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$l_{1-2}$ GLasso: $L_{1-2}$ Regularized Multi-task Graphical Lasso for Joint Estimation of eQTL Mapping and Gene Network

Wei Miao, Lan Yao

arXiv:2301.02225v1stat.MLmath.STq-bio.QMstat.TH

TL;DR

The paper addresses joint estimation of SNP–gene eQTL associations and gene-network structure. It proposes an L1−2 regularized graphical-lasso model to share information between these tasks. The authors report improved recovery on synthetic data and sparser, more accurate solutions on real data than commonly used methods.

  • Problem

    The paper addresses how to jointly estimate SNP–gene associations and gene–gene relationships in order to study gene-expression regulation.

    Source: §1 · p. 2

  • Method

    The proposed L1−2 graphical lasso jointly estimates regression coefficients B and output structure Θ, using a new sparsity penalty for B and a graph-guided fusion penalty.

    Source: §3 · pp. 5–6, §3.2 · p. 6

  • Results

    The authors report that L1−2 GLasso outperforms other models in synthetic recovery of eQTL associations and gene-network structure, with real-data results showing superiority and a sparser, more accurate solution than baseline methods.

    Source: §1 · p. 2, §6 · p. 18

  • Takeaways & Limitations

    The reported synthetic and real-data results support L1−2 GLasso as a method for recovering both eQTL associations and gene-network structure.

    Source: §6 · p. 18

    The authors identify improving solution efficiency and decomposing large-scale problems as future work.

    Source: §6 · p. 18

Abstract

from arXiv · show

A critical problem in genetics is to discover how gene expression is regulated within cells. Two major tasks of regulatory association learning are : (i) identifying SNP-gene relationships, known as eQTL mapping, and (ii) determining gene-gene relationships, known as gene network estimation. To share information between these two tasks, we focus on the unified model for joint estimation of eQTL mapping and gene network, and propose a $L_{1-2}$ regularized multi-task graphical lasso, named $L_{1-2}$ GLasso. Numerical experiments on artificial datasets demonstrate the competitive performance of $L_{1-2}$ GLasso on capturing the true sparse structure of eQTL mapping and gene network. $L_{1-2}$ GLasso is further applied to real dataset of ADNI-1 and experimental results show that $L_{1 -2}$ GLasso can obtain sparser and more accurate solutions than other commonly-used methods.

1. Introduction.

The introduction frames joint eQTL mapping and gene-network estimation as structured-sparsity problems in high-dimensional genetic data. It presents an L1−L2 graphical-lasso model intended to produce sparser regression estimates and improve recovery of output structure.

1. Introduction.

  • eQTL methods progressed from testing a gene–SNP association to modeling multiple SNPs with one gene and then multiple SNPs with multiple genes.The introduction also describes gene networks as gene relationships represented by edges, including dependencies modeled through Gaussian graphical models.Source: §1 · pp. 1–2
  • High-dimensional genetic data motivate sparse learning, valued for sparse solutions, interpretability, and convenient computation.Structured-sparse models combine sparse learning with regularization based on assumed sparsity and problem structure.Source: §1 · p. 2
  • Structured-sparsity approaches jointly estimate regression coefficients and output structure, but several require prior knowledge of relationships among outputs.Other approaches estimate conditional covariance rather than output covariance; noisy data may prevent recovery of the true output structure.Source: §1 · p. 2
  • The L1−L2 penalty has been studied as a robust sparse-solution penalty and applied in settings including nonnegative least squares, signal reconstruction, and compressed sensing.Prior work also describes it as an approximation to L0-based sparsity.Source: §1 · p. 2
  • The proposed model adds a difference-of-L1-and-L2 penalty and is reported to yield sparser regression coefficients and better output-structure recovery than baseline methods.The introduction also reports a faster optimization algorithm.Source: §1 · p. 2

2. Background.

The background introduces regression and graphical-model components for jointly estimating SNP–gene coefficients and gene-output structure. It reviews Multi-Task Lasso, MRCE, and ICLasso, then motivates replacing the L1 penalty with a sparsity-promoting alternative.

