uGLAD: A deep learning model to recover conditional independence graphs
Authors: Harsh Shrivastava
Urszula Chajewska
Robin Abraham
Xinshi Chen
Venue: Submitted to ICLR 2023 (2023)
Keywords: Graphical Lasso
Deep Learning
Unrolled Algorithms
Conditional Independence graphs
Sparse graphs

Abstract: Probabilistic Graphical Models are generative models of complex systems. They rely on conditional independence assumptions between variables to learn sparse representations which can be visualized in a form of a graph. Such models are used for domain exploration and structure discovery in poorly understood domains. This work introduces a novel technique to perform sparse graph recovery by optimizing deep unrolled networks. Assuming that the input data $X\in\mathbb{R}^{M\times D}$ comes from an underlying multivariate Gaussian distribution, we apply a deep model on $X$ that outputs the precision matrix $\Theta$. Then, the partial correlation matrix \mathrm{P} is calculated which can also be interpreted as providing a list of conditional independence assertions holding in the input distribution. Our model, \texttt{uGLAD}, builds upon and extends the state-of-the-art model \texttt{GLAD} to the unsupervised setting. The key benefits of our model are (1) \texttt{uGLAD} automatically optimizes sparsity-related regularization parameters leading to better performance than existing algorithms. (2) We introduce multi-task learning based `consensus' strategy for robust handling of missing data in an unsupervised setting. We evaluate performance on synthetic Gaussian, non-Gaussian data generated from Gene Regulatory Networks, and present case studies in anaerobic digestion and infant mortality.

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UGLAD: A DEEP LEARNING MODEL TO RECOVER CONDITIONAL INDEPENDENCE GRAPHS
Anonymous authors Paper under double-blind review
ABSTRACT
Probabilistic Graphical Models are generative models of complex systems. They rely on conditional independence assumptions between variables to learn sparse representations which can be visualized in a form of graph. Such models are used for domain exploration and structure discovery in poorly understood domains. This work introduces a novel technique to perform sparse graph recovery by opti- mizing deep unrolled networks. Assuming that the input data X E IR IRM comes from an underlying multivariate Gaussian distribution, we apply deep model on X that outputs the precision matrix θ Then, the partial correlation matrix P is calculated which can also be interpreted as providing list of conditional independence assertions holding in the input distribution. Our model, uGLAD' builds upon and extends the sfte-of-thhe-rr model GLAD Shrivastava al. (2020b) to the unsupervised setting. The key benefits of our model are (1) uGLAD auto- matically optimizes sparsity-related regularization parameters leading to better performance than existing algorithms. (2) We introduce multi-task learning based 'consensus' strategy for robust handling of missing data in an unsupervised setting. We evaluate performance synthetic Gaussian, non-Gaussian data generated from Gene Regulatory Networks, and present case studies in anaerobic digestion and infant mortality.
Keywords: Graphical Lasso, Deep Learning, Unrolled Algorithms, Conditional Independence graphs, Sparse graphs
INTRODUCTION
Probabilistic graphical models (PGMs) Pearl (1988); Koller & Friedman (2009) are generative models of complex systems, used to describe dependencies within a set of random variables and visualize the structure of domain. The models rely on conditional independence assumptions between variables, which result in sparse representation and enable efficient inference. In the graphical representation of the models, conditional independence is indicated by absence of an edge between two variables. Such models can be learned from observational data Heckerman (1995); Friedman et al. (2008). Structure discovery enabled by PGMs is important for new and poorly understood domains where relationships between variables are not known. PGMs have been used in various domains, including medical diagnosis Heckerman al. (1992); Heckerman & Nathwant (1992), fault diagnosis. genomic data analysis via gene regulatory networks Moerman et al. (2019); Pratapa et al. (2020); Aluru et al. (2021): Shrivastava al. (2022), speech recognition, and in finance Hallac al. (2017).
The problem of recovering the structure from observational data is particularly difficult in high- dimensional settings, where the number of features may be larger than the number of observations. We focus specifically on learning undirected models where the data is assumed to have been generated from a multivariate Gaussian distribution Friedman et (2008); Belilovsky al. (2017); Zheng et (2018); Yu et al. (2019); Shrivastava et al. (2020b). In such cases, the goal is to estimate a sparse inverse covariance matrix. Sparsity is typically enforced by the use of E (lasso) regularization.
