# TRACE: Trajectory Recovery for Continuous Mechanism Evolution in Causal Representation Learning

arXiv: https://arxiv.org/abs/2601.21135 · HTML: https://mlsanity.com/paper/2601.21135/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 69 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Builds on (most cited first, 30 of 39)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [DeepSeekMoE: Towards Ultimate Expert Specialization in Mixture-of-Experts Language Models](https://mlsanity.com/paper/2401.06066/) (2024)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Return of Frustratingly Easy Domain Adaptation](https://mlsanity.com/paper/1511.05547/) (2015)
- [Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learning](https://mlsanity.com/paper/1805.08651/) (2018)
- [Unsupervised Feature Extraction by Time-Contrastive Learning and Nonlinear ICA](https://mlsanity.com/paper/1605.06336/) (2016)
- [The Unmanned Aerial Vehicle Benchmark: Object Detection and Tracking](https://mlsanity.com/paper/1804.00518/) (2018)
- [OpenMoE: An Early Effort on Open Mixture-of-Experts Language Models](https://mlsanity.com/paper/2402.01739/) (2024)
- [Causal Discovery from Heterogeneous/Nonstationary Data with Independent Changes](https://mlsanity.com/paper/1903.01672/) (2019)
- [CITRIS: Causal Identifiability from Temporal Intervened Sequences](https://mlsanity.com/paper/2202.03169/) (2022)
- [Learning Temporally Causal Latent Processes from General Temporal Data](https://mlsanity.com/paper/2110.05428/) (2021)
- [Continuous-Time Sequential Recommendation with Temporal Graph Collaborative Transformer](https://mlsanity.com/paper/2108.06625/) (2021)
- [Causal Representation Learning from Multiple Distributions: A General Setting](https://mlsanity.com/paper/2402.05052/) (2024)
- [Temporally Disentangled Representation Learning](https://mlsanity.com/paper/2210.13647/) (2022)
- [Adaptive piecewise polynomial estimation via trend filtering](https://mlsanity.com/paper/1304.2986/) (2013)
- [Collapsed Amortized Variational Inference for Switching Nonlinear Dynamical Systems](https://mlsanity.com/paper/1910.09588/) (2019)
- [CaRiNG: Learning Temporal Causal Representation under Non-Invertible Generation Process](https://mlsanity.com/paper/2401.14535/) (2024)
- [Temporally Disentangled Representation Learning under Unknown Nonstationarity](https://mlsanity.com/paper/2310.18615/) (2023)
- [Bridging Past and Future: Distribution-Aware Alignment for Time Series Forecasting](https://mlsanity.com/paper/2509.14181/) (2025)
- [On the Identification of Temporally Causal Representation with Instantaneous Dependence](https://mlsanity.com/paper/2405.15325/) (2024)
- [Causal Temporal Representation Learning with Nonstationary Sparse Transition](https://mlsanity.com/paper/2409.03142/) (2024)
- [Causal Representation Learning from Multimodal Biomedical Observations](https://mlsanity.com/paper/2411.06518/) (2024)
- [Interpolating Video-LLMs: Toward Longer-sequence LMMs in a Training-free Manner](https://mlsanity.com/paper/2409.12963/) (2024)
- [Infant Agent: A Tool-Integrated, Logic-Driven Agent with Cost-Effective API Usage](https://mlsanity.com/paper/2411.01114/) (2024)
- [Glocal Information Bottleneck for Time Series Imputation](https://mlsanity.com/paper/2510.04910/) (2025)

## Built on by (most cited first, 4 of 4)

- [MOSAIC: Module Discovery via Sparse Additive Identifiable Causal Learning for Scientific Time Series](https://mlsanity.com/paper/2605.05524/) (2026)
- [Mind Dreamer: Untethering Imagination via Active Causal Intervention on Latent Manifolds](https://mlsanity.com/paper/2605.16030/) (2026)
- [Disentangling Continuous-Time Latent Dynamics: Identifiability of Latent SDEs via Diffusion Shifts](https://mlsanity.com/paper/2606.28228/) (2026)
- [TimeEvo: Failure-Driven Self-Evolution of a Time Series Agent](https://mlsanity.com/paper/2609.27277/) (2026)

## Abstract (arXiv)

Temporal causal representation learning methods assume that causal mechanisms switch instantaneously between discrete domains, yet real-world systems often exhibit continuous mechanism transitions. For example, a vehicle's dynamics evolve gradually through a turning maneuver, and human gait shifts smoothly from walking to running. We formalize this setting by modeling transitional mechanisms as convex combinations of finitely many atomic mechanisms, governed by time-varying mixing coefficients. Our theoretical contributions establish that both the latent causal variables and the continuous mixing trajectory are jointly identifiable. We further propose TRACE, a Mixture-of-Experts framework where each expert learns one atomic mechanism during training, enabling recovery of mechanism trajectories at test time. This formulation generalizes to intermediate mechanism states never observed during training. Experiments on synthetic and real-world data demonstrate that TRACE recovers mixing trajectories with up to 0.99 correlation, substantially outperforming discrete-switching baselines.