# LOCAL: Learning with Orientation Matrix to Infer Causal Structure from Time Series Data

arXiv: https://arxiv.org/abs/2410.19464 · HTML: https://mlsanity.com/paper/2410.19464/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 43 references that are in the map.
- First posted on arXiv 25 Oct 2024.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [DYNOTEARS: Structure Learning from Time-Series Data](https://mlsanity.com/paper/2002.00498/) (2020)
- [On the Role of Sparsity and DAG Constraints for Learning Linear DAGs](https://mlsanity.com/paper/2006.10201/) (2020)
- [DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization](https://mlsanity.com/paper/2209.08037/) (2022)
- [Masked Gradient-Based Causal Structure Learning](https://mlsanity.com/paper/1910.08527/) (2019)
- [Economy Statistical Recurrent Units For Inferring Nonlinear Granger Causality](https://mlsanity.com/paper/1911.09879/) (2019)
- [DAGs with No Curl: An Efficient DAG Structure Learning Approach](https://mlsanity.com/paper/2106.07197/) (2021)
- [Neural graphical modelling in continuous-time: consistency guarantees and algorithms](https://mlsanity.com/paper/2105.02522/) (2021)
- [CUTS+: High-dimensional Causal Discovery from Irregular Time-series](https://mlsanity.com/paper/2305.05890/) (2023)
- [EigenTrajectory: Low-Rank Descriptors for Multi-Modal Trajectory Forecasting](https://mlsanity.com/paper/2307.09306/) (2023)
- [CausalTime: Realistically Generated Time-series for Benchmarking of Causal Discovery](https://mlsanity.com/paper/2310.01753/) (2023)
- [NTS-NOTEARS: Learning Nonparametric DBNs With Prior Knowledge](https://mlsanity.com/paper/2109.04286/) (2021)
- [CUTS: Neural Causal Discovery from Irregular Time-Series Data](https://mlsanity.com/paper/2302.07458/) (2023)
- [BayesDAG: Gradient-Based Posterior Inference for Causal Discovery](https://mlsanity.com/paper/2307.13917/) (2023)
- [Rhino: Deep Causal Temporal Relationship Learning With History-dependent Noise](https://mlsanity.com/paper/2210.14706/) (2022)
- [Constraint-Free Structure Learning with Smooth Acyclic Orientations](https://mlsanity.com/paper/2309.08406/) (2023)
- [Jacobian Regularizer-based Neural Granger Causality](https://mlsanity.com/paper/2405.08779/) (2024)
- [Learning DAGs from Data with Few Root Causes](https://mlsanity.com/paper/2305.15936/) (2023)
- [From graphs to DAGs: a low-complexity model and a scalable algorithm](https://mlsanity.com/paper/2204.04644/) (2022)
- [Directed Acyclic Graph Structure Learning from Dynamic Graphs](https://mlsanity.com/paper/2211.17029/) (2022)
- [CausalMMM: Learning Causal Structure for Marketing Mix Modeling](https://mlsanity.com/paper/2406.16728/) (2024)
- [Recovering Linear Causal Models with Latent Variables via Cholesky Factorization of Covariance Matrix](https://mlsanity.com/paper/2311.00674/) (2023)

## Built on by (most cited first, 2 of 2)

- [Toward Reasoning-Centric Time-Series Analysis](https://mlsanity.com/paper/2510.13029/) (2025)
- [AutoCause: A Python framework that automates expert decisions in environmental time-series causal discovery](https://mlsanity.com/paper/2608.00198/) (2026)

## Abstract (arXiv)

Discovering the underlying Directed Acyclic Graph (DAG) from time series observational data is highly challenging due to the dynamic nature and complex nonlinear interactions between variables. Existing methods typically search for the optimal DAG by optimizing an objective function but face scalability challenges, as their computational demands grow exponentially with the dimensional expansion of variables. To this end, we propose LOCAL, a highly efficient, easy-to-implement, and constraint-free method for recovering dynamic causal structures. LOCAL is the first attempt to formulate a quasi-maximum likelihood-based score function for learning the dynamic DAG equivalent to the ground truth. Building on this, we introduce two adaptive modules that enhance the algebraic characterization of acyclicity: Asymptotic Causal Mask Learning (ACML) and Dynamic Graph Parameter Learning (DGPL). ACML constructs causal masks using learnable priority vectors and the Gumbel-Sigmoid function, ensuring DAG formation while optimizing computational efficiency. DGPL transforms causal learning into decomposed matrix products, capturing dynamic causal structure in high-dimensional data and improving interpretability. Extensive experiments on synthetic and real-world datasets demonstrate that LOCAL significantly outperforms existing methods and highlight LOCAL's potential as a robust and efficient method for dynamic causal discovery.