Assume we have d-dimensional multivariate Gaussian random variable X IX....., Xa] with m observations. The goal is to estimate its covariance matrix D and precision matrix O- = (2*)-1 O encodes conditional independence assumptions between variables: the ij component is zero
UGLAD code link (Provided in Supplementary)

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if and only if X; and X, conditionally independent given the other variables (Xx) (X This problem is known as the sparse graph recovery problem and usually formulated (following Friedman al. (2008)): as the f /--regularized maximum likelihood estimation
θ arg min - -log(det @) + +rr(ee) + p1|@ll
(1)
where is the empirical covariance matrix based on m samples, S is the space of d d symmetric positive definite matrices (SPD), and ||6|| o = Lit, 1Qi is the off-diagonal L regularizer with regularization parameter The use of this estimator is justified even for non Non-Gaussian X, since it minimizing an l --penalized log-determinant Bregman divergence Ravikumar et al. (2011), The problem in Eq. is convex optimization problem. It can be solved by many algorithms, see Section 2 for examples.
However, classic approaches have their limitations in both the statistical aspect and computational aspect. Statistically, the classic formulation uses single regularization parameter for all entries in the precision matrix θ which may not be optimal. A recent theoretical work Sun et al. (2018) validates the use of adaptive parameters. Shrivastava et al. (2020b) has proposed a model with multiple regularization parameters called GLAD and has shown emprical evidence that such model pushes the sample complexity limits. Based on that evidence, we hypothesise that one may obtain better recovery results by allowing the regularization parameters vary across the entries in the precision matrix. However, it is hard for traditional approaches to search over a large number of hyperparameters. Computationally. the complexity of solving the optimization depends on the convexity of the objective, the step sizes of the algorithm, the initialization, the design of the update steps, etc. Different problems may require different designs of the algorithm achieve better efficiency. A unified design for all problems may not be optimal.
In this work, we propose uGLAD (unsupervised-GLLA)) which is deep model that can recover sparse graphs in an unsupervised manner. As the name suggests, it builds upon and extends the GLAD model, which recovers sparse graphs under supervision. uGLAD uses the same objective function as GLAD Using additional variable Z, the reguulrrrd maximum likelihood from Eq. can be written as
e arg min - log(det 9) - tree) + p||
stt. sst. Z =
Now, including the constraint as squared penalty term À we obtain the reformulated objective as
θ 2 arg mins, log(det Q) + tr (īe) + + 2II, + 1 - 21
(2)
Note that introducing the variable Z helps in splitting the objective into parts and those can be optimized alternately using the Alternating Minimization algorithm. A Conditional Independence (CI) graph can be obtained from this precision matrix by calculating the partial correlation matrix. The CI graphs very informative they describe how the features in domain interact with each other. Furthermore, they can additionally capture negative partial correlations between features which are usually not modeled by traditional graphs
Key contributions of this work:
Extending GLAD to unsupervised setting: The uGLAD docsn't rely on availability of ground truth
to do graph recovery.
Adaptive hyperparameters: The uGLAD architecture design enables the hyperparameters to opti- mally adapt at each step of the unrolled Alternating Minimization (AM) algorithm Shrivastava et al. (2020b) that leads its superior performance.
Automatically decide optimum sparsity parameters: The sparsity of the recovered graph is highly sensitive to the choice the regularization hyperparameters. Instead, uGLAD models hyperparam- cters within the neural network framework and they are directly optimized for the uGLAD objective defined above. So, there is no need to separately optimize the sparsity byperparameters which otherwise computationally expensive process.
Runtime efficiency: The uGLAD software can run on GPUs for higher time efficiency and scalability. Missing data handling: The uGLAD framework can also be used for multi-task learning. We leverage this property further to develop a novel 'consensus' strategy to robustly handle missing data.

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2 RELATED WORK
Traditional algorithms for graphical lasso: These are primarily iterative methods for optimizing the graphical lasso objective. Main methods developed for this problem in the last two decades are dctailed in this survey paper Witten ct al. (2011). The variant of the Block Coordinate Descent (BCD) algorithm by Friedman et al. (2008) is widely used to recover graphs using the graphical lasso method. This method is also implemented in the popular python sklearn package. The G-ISTA algorithm by Guillot al. (2012) based on the iterative shrinkage thresholding procedure. is one of the prominent methods based using the proximal gradient descent approach. The Alternating Direction Method of Multipliers (ADMM) Danaher et al. (2014) has also been successfully used in various graphical lasso based applications.
Deep Learning approaches for graph recovery: DeepGraph (DG) Belilovsky et al. (2017). supervised deep learning method that takes in the input samples and outputs the corresponding adjacency matrix which shows the connections between input features. DeepGraph architecture consists of many convolutional layers followed by multi-layer perceptrons that finally decides whether an edge is present between every combination of features. Another relevant DL method roughly based on modeling the input data with a Variational Autoencoder for graph recovery is DAG-GNN Yu et al. (2019). These deep architectures have very high number of learnable parameters, which is a significant drawback. Hence, we pursue different line of research (using inductive biases) which gives similar performance with significantly reduced number of learnable parameters and brings more interpretability as shown in Shrivastava et al. (2020b).