2. Background.

  • The background represents gene-expression values as a matrix Y with n samples and q genes, and SNP genotypes as a matrix X with n samples and p SNPs.The matrix form relates X, a coefficient matrix B, and Y.Source: §2 · p. 3

2.1. Multi-Task Lasso.

  • Multi-Task Lasso uses multivariate linear regression to detect SNPs significantly associated with genes, with B denoting regression coefficients and λ controlling sparsity.The passage assumes normally distributed errors.Source: §2.1 · p. 3

2.2. Multivariate regression with covariance estimation.

  • MRCE jointly estimates regression coefficients and output structure, but its Ω represents conditional inverse covariance of Y|X and cannot capture the exact relationship between B and output structure.The passage characterizes Ω as related to the Gaussian noise matrix.Source: §2.2 · p. 3, §2.1 · p. 3

2.3. Inverse-Covariance-Fused Lasso.

  • ICLasso jointly estimates B and the marginal inverse covariance Θ; under simplifying covariance assumptions, Θ^-1 is proportional to B^T B.Its objective combines Multi-Task Lasso, sparse inverse-covariance estimation, and a graph-guided fusion penalty that encourages information sharing between B and Θ.Source: §2.3 · p. 4

2.4. Estimating Model Parameters with a Fusion Penalty.

  • The background describes the fusion-penalty objective as combining conditional and marginal negative log-likelihood terms with a penalty encouraging shared information between B and Θ.Because L1 penalties induce sparsity, the authors motivate generalizing the penalty to obtain sparser solutions and introduce their L1−2 model.Source: §2.4 · p. 5, §3 · p. 5

3. The l1−2 Graphical Lasso.

The section introduces an l1−2 regularized graphical lasso for jointly modeling regression coefficients and output relationships, then describes its alternating and coordinate-based optimization. The method uses sparsity and graph-guided fusion penalties, with synthetic-data comparisons set up against established alternatives.

3. The l1−2 Graphical Lasso.

  • The l1−2 penalty promotes sparse regression coefficients and is described as closer to the l0 norm than l1, while graph-guided fusion encourages coefficients of related outputs to be similar.The fusion penalty scales with partial correlations, with its sign reflecting positive or negative relationships.Source: §3 · p. 6
  • The model assumes x ∼ N(0,T) and y|x ∼ N(x^T B,E); compared with ICLasso, it uses the l1−2 penalty to seek sparser solutions while maintaining regression error.The paper states that suitable λ and τ can yield a solution close to one with an l0 penalty.Source: §3 · p. 5, §3.1 · p. 6

3.1. Relationship to ICLasso.

  • Because the graph-guided fusion term is bi-convex, the authors use alternating minimization to solve the l1−2-GLasso problem.They describe this strategy as building on an approach developed by Marchetti-Bowick et al. (2019).Source: §3.2 · p. 6

3.2. Optimization.

  • The new regularizer is decomposed into differentiable and non-differentiable parts, and the convex part is solved with PA-PG, which the authors say converges consistently faster than proximal gradient descent.The text separately states that the subgradient method converges in at most O(1/ε) steps for accuracy ε > 0.Source: §3.2 · pp. 6–7
  • The optimization uses block coordinate descent, coordinate-wise updates, and a variant of soft thresholding; the text also reports a closed-form solution for β.With Θ fixed, the B subproblem includes a gradient step using X^T(XB − Y) − τΣ.Source: §3.2 · p. 7
  • On synthetic data with known B and Θ, the study compares GFLasso, MRCE, ICLasso, and l1−2-GLasso, selecting each model’s hyperparameters by held-out validation error.The setup is intended to measure recovery of the true parameter values directly.Source: §4 · p. 7

4. Simulation Study.

The simulation study tests recovery of the regression matrix B and output structure Θ, as well as prediction error, across synthetic settings. Across the four cases considered, l1-2-GLasso outperforms the baselines on the reported results.

