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Learning DAGs from Data with Few Root Causes

Panagiotis Misiakos, Chris Wendler, Markus Püschel

TL;DR

A novel perspective and algorithm for learning directed acyclic graphs (DAGs) from data generated by a linear structural equation model (SEM) is presented and it is proved that the true DAG is the global minimizer of the $L^0$-norm of the vector of root causes.

Abstract

We present a novel perspective and algorithm for learning directed acyclic graphs (DAGs) from data generated by a linear structural equation model (SEM). First, we show that a linear SEM can be viewed as a linear transform that, in prior work, computes the data from a dense input vector of random valued root causes (as we will call them) associated with the nodes. Instead, we consider the case of (approximately) few root causes and also introduce noise in the measurement of the data. Intuitively, this means that the DAG data is produced by few data-generating events whose effect percolates through the DAG. We prove identifiability in this new setting and show that the true DAG is the global minimizer of the $L^0$-norm of the vector of root causes. For data with few root causes, with and without noise, we show superior performance compared to prior DAG learning methods.

Learning DAGs from Data with Few Root Causes

TL;DR

A novel perspective and algorithm for learning directed acyclic graphs (DAGs) from data generated by a linear structural equation model (SEM) is presented and it is proved that the true DAG is the global minimizer of the -norm of the vector of root causes.

Abstract

We present a novel perspective and algorithm for learning directed acyclic graphs (DAGs) from data generated by a linear structural equation model (SEM). First, we show that a linear SEM can be viewed as a linear transform that, in prior work, computes the data from a dense input vector of random valued root causes (as we will call them) associated with the nodes. Instead, we consider the case of (approximately) few root causes and also introduce noise in the measurement of the data. Intuitively, this means that the DAG data is produced by few data-generating events whose effect percolates through the DAG. We prove identifiability in this new setting and show that the true DAG is the global minimizer of the -norm of the vector of root causes. For data with few root causes, with and without noise, we show superior performance compared to prior DAG learning methods.
Paper Structure (23 sections, 15 theorems, 51 equations, 6 figures, 9 tables)

This paper contains 23 sections, 15 theorems, 51 equations, 6 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Linear SEMs and Root Causes
  3. Transitive closure and root causes
  4. Example: Pollution model
  5. Learning the DAG
  6. Identifiability
  7. $L^0$ minimization problem and global minimizer
  8. Continuous relaxation
  9. Related work
  10. Experiments
  11. Evaluation on data with few root causes
  12. Evaluation on a real dataset
  13. Broader Impact and Limitations
  14. Conclusion
  15. Few Root Causes Assumption
  16. ...and 8 more sections

Key Result

Lemma 2.1

The linear SEM eq:SEMrecursive computes data $\mathbf{X}$ as In words, the data values in $\mathbf{X}$ are computed as linear combinations of the noise values $\mathbf{N}$ of all predecessor nodes with weights given by the reflexive-transitive closure $\mathbf{I} + \overline{\mathbf{A}}$.

Figures (6)

  • Figure 1: (a) A DAG for a river network. The weights capture fractions of pollution transported between adjacent nodes. (b) The transitive closure. The weights are fractions of pollution transported between all pairs of connected nodes. (c) A possible vector measuring pollution, and (d) the root causes of the pollution, sparse in this case.
  • Figure 2: Performance report on the default settings. (a,b,c) illustrate SHD, SID and runtime (lower is better) while varying the number of nodes with $1000$ samples (first row) or varying the number of samples with $100$ nodes (second row). (d, e) illustrate TPR and FPR of the estimated support of $\mathbf{C}$, and (f) reports the accuracy of estimating $\mathbf{C}$ as NMSE.
  • Figure 3: Evaluation of the top-performing methods on denser DAGs.
  • Figure 4: Plots illustrating performance metrics (a) SHD (lower is better), (b) SID (lower is better), (c) Time [seconds], (d) TPR (higher is better), (e) Total number of proposed edges and (f) NMSE (lower is better). Each metric is evaluated in two experimental scenarios: when varying the number of rows (upper figure) and when varying the number of samples (lower figure).
  • Figure 5: Evaluation of LiNGAM's performance when $\mathbf{N}_x=\bm{0}$.
  • ...and 1 more figures

Theorems & Definitions (37)

  • Lemma 2.1
  • Definition 2.2: Few Root Causes assumption
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma A.1
  • proof
  • Theorem C.1
  • Theorem C.2
  • ...and 27 more