Deep learning models using inductive biases: domain knowledge in the desigan/nitialization of deep learning architectures. For instance, Shrivastava al. (2019) presents generic technique to use probabilistic graphical model prior to design a deep model. The authors were able to show enhanced performance on the document classification task by leveraging the Latent Dirichlet Allocation prior. Another way of including prior knowledge about the domain is using an optimization algorithm for a related objective function as a template to design the deep architecture. Unrolling the optimization algorithms and parameterizing the step updates using neural networks have been fairly successful for many tasks Liu & Chen (2019); Chen al. (2020); Shrivastava (2020): Pu et al. (2021); Shrivastava et al. (2022).
Improved performance often results from including
This work focuses on recovering undirected graphs, specifically based on optimizing the graphical lasso objective function. Hence, we skipped discussing the methods developed specifically to recover Directed Acyclic Graphs (DAGs). The work most closely related to ours is the GLAD Shrivastava al. (2020b) model. Since our algorithm builds upon GLAD's architecture we are going to describe it in detail in Section 3, while pointing out ways in which uGLAD differs from GLAD.
3 THE UGLAD MODEL
Given imput data X E RMMDD, with M samples with D features, the task is to recover a sparse graph showing partial correlations between the features. Recovering the sparse graph (the adjacency matrix) corresponds to obtaining the precision matrix Q and the partial correlation matrix P of the underlying multivariate Gaussian distribution.
3.1 UNDERSTANDING THE GLAD ARCHITECTURE
uCLAD uses the same architecture as GLAD We we have function θ = fnn fX), implemented the GLAD architecture Shrivastava et al. (2020b). The GLAD model uses the Alternating Minimization algorithm updates, unrolled some iterations, for the maximum likelihood objective as a template for its deep architecture design. Penalty constants (p. A) are replaced by problem dependent neural networks, Pun and Ann- These neural networks are minimalist terms of the number of parameters as the input dimensions are mere {3, 2} for {pmm Ann and outputs single value. Appendix A summarizes the update equations for the unrolled AM based model, GLAD. This unrolled algorithm with neural network augmentation can be viewed as a highly structured recurrent architecture, see Fig.
Note that the GLAD model has a low number of learnable parameters and maintains the permutation invariance w.r.t. the input. The design prior of the AM algorithm enforces the positive definite

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constraint of the precision matrix every step of its optimization. One can the state of the recovered graph at each unrolled step of the AM optimization, thus giving insights about the learning process. The AM algorithm converges linearly for the graphical lasso objective. The architecture described in more detail in in Appendix A and in the original GLAD paper Shrivastava et al. (2020b).
3.2 THE OGLAD LOSS FUNCTION
To learn the parameters in GLAD architecture, the authors used supervision in form of the true underlying graphs. They leveraged the interpretable nature the GLAD's deep architecture to define the loss for training. Specifically, each iteration of the model will output valid precision matrix cstimation and this allowed them to add auxiliary losses to regularize the intermediate results of GLAD, guiding it to learn parameters which generate a smooth solution trajectory.
The authors used Frobenius norm in their loss function:
Samples
CELAD = n
K
0 0-
1
(3)
A
where (0 2 ) GLADcell, (5 x 2, (() is the out- put of the recurrent unit GLADcell at k-th iteration, K is number of unrolled iterations, Y < 1 is a discounting factor and e is the ground truth precision matrix. Then the stochastic gradient descent algorithm was used to train the parameters f in the GLADcell.
Aan
Glosnd Purrs
In contrast to GLAD, uGLAD is designed to work without supervision, cannot take advantage of the underlying true generating distribution in the form of the true covariance matrix. Thus, we need replace GLAD's MSE loss from Equation 3 with new loss, Given function è = fnn (X), we optimize the log likelihood function given by
Pnn
LUGLAD (S, ()) log| θ + ()
(4)
Threshotenng
Σ Zest Gz
LEGLAD (X) = loglfnn (X)) (oco((())))) fnn(
(5)
where S cov(X) is the covariance matrix. We take the function fun the GLAD model and substitute it in the uGLAD loss function.
Graphs
Figure 1: The recurrent unit GLADcell. (Taken from Shrivastava et al. (2020b), more details Appendix A)
Algorithm 1 gives the pseudo code for learning the uGLAD model for doing sparse graph recovery. Appendix A contains more information about the parameter choices of the uGLAD and the GLAD model.
3.3 CONVERGENCE PROPERTIES OF GLAD AND UGLAD
The GLAD paper Shrivastava et (2020b) evaluates convergence propertics for the GLAD algorithm using normalized mean square error (NMSE) and probability of success (PS) to evaluate the algorithm performance. NMSE is log (E | 11 /E ||0"1(?) and PS is the probability of correct signed edge-set recovery, i.e., [s) sign(O;,) v(i,j) E B(O*)], where E(O*) is the true edge set. Optimization objective always converges. However, eiTors of recovering true precision matrices measured by NMSE have very different behaviors given different regularity parameter p, which indicates the necessity of directly optimizing NMSE and hyperparameter tuning. NMSE values are very sensitive to both م and the quadratic penalty \ of ADMM method. In GLAD and uGLAD, p and are not fixed, but are optimized together with the rest of network parameters, leading to smooth convergence.