Simulation Study

  • The simulations assess recovery of sparse structures with F1 scores and assess regression error by using estimated B to predict Y from X.For F1 scoring, a threshold classifies each element as zero or nonzero, then precision and recall are calculated.Source: §4 · p. 7
  • The synthetic design uses block-structured output networks, with each non-overlapping gene group forming a fully connected subgraph.The data-generation procedure fixes sparse structures, generates coefficient values, and then samples X and Y.Source: §4 · pp. 7–8
  • The coefficient-generation design assigns each module a primary gene and gives its associated SNPs strengths drawn from Uniform(0,1).Other genes in the module receive strengths drawn from Uniform(βjk, ρ2), with ρ = 0.1 in the described setting.Source: §4 · p. 8
  • For the single synthetic dataset, l1-2-GLasso’s estimated B is closer to the real B than the other models’ estimates.The dataset uses n = 120, p = 60, and q = 60, with E = Iqq and T not equal to Ipp.Source: §4 · p. 8, Figure 1 · p. 8
  • Across the four cases considered, l1-2-GLasso outperforms the baselines on recovery and regression-error results.The reported comparisons cover F1 scores for B and Θ and regression error on Y.Source: §4 · p. 9, Figure 3 · p. 9, Figure 4 · p. 10
  • Only l1-2-GLasso and ICLasso recover Θ more accurately in the single-dataset comparison.The real Θ and model estimates are compared in Figure 2.Source: §4 · p. 9, Figure 2 · p. 9
  • For 15 synthetic datasets, the study compares F1 scores for B and Θ and regression error across covariance structures and SNP counts.The covariance settings vary T and E; results are averaged over 15 datasets per case.Source: §4 · pp. 8, 10, Figure 3 · p. 9, Figure 4 · p. 10
  • The study also explores how the new hyperparameter τ affects regression error, using the Figure 1 dataset and different λ1 ratios.The supplied results state that error is minimized for τ < 3 and becomes increasingly stable for τ ≥ 3.Source: §4 · p. 11, Figure 6 · p. 11

5. Real dataset From eQTL Mapping.

On the ADNI-1 dataset, l1-2-GLasso selected more experimentally identified SNPs than the compared models and produced the smallest regression error. The results also indicate sparser regression coefficients and recovery of larger gene-network subnetworks.

5. Real dataset From eQTL Mapping.

  • The ADNI-1 analysis used 632 subjects and 486 SNPs from 15 candidate genes after quality control and imputation.The included SNPs were restricted to those belonging to the top 15 candidate genes listed in the AlzGene database.Source: §5 · p. 12
  • Across 10 datasets formed by randomly selecting 480 samples for fitting and using the remaining 152 to compute regression error, l1-2-GLasso produced the smallest regression error.Source: §5 · p. 12
  • Both l1-2-GLasso and ICLasso estimated the two larger gene-network subnetworks, which the authors say biological experiments had demonstrated.The authors also identify rs405509 and rs10787010 as potentially associated with the largest number of ROIs.Source: §5 · p. 13
  • For SNP rs4311, l1-2-GLasso produced a sparser coefficient pattern than ICLasso, assigning zero coefficients to four of five listed additional gene associations.For AmygVol (L), CerebCtx (L), LatVent (L), EntCtx (L), and Fusiform (L), the reported coefficients were 0.019947, 0, 0, 0, and 0, respectively; Figure 8 displays the coefficient matrices.Source: §5 · p. 13, Figure 8 · p. 14
  • Among the 24 SNPs identified in biological experiments, l1-2-GLasso selected 22, compared with 14 for GFLasso, 16 for MRCE, and 17 for ICLasso.Source: §5 · p. 13

6. Conclusion.

The conclusion points to solver efficiency and large-scale decomposition as future work, and notes possible extension to financial data. The analyzed data are available in the bgsmtr R package.

6. Conclusion.

  • Future work will seek a more efficient solution algorithm and decomposition for large-scale problems.Source: §6 · p. 18
  • The authors identify financial data as a possible extension domain for the model.Source: §6 · p. 18
  • The analyzed data are available in the R package bgsmtr.The passage gives R-console commands for loading the example data.Source: Data availability · p. 18
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