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Non-negative DAG Learning from Time-Series Data

Samuel Rey, Gonzalo Mateos

TL;DR

The paper addresses learning instantaneous DAG structure from multivariate time-series modeled by SVARM, where traditional acyclicity constraints yield non-convex optimization. It introduces a convex formulation by enforcing non-negativity on the instantaneous weights and using a convex log-determinant acyclicity function, solved via a method-of-multipliers augmented Lagrangian to guarantee a global optimum. Empirical results on synthetic data show that the proposed CVX-DYN method outperforms DYNOTEARS in recovering both instantaneous and lagged dependencies, with strong performance even in high-dimensional regimes. This approach offers reliable, scalable causal structure learning from time-series with theoretical guarantees on optimality and practical benefits for downstream applications.

Abstract

This work aims to learn the directed acyclic graph (DAG) that captures the instantaneous dependencies underlying a multivariate time series. The observed data follow a linear structural vector autoregressive model (SVARM) with both instantaneous and time-lagged dependencies, where the instantaneous structure is modeled by a DAG to reflect potential causal relationships. While recent continuous relaxation approaches impose acyclicity through smooth constraint functions involving powers of the adjacency matrix, they lead to non-convex optimization problems that are challenging to solve. In contrast, we assume that the underlying DAG has only non-negative edge weights, and leverage this additional structure to impose acyclicity via a convex constraint. This enables us to cast the problem of non-negative DAG recovery from multivariate time-series data as a convex optimization problem in abstract form, which we solve using the method of multipliers. Crucially, the convex formulation guarantees global optimality of the solution. Finally, we assess the performance of the proposed method on synthetic time-series data, where it outperforms existing alternatives.

Non-negative DAG Learning from Time-Series Data

TL;DR

The paper addresses learning instantaneous DAG structure from multivariate time-series modeled by SVARM, where traditional acyclicity constraints yield non-convex optimization. It introduces a convex formulation by enforcing non-negativity on the instantaneous weights and using a convex log-determinant acyclicity function, solved via a method-of-multipliers augmented Lagrangian to guarantee a global optimum. Empirical results on synthetic data show that the proposed CVX-DYN method outperforms DYNOTEARS in recovering both instantaneous and lagged dependencies, with strong performance even in high-dimensional regimes. This approach offers reliable, scalable causal structure learning from time-series with theoretical guarantees on optimality and practical benefits for downstream applications.

Abstract

This work aims to learn the directed acyclic graph (DAG) that captures the instantaneous dependencies underlying a multivariate time series. The observed data follow a linear structural vector autoregressive model (SVARM) with both instantaneous and time-lagged dependencies, where the instantaneous structure is modeled by a DAG to reflect potential causal relationships. While recent continuous relaxation approaches impose acyclicity through smooth constraint functions involving powers of the adjacency matrix, they lead to non-convex optimization problems that are challenging to solve. In contrast, we assume that the underlying DAG has only non-negative edge weights, and leverage this additional structure to impose acyclicity via a convex constraint. This enables us to cast the problem of non-negative DAG recovery from multivariate time-series data as a convex optimization problem in abstract form, which we solve using the method of multipliers. Crucially, the convex formulation guarantees global optimality of the solution. Finally, we assess the performance of the proposed method on synthetic time-series data, where it outperforms existing alternatives.

Paper Structure

This paper contains 6 sections, 10 equations, 1 figure.

Table of Contents

  1. Introduction
  2. DAG structure learning from time series
  3. Learning non-negative DAGs from SVAR data
  4. DAG structure learning via the method of multipliers
  5. Numerical experiments
  6. Conclusions

Figures (1)

  • Figure 1: Performance evaluation of the proposed method (CVX-DYN) compared to DYNOTEARS (we use DYNO in the legends for clarity) across different scenarios. (a) NFE of the estimated weights ${\hat{\mathbf W} }$ and ${\hat{\mathbf A} }$ as a function of the time-series length $T$. (b) F1-score of ${\hat{\mathbf W} }$ and ${\hat{\mathbf A} }$ as the number of nodes $N$ increases. (c) NFE for varying autoregressive orders $P$.