In experiments evaluating edge recovery success, GLAD consistently outperforms traditional methods in terms of sample complexity as recovers the true edges with considerably fewer number of samples. Since in uGLAD we are still using the AM minimization based GLAD architecture which is also based on optimizing the Eq. 1, we expect the linear convergence properties of the AM algorithm will hold for uGLAD as well. The synthetic experiments in Sec. 6 show the results obtained from UGL AD are better or even surpass comparison to block coordinate descent based approach.

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3.4 OBTAINING CONDITIONAL INDEPENDENCE GRAPHS
The CI graph shows the partial correlations between the input features. A non-zero partial correlation between 2 features fB) indicates a direct dependence. So, all the other features are fixed, a positive partial correlation will indicate that increasing value of fA will increase the value of fB and vice-versa for the negative correlation. To obtain this CI graph, we calculate the partial correlation matrix from the precision matrix. Each entry of the partial correlation matrix Piς shows the correlation of the feature 2;, 2 given the values of the other features are observed. This helps us B,
obtain the direct dependence of the features, Piş ve,00
We can then visualize the graphs and
study the positive & negative correlations, with edge weights corresponding to correlation strengths.
4 MULTI-TASK LEARNING FOR PRECISION MATRIX RECOVERY
Most of the work in learning Gaussian graphical mod- els has focused on estimating a single model. recent years, the framework was extended to jointly fitting collection of such models, based on data that share the same variables, with dependency structure varying with some external category. For example, in NLP applica- tion, can encounter different styles, which induce links between some concepts, even as the underlying grammar and semantics of the language stay the same.
Algorithm 1: Optimizing uGLAD
Input: Observations X RRXx S = cov(X); for 1,... Eo
Ô. GLAD(S) unrolled for L iterations;
Compute loss LSGLAD (S,));
Backprop to update GLAD params; end for return θ
There have been extensive studies on the joint estimation of multiple undirected Gaussian graphical models Song et al. (2009); Honorio & Samaras (2010); Guo et al. (2011); Cai al. (2011); 1); Oyen Lane (2012): Danaher al. (2014); Kolar et al. (2010); Mohan et al. (2014); Peterson et al. (2015); Yang et al. (2015); Gonçalves et al. (2016); Varici et al. (2021). Most traditional algorithms construct joint objective for multiple estimation tasks. This objective typically incorporates similarities among various tasks by adding group norms other regularizations. However, in many practical problems, we only know that multiple tasks are related, without knowing how they are similar to each other quantitatively. Manually constructing the joint objective may not best reflect the actual similarity.
In contrast to traditional algorithms, in uGLAD, do not need to pre-assume the specific sim- ilarity among different tasks. Instead, we use a single network uGLAD to solve multiple tasks. Since the parameters in uGLAD shared across different tasks, the similarity among the tasks is automatically learned from data. More specifically, given samples from K different models, XK [X1, X2. Xk). we optimize the following objective
LUBLAD mallitnsk (Xk)
(covn Fnn Fnn) (Xk))
(6)
K
1=1
Alternatively, we can use the cross-validation split (Xk train
XK val +- xxk for training.
5 HANDLING MISSING VALUES
The missing data problem is ubiquitous in all data problems. uGLAD can easily be extended to handle this problem. If we observe that some specific feature columns have missing values and have reasons to believe that these values are missing at random, we can run imputation algorithms for those columns predict the missing entries. Then, we will have the complete imputed input data Xamp E IR M D over which we can run the uGLAD model and obtain the underlying precision matrix.
can often happen that there are technical errors or human mistakes in collecting samples, which can often lead to missing values or noise seeping into the sample. Also, we assume that the samples are independent and identically distributed (IID), so we cannot make use of the imputation techniques discussed in the case above. For this case, we propose novel multi-task learning technique based on utilizing the UGLAD'S s ability to optimize over a batch of input samples.
5

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AUPR
Method M=10
M=25
M=50
BCD
0.112:00013
0.132=0.012
0.219-40.085
UGLAD
0.15910.029
0.174=0.018
0.223+0.002
AUC
Method
M=10
M-25
M=50
BCD
0.505:00:007
0.332+0.031
0.617
uGLAD
0.572:10.027
0.595:10.04
0.651=0.021
Table 1: Synthetic data: Gaussian. AUPR and AUC on 20 test graphs for number of features D 25 and varying number of samples M. Gaussian random graphs with sparsity p 0.1 were chosen and edge values were sampled from ~ ( 1). We can observe that uGLAD (CV mode) significantly outperforms the BCD algorithm (sklearn's s graphicalLassoCV) for samples/features ratio and gives comparable performance as the number of samples increases.
