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Efficient learning of differential network in multi-source non-paranormal graphical models

Mojtaba Nikahd, Seyed Abolfazl Motahari

TL;DR

An efficient approach is proposed that outputs the exact solution path, outperforming the previous methods that only sample from the solution path in pre-selected regularization parameters and is shown to be very effective in inferring differential network in real-world problems.

Abstract

This paper addresses learning of sparse structural changes or differential network between two classes of non-paranormal graphical models. We assume a multi-source and heterogeneous dataset is available for each class, where the covariance matrices are identical for all non-paranormal graphical models. The differential network, which are encoded by the difference precision matrix, can then be decoded by optimizing a lasso penalized D-trace loss function. To this aim, an efficient approach is proposed that outputs the exact solution path, outperforming the previous methods that only sample from the solution path in pre-selected regularization parameters. Notably, our proposed method has low computational complexity, especially when the differential network are sparse. Our simulations on synthetic data demonstrate a superior performance for our strategy in terms of speed and accuracy compared to an existing method. Moreover, our strategy in combining datasets from multiple sources is shown to be very effective in inferring differential network in real-world problems. This is backed by our experimental results on drug resistance in tumor cancers. In the latter case, our strategy outputs important genes for drug resistance which are already confirmed by various independent studies.

Efficient learning of differential network in multi-source non-paranormal graphical models

TL;DR

An efficient approach is proposed that outputs the exact solution path, outperforming the previous methods that only sample from the solution path in pre-selected regularization parameters and is shown to be very effective in inferring differential network in real-world problems.

Abstract

This paper addresses learning of sparse structural changes or differential network between two classes of non-paranormal graphical models. We assume a multi-source and heterogeneous dataset is available for each class, where the covariance matrices are identical for all non-paranormal graphical models. The differential network, which are encoded by the difference precision matrix, can then be decoded by optimizing a lasso penalized D-trace loss function. To this aim, an efficient approach is proposed that outputs the exact solution path, outperforming the previous methods that only sample from the solution path in pre-selected regularization parameters. Notably, our proposed method has low computational complexity, especially when the differential network are sparse. Our simulations on synthetic data demonstrate a superior performance for our strategy in terms of speed and accuracy compared to an existing method. Moreover, our strategy in combining datasets from multiple sources is shown to be very effective in inferring differential network in real-world problems. This is backed by our experimental results on drug resistance in tumor cancers. In the latter case, our strategy outputs important genes for drug resistance which are already confirmed by various independent studies.
Paper Structure (15 sections, 4 theorems, 37 equations, 3 figures, 1 algorithm)

This paper contains 15 sections, 4 theorems, 37 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations
  3. Material and methods
  4. Non-paranormal graphical models
  5. Problem setting
  6. Algorithm
  7. Covariance matrix estimation
  8. Results
  9. Synthetic data
  10. Real data
  11. Conclusions
  12. Proof of Theorem \ref{['piecewise linear solution path']}
  13. Proof of Lemma \ref{['errorBoundOfCorrelationEstimator']}
  14. Proof of Theorem \ref{['errorBoundofDeltaEstimator']}
  15. Proof of Theorem \ref{['signConsistencyofDeltaEstimator']}

Key Result

Theorem 2.1

There are $\infty = \lambda_0 > \lambda_1 > \lambda_2 > \dots > \lambda_T \ge 0$, such that $\hat{\Delta}\left(\lambda\right)$ for any $\lambda \in \left(\lambda_T,\infty\right)$ is given by

Figures (3)

  • Figure 1: Performance of different approaches on synthetic data with 20 differential edges between two subjects, and different combinations of the number of iterations. The lines color and type indicate the method type and the number of iterations in iterative methods, respectively. Also, each point corresponds to the solution of an iterative method for a specific value of $\lambda$. (A) Accuracy of different method. (B) Speed of different methods.
  • Figure 2: Precision-recall curves of the single dataset, homogeneous dataset, and heterogeneous datasets.
  • Figure 3: The inferred differential network between platinum-sensitive and platinum-resistant ovarian tumors with the regularization parameter are obtained by StARS method. The node size and color correspond to its degree.

Theorems & Definitions (5)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Remark 1
  • Theorem 2.4