com/paper/2509.10613/) (2025)
- [Novelty Detection on Radio Astronomy Data using Signatures](https://mlsanity.com/paper/2402.14892/) (2024)
- [ParallelFlow: Parallelizing Linear Transformers via Flow Discretization](https://mlsanity.com/paper/2504.00492/) (2025)
- [Learning the Dynamics of Sparsely Observed Interacting Systems](https://mlsanity.com/paper/2301.11647/) (2023)
- [Learning to Embed Distributions via Maximum Kernel Entropy](https://mlsanity.com/paper/2408.00549/) (2024)
- [Numerical Schemes for Signature Kernels](https://mlsanity.com/paper/2502.08470/) (2025)
- [Signature Maximum Mean Discrepancy Two-Sample Statistical Tests](https://mlsanity.com/paper/2506.01718/) (2025)
- [Non-parametric online market regime detection and regime clustering for multidimensional and path-dependent data structures](https://mlsanity.com/paper/2306.15835/) (2023)
- [Path Signature Area-Based Causal Discovery in Coupled Time Series](https://mlsanity.com/paper/2110.12288/) (2021)
- [Deep incremental learning models for financial temporal tabular datasets with distribution shifts](https://mlsanity.com/paper/2303.07925/) (2023)
- [Explicit and Effectively Symmetric Schemes for Neural SDEs on Lie Groups](https://mlsanity.com/paper/2509.20599/) (2025)
- [Path Signatures Enable Model-Free Mapping of RNA Modifications](https://mlsanity.com/paper/2511.08855/) (2025)
- [Generating drawdown-realistic financial price paths using path signatures](https://mlsanity.com/paper/2309.04507/) (2023)
- [Gaussian processes based data augmentation and expected signature for time series classification](https://mlsanity.com/paper/2310.10836/) (2023)
- [Log-PDE Methods for Rough Signature Kernels](https://mlsanity.com/paper/2404.02926/) (2024)
- [Scalable Signature-Based Distribution Regression via Reference Sets](https://mlsanity.com/paper/2410.09196/) (2024)
- [Novelty detection on path space](https://mlsanity.com/paper/2512.03243/) (2025)
- [Random Controlled Differential Equations](https://mlsanity.com/paper/2512.23670/) (2025)
- [Branched Signature Kernel Solvers for ODEs with rough Single-Trajectory signals](https://mlsanity.com/paper/2605.25826/) (2026)
- [Faithful Embeddings of Irregular and Asynchronous Data for Online Log-NCDEs](https://mlsanity.com/paper/2605.30213/) (2026)

## Abstract (arXiv)

Distribution regression refers to the supervised learning problem where labels are only available for groups of inputs instead of individual inputs. In this paper, we develop a rigorous mathematical framework for distribution regression where inputs are complex data streams. Leveraging properties of the expected signature and a recent signature kernel trick for sequential data from stochastic analysis, we introduce two new learning techniques, one feature-based and the other kernel-based. Each is suited to a different data regime in terms of the number of data streams and the dimensionality of the individual streams. We provide theoretical results on the universality of both approaches and demonstrate empirically their robustness to irregularly sampled multivariate time-series, achieving state-of-the-art performance on both synthetic and real-world examples from thermodynamics, mathematical finance and agricultural science.