Consensus strategy: Multi-task learning row-subsampled input
The key idea is to create a batch of row subsamples of the input data X E RMxx, Since all of these subsamples come from the same underlying distribution, we should ideally recover the same precision matrix for the entire batch. Thus, if we have a model that can be jointly optimized over the entire batch for the uGLAD objective, resulting in the recovered precision matrix being robust against erroneous or noisy samples.
Steps for the multi-task learning approach train the uGLAD model for handling missing data:
1. Statistical imputation for the input: Replace all the missing entries of the input data X with their respective column mean (the mean is calculated ignoring the missing entries) X c] = mean [X[:
c)). Replacing by mean is usually preferred approach as its contribution zeros out (X px)
while centering the data for the covariance matrix calculation.
2. Getting the batches: Perform stratified K-fold sampling to distribute the rows with missing values
evenly among different batches. Say, we have XK [X1, X2, Xx] batches with each
X₁ E MM "1 >xD Thus, the batch input for the uGLAD model is XK E R Kx K ( )DD
Optimizing uGLAD: It becomes a multi-task learning setting as we are jointly optimizing over a batch input Xk. The uGLAD model takes in the batch input Xx and outputs the corresponding K precision matrices. Since, the entire batch data is coming from the same underlying distribution, use the entire data X E MDD for the uGLAD loss to optimize the parameters of the uGLAD model. Mathematically, we are minimizing the uGLAD loss over the batch as
1 Σ CUELAD (cov fnn
LoGLAD-metA (XK)
4. Consensus among the batch to obtain the final precision matrix: After optimizing the uGLAD for the batch input, we will obtain K different precision matrices θ E RKxDxD Ideally, all the precision matrices should be the same but there will be some discrepancies we are working with missing values. Our 'consensus' strategy to obtain the final precision matrix O is to find the common edges with their correlation type (positive negative) from the batch precision matrices. Mathematically, we can obtain each entry [ of the final precision matrix as
0 = max-count 1...... (sign O* (,y) min 10
(8)
Here, the 'max-count term determines whether the correlation among the batches for that entry is positive or negative. The 2nd term chooses the minimum absolute value for that entry among the batches this facilitates sparsity and is conservative in terms of the strength of an edge.
EXPERIMENTS
Our software package hosted on the GitHub website, details given Appendix 7. Its function signature is very much akin to the skleamn' 's GraphicalLassoff intended to make easier for the users to try out our method with minimal change their existing code pipeline. Appendix D lists some more potential applications of the uGLAD model. We believe that our model can be seamlessly integrated with the existing pipeline for these applications and we
Pedregosa et al. (2011). This was

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AUPR
Method
M=20
M=100
M=1000
BCD
0.16310.028
0.24110.014
0.523:10.011
uGLAD
0.206.10.035
0.272.10.024
0.569.10.048
AUC
Method
M=20
M=100
M=1000
BCD
0.67010.013
0.718
0.83910.006
uGLAD
0.774
0.812-40.049
0.909-10.040
Table 2: GRN data: non-Gaussian. AUPR and AUC on 20 test graphs for D = 100 nodes and varying samples M. Graphs were sampled from the SERGIO simulator for the Gene Regulatory network recovery task. We can observe that the uGLAD model is more adaptive in non-Gaussian settings. A post-hoc masking operation was done to remove all the edges not containing transcription factor. This was done for all the methods.
AUPR
Method
M=10
M-25
M=50
BCD-avg
0.137=0.099
0.179:0.007
0.241=0.004
UGLAD-multi
0.18:::00028
0.204:10.044
0.279::0.027
AUC
Method
M=10
M=25
M=50
BCD-avg
0.508=00.224
0.538:10.02
0.24
0.597=10047
uGLAD-mult
0.552+00048
0.573 047
0.626+100022
Table 3: Multi-task learning. Average AUPR and AUC over K 10 graphs coming from sparsity~ [0.05, 0.2). The number of nodes D = 25 with varying samples M = [10, 25, 50). The BCD-avg considers each instance of the batch as separate task and reports the average results over the batch. uGLAD-multi used to recover the graphs jointly using a single model.
AUPR
Method
dp=0.25
dp=0.50
dp=0.75
BCD-mean
0.583=0.082
0.335-4.012
0.100+0.009
mean
0.605 101
0.357:40034
0.113:00016
uGTAD -misiing
0.612 0.100
0.375:
0.132+0:007
AUC
Method
dp=0.225
dp=0.50
dp=0.75
BCD-mean
0.792.00.45
0 0.64910.005
0.508
uGLAAD-mean
0.806=0.019
0.527+0.011 0.691+0.025
uGLAD-missing
0.813=0.010
0.71810.002
0.560=0.0:41
Table 4: Missing data: Gaussian. AUPR AUC on 20 test graphs for D 25 nodes and samples = 500. Gaussian random graphs were generated as described Sec. 6.1. Increasing fraction of dropouts were introduced to observe the robustness of handling missing data. We can observe that the AGLAD-missing model is more robust, especially in high dropout settings.
AUPR
Method
dp=0.50
dp=0.75
dp=0.90
BCD-mean
0.468=0.015
0.32310.008
0.04240.017
UGLAV-menn
0.503=0.011
0.34610:021
0.090-10.069
uCLAD missing
0.523 0.004
0.36110.044
0.117:10.033
AUC
Method
dp=0.30
dp=0.75
dp=0.90
BCD-mean
0.819=0.005
0.794-1.042
0.510+0.010
uGLAD menn
0.89720.009
0.821
0.59811.079
uGLAD-missing
0.906=00007
0.876:0.013
0.706:10.206
Table 5: Missing data: GRN. AUPR and AUC on 20 test graphs for 100 nodes and samples M 1000. Gene regulatory network data was used as described in Sec. 6.2. Increasing fraction of dropouts were introduced to the observed samples of the microarra expression data. uGLAD-missing model is more robust for high dropout settings. Such high dropout ratios are quite common in collecting samples for microarray gene expression data.

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hopeful that it improves results over state-of-thh-arr methods. Appendix B applies uGLAD to analyse the actual data collected for anaerobic digestion and Appendix C to analyse data in the infant mortality domain.
We use AUC (area under the ROC curve) and AUPR (area under the precision-recall curve) as primary evaluation metries. The sparsity of the graph leads very few positive edges. These two metrics account for such imbalance in the data. In addition, they have the advantage of working without specifically sctting a threshold for non-zero entries. Their values reporied in this work have the mean and the associated standard deviation values listed.
We primarily compare against the BCD algorithm. has been shown that the other traditional graphical lasso methods like ADMM and G-ISTA gave similar performance as BCD Shrivastava al. (2020b). We are not aware of any unsupervised deep learning approaches that optimize for the graphical lasso objective.
6.1 PERFORMANCE ON SYNTHETICALLY GENERATED GAUSSIAN SAMPLES
The synthetic data was generated based on the procedure similar to the one described in Guillot et al. (2012). A d-dimensional precision matrix O was generated by initializing d x d matrix with its off-diagonal entries sampled i.i.d. from a uniform distribution ,, ~ (1,1,1). These entries were then set zero based on the sparsity pattern of the corresponding Erdos-Renyi random graph with a certain probability p. Finally, an appropriate multiple of the identity matrix was added to the current matrix, so that the resulting matrix had the smallest eigenvalue of 1. In this way, θ was ensured to be a well-conditioned, sparse and positive definite matrix and was used in the multivariate Gaussian distribution (0, 1-1), ), to obtain M i.dd samples. Table shows the results on this synthetic data.
6.2 RECOVERY OF GENE REGULATORY NETWORKS
We conducted exploratory study to gauge the generalization ability of uGLAD non-Gaussian distributions. We chose the GRN inference task for this purpose. To this end, we use the SERGIO simulator Dibacinia & Sinha (2020) that provides a list of parameters simulate cells from different types of biological processes and gene-cxpression levels with various amounts of intrinsic and technical noise. For evaluation purposes in this work, we created random graphs (GRNs) that were used as input to SERGIO and will act as ground truth for evaluation. First, we set the number of Transcription Factors (TFs) or master regulators. Then, we randomly added edges between the TF's and the other genes based on sparsity requirements. Also, we randomly added some edges between the TFs themselves but excluded any self-regulation edges and maintained connectivity of the graph. When simulating data with no technical noise (clean data), we set the following parameters: sampling-ste statiing-tttt------ 15 (determines the number of steps of simulations for each stcady state); noise-param U00.1, 0.3] (controls the amount of intrinsic noise); noise-type = 'dpd' (the type of intrinsic noise is dual production decay noise, which is the most complex out of all types provided); set genes decay parameter 1. The parameters required to decide the master regulators' basal production cell rate for all cell types: low expression range of production cell rate ~ U[0.2, 0.5] and high expression range of cell rate U[0.7,1). We chose K~ U[1, 5), where denotes the maximum interaction strength between master regulators and target genes. Positive strength values indicate activating interactions and negative strength values indicate repressive interactions and signs are randomly assigned. The parameters were configured such that the statistical properties of these synthetic data (clean) set are comparable with the mouse brain Zeisel et al. (2015). Table shows the comparison of the different graph recovery methods on the simulated data generated by SERGIO for cell types C. = 5 and clean data setting.
6.3 MULTI-TASK LEARNING
This experiment verifies the ability of uGLAD to do multi-task learning. We chose collection of tasks as a set of data coming from graphs with varying sparsity. For K different tasks, our input data We run the uGLAD model in multi-task learning mode as described in Sec. 4 and recover K different precision matrices θ E R DxD that are optimized for the loss function Legran--muliinsk
X E x
(Xxx (Xx) given by equation Table shows results from a single uGADD model in multi-

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task setting which is run the synthetic data generated as described in Sec. 6.1. uGLAD recovers multiple graphs with varying sparsity and shows promise for multitask learning problems.
6.4 ROBUSTNESS TESTING FOR MISSING DATA
We artificially introduced missing values or 'dropouts' in the input data X E RRMx to create noisy data. Our aim is to study the effectiveness the 'consensus' strategy discussed in Sec. 5. We compare it with the baseline statistical imputation technique that does column-wise (or feature-wise) mean imputation as preprocessing step, BCD-mean and uGLAD-mean, report the results of running the corresponding methods on the column mean imputed data while the uGLAD-min BOLAD-missing uses the 'consensus' strategy. Tables 4&5 show the robustness of the 'consensus' strategy introduced in this work for synthetic Gaussian data as well as for the Gene regulatory networks recovery tasks.
SOFTWARE DETAILS: OPTIMIZING MODES OF UGLAD
To optimize uGLAD for the uGLAD loss function, we have introduced 4 different modes of training, In the software package, these modes can be switched from one the other using an indicator flag-
Direct mode: The input to the uGLAD model is complete data X and the output precision matrix = Snn (X) optimized to reduce the uGLAD loss LUSIAD (X), as defined in Eq.
CV mode (recommended): In the k-fold cross validation mode, we split the input samples X (Xtrain: Xealia). We use the Xtrain as input and optimize for the uGLAD loss LaCtaD (actaa((iran)) (Xiran). Then, we select the best model that minimizes the LUGLAD (Xvatia) for the Xvatid samples.
Missing data mode: We give the entire data as input with a 'NaN' indicator for the entries where the values are missing. The software then follows the 'consensus' strategy for handling of missing data given in Sec. 5 and outputs the final precision matrix O
Mask mode: Given a batch of imput data X E RXXMx we jointly optimize them for the moxx> uGLAD loss objective to obtain K different precision matrices θ E IR RXDDxD refer to Sec. 4.
CONCLUSIONS
We introduced novel technique uGLAD to perform sparse graph recovery by optimizing deep unrolled network for the graphical lasso objective. This is an extension to the previous GLAD model that designed to use supervision. The key advantages of using model over the state-of-the-as algorithms for the graphical lasso problems are (1) Sparsity related hyperparameters are modeled using neural networks which are automatically learned during the optimization. We thus address the sensitivity issue of choosing the right sparsity parameters which is usually tedious task and often manually set for the other algorithms. (2) By design, neural networks of uCLAD enable the sparsity regularization to be adaptive over the iterations of the optimization leading to superior performance. (3) Our software implementation supports GPU based acceleration and thus can be scaled efficiently to meet the runtime requirements. (4) The uGLAD framework can do efficient multi-task learning. The proposed 'consensus' strategy based on leveraging this property works well robustly handle missing data. As our experience with anaerobic digestion (see Appendix B) demonstrates, uGLAD can be successfully used as a tool for generating insight into growth dynamics of organisms in a digester and (hopefully) into domain structure in many other applications. We hope that model becomes one the widely used algorithms to solve the graphical lasso objective.
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12


Reviews:
[{"review_id": "sPTFyq4GVHu", "reviewer": "Reviewer_HUi2", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper extends previous work (GLAD) and proposes a novel deep model, uGLAD, to recover sparse graphs from Gaussian data. The paper introduces the following extensions over the GLAD framework - (1) it extends the framework to recover graphs in unsupervised learning by replacing the loss function (2) it utilizes adaptive regularization parameters (3) by using uGALD in a multi-task setting, the paper describes a consensus strategy to handle missing data. The experiments in the paper show the uGLAD model outperforms traditional block coordinate descent (BCD) on graph recovery of both Gaussian and non-Gaussian (simulated) data.", "summary_of_the_review": "Overall, the ideas presented are interesting and useful. The paper, however, requires a more descriptive and robust experimental design and results to contribute significantly to the community. \n", "strength_and_weaknesses": "Strength:\n+ This work extends the GLAD model to unsupervised and multi-task settings. The adaptive parameter for each precision matrix element provides more thorough controls, and the inductive-biases-based model eases computational burdens. Overall, the method looks promising in theory, and the reasoning behind the extensions is valid and important for real-world applications\n\nWeakness: \n\n- The paper requires a more extensive and robust experimental setup to support its claims. \n\n- For example, in section (page 8), the authors state that the GLAD paper has shown that “ADMM and G-ISTA gave a similar performance as BCD.” So the paper only compares uGLAD with BCD. But there are more sparse Gaussian graphical model estimators. For example, M-estimators [1] and thresholding [2,3] are other types of methods. Although some of these recently proposed methods induce the same regularization coefficients for all precision matrix elements, comparing uGLAD with them is worthwhile to validate adaptive parameter performs better. \n\n- Moreover, this paper tested uGLAD on multi-task data and data with missing values. However, uGLAD is also only compared with BCD. But there are more recent works for these two tasks. For instance, JGL[4] and FASJEM[5] estimate multi-task graphs. MissGLasso[6] and [7] estimate inverse covariance matrices from data with missing values. \n\n- The choice of the simulation is not well justified, and the experimental settings and the results need better descriptions to connect to the paper's contributions. \n\n- Real-world gene expression datasets now have features >25 (in the order of 100s and 1000s). How does the proposed method scale to the large feature size and more noise?\n\nMinor:\n\n- The paper has presented a paper directly from another work (with citation) - this practice is not common in the field and is often discouraged. It would be useful to modify the figure to highlight the novelty and contributions of this work. \n\n- References should have brackets around them when they are not being used as subjects of the sentence. \n\nReferences:\n[1] Yang, Eunho, Aurélie C. Lozano, and Pradeep K. Ravikumar. \"Elementary estimators for graphical models.\" Advances in neural information processing systems 27 (2014).\n\n[2] Sojoudi, Somayeh. \"Equivalence of graphical lasso and thresholding for sparse graphs.\" The Journal of Machine Learning Research 17.1 (2016): 3943-3963.\n\n[3] Zhang, Richard, Salar Fattahi, and Somayeh Sojoudi. \"Large-scale sparse inverse covariance estimation via thresholding and max-det matrix completion.\" International Conference on Machine Learning. PMLR, 2018.\n\n[4] Danaher, Patrick, Pei Wang, and Daniela M. Witten. \"The joint graphical lasso for inverse covariance estimation across multiple classes.\" Journal of the Royal Statistical Society: Series B (Statistical Methodology) 76.2 (2014): 373-397.\n\n[5] Wang, Beilun, Ji Gao, and Yanjun Qi. \"A fast and scalable joint estimator for learning multiple related sparse gaussian graphical models.\" Artificial Intelligence and Statistics. PMLR, 2017.\n\n[6] Städler, Nicolas, and Peter Bühlmann. \"Missing values: sparse inverse covariance estimation and an extension to sparse regression.\" Statistics and Computing 22.1 (2012): 219-235.\n\n[7] Loh, Po-Ling, and Martin J. Wainwright. \"High-dimensional regression with noisy and missing data: Provable guarantees with non-convexity.\" Advances in neural information processing systems 24 (2011).\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "kh_RkeyalQ6", "reviewer": "Reviewer_32py", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper considered the problem of learning graph models in an unsupervised setting. The authors extended the state-of-the-art GLAD method to the unsupervised setting, and show its effectiveness in handling missing data by numerical experiments.\n", "summary_of_the_review": "The authors extended the state-of-the-art method GLAD to the unsupervised setting. However, such an extension is very simple to conduct, and it is not clear about its significance. The contributions and novelty of this paper are not significant, and experimental comparisons are not sufficient.", "strength_and_weaknesses": "Strength:\n\nThe authors extended the state-of-the-art GLAD method to the unsupervised setting. Numerical results demonstrated that the proposed method can robustly handle missing data.\n\nWeakness:\n\n1. The extension from GLAD to uGLAD is simple without the need to change the architecture. Therefore, the contributions of this paper are not significant. \n\n2. There are a large number of methods proposed for learning graph models in the literature. More compared methods are needed in experiments.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "Txw7U1ffHQA", "reviewer": "Reviewer_aant", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes uGLAD, an unsupervised version of previously proposed GLAD paper, to learn the graph structure for undirected (Gaussian) graphical models. . Authors also added several functionalities on top, including the capability to handle multi-task and missing values. Empirical results show good performance. ", "summary_of_the_review": "incremental improvement over GLAD. ", "strength_and_weaknesses": "Strength:\nEmpirical improvement seems significant. \n\nWeakness:\n- novelty: although the paper makes several contribution, they are mostly incremental or straightforward. New additions on unsupervised loss function, multi-task extension, and missing value imputation are standard and do not offer any new insight. \n- organization: related to novelty, it is not fully clear what problem authors are tackling after reading the paper. Different sections seem to solve different problems, and they seem to be put together just to fill the content of the paper. \n- Experiments are only done via one baseline method. Despite authors' claim on other baselines have similar performance, it was only done with specific datasets. Other baselines' results are important here.\n\nOther:\n1. citations are not done correctly.\n2. \"...we expect the linear convergence property holds...\" since you change the loss function, how can you ensure this is still the case?\n3. \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject