SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Since the true SCM is unknown, fitting a linear model  $X_{B} \sim \omega_{1}X_{F} + \omega_{2}X_{D}$  would result in ambiguous coefficients  $\omega_{1}, \omega_{2} \in [0,1]$  with  $\omega_{1} + \omega_{2} = 1$ . By fitting  $X_{D} \sim cX_{F}$  and obtaining  $c$  close to 1, we have the estimated causal graph as  $X_{F} \xrightarrow{\omega_{1}} X_{B}$ , and  $X_{F} \xrightarrow{c} X_{D} \xrightarrow{\omega_{2}} X_{B}$ . Based on Def. 4.5, we estimate the corresponding causal effects in Table 1. It can be observed that the total effect of  $X_{F}$  on  $X_{B}$  surpasses the total effect from  $X_{D}$ , indicating the spurious nature of the diaper purchasing, which should be removed to form the desired NSCG. Yet, using the direct effect may not be able to distinguish their differences due to the high correlation between  $X_{F}$  and  $X_{D}$  if  $\omega_{1} < \omega_{2}$ .
5 Necessary and Sufficient Causal Structural Learning
In this section, we formally present how to learn NSCG. Based on Thms. 4.4 and 4.6, a simple solution is to first use a pre-screening process to find necessary and sufficient features from  $Z$  that achieve high scores of causation, and then estimate the causal graph among the selected nodes and  $Y$  to approximate  $\mathcal{G}_V$ . This approach works for general SCMs while may suffer from overfitting. Instead of using such a two-step learning, we propose to learn necessary and sufficient features and the causal graph simultaneously through a single-step optimization. To this end, in § 5.1, we first introduce the structural equation model in order to provide the closed-form expressions of the proposed causal quantities. The main algorithm based on causal effects is presented in § 5.2 for the linear model, with the POC-based version available in App. B for the nonlinear model.
5.1 Structural Equation Model and Close Form of Causal Effects
Structural equation model. We define a selection function  $g$  that maps the feature set  $Z$  to a subset, aiming to maintain good interpretability. That is,  $g: Z \in \mathbb{R}^p \to g(Z) \in \mathbb{R}^d$  where  $d \ll p$ , and we denote the  $i$ -th dimension of  $g(Z)$  as  $g_i(Z)$ . Following the causal structure learning literature [35; 25; 46; 44; 48; 5], we assume the Markov and faithfulness conditions and consider a linear structural equation model (LSEM) such that  $\{g(Z), Y\}$  is characterized by the pair  $(B, \epsilon)$  as
$$
\left[ \begin{array}{l} g (\boldsymbol {Z}) \\ Y \end{array} \right] \leftarrow \boldsymbol {B} \left[ \begin{array}{l} g (\boldsymbol {Z}) \\ Y \end{array} \right] + \boldsymbol {\epsilon} \equiv \left[ \begin{array}{l l} \boldsymbol {B} _ {g} & 0 \\ \boldsymbol {\theta} & 0 \end{array} \right] \left[ \begin{array}{l} g (\boldsymbol {Z}) \\ Y \end{array} \right] + \left[ \begin{array}{l} \boldsymbol {\epsilon} _ {\boldsymbol {Z}} \\ \epsilon_ {Y} \end{array} \right], \tag {1}
$$
where  $\pmb{B}$  is a  $(d + 1)\times (d + 1)$  weighted adjacent matrix that characterizes the causal relationship among  $\{g(Z),Y\}$ , and  $\epsilon \equiv [\epsilon_{Z}^{\top},\epsilon_{Y}]^{\top}$  is a  $d + 1$  dimensional random vector of jointly independent errors. Here,  $\pmb{B}$  consists of three components: (1). a  $d\times d$  matrix  $B_{g} = \{b_{i,j}\}_{1\leq i\leq d,1\leq j\leq d}$  with  $b_{i,j}$  as the weight of the edge  $g_{i}(Z_{i})\to g_{j}(Z_{i})$  if exists and  $b_{i,j} = 0$  otherwise; (2) a  $1\times d$  vector  $\pmb {\theta} = [\theta_1,\dots ,\theta_p]$  for  $\theta_{i}$  presenting the weight of the direct edge  $g_{i}(Z)\rightarrow Y$ ; and (3). a  $(d + 1)\times 1$  zero vector indicating the outcome of interest  $Y$  cannot be any parent of the features. Without further assumptions, the model in (1) given a particular selector  $g$  can be identified only up to a Markov equivalence class (MEC) [35; 25]. In the following, we focus on cases where the DAG can be uniquely identifiable, such as LSEM with Gaussian noises of equal variance [35; 26; 24], and linear model with non-Gaussian noise [34; 47]. See more details and extensions to MEC in App. C.1.
Close form and estimation of causal effects. We next provide the close forms of the causal effects in Def. 4.5 under the model in (1). Recall that  $\theta_{i}$  presents the weight of the direct edge  $g(Z)_i\to Y$ . According to (1) and Def. 4.5, we have
$$
D E _ {i} (\boldsymbol {B}; g) = \theta_ {i}.
$$
The total causal effect can be quantified by the path method [see e.g., 43; 20]. Specifically, the causal effect of  $g_{i}(\mathbf{Z})$  on  $g_{j}(\mathbf{Z})$  along a directed path from  $g_{i}(\mathbf{Z}) \to g_{j}(\mathbf{Z})$  in  $\mathcal{G}$  can be calculated by multiplying all edge weights along the path, under LSEM. Denote the set of directed paths that starts with  $g_{i}(\mathbf{Z})$  and ends with  $Y$  as  $\pi_{i} = \{g_{i}(\mathbf{Z}) \to \dots \to Y\}$  with the size as  $m_{i}$ . Then the causal effect of  $g_{i}(\mathbf{Z})$  on  $Y$  through the directed path  $\pi_{i}^{(k)} = \{i, l_{1}, \dots, l_{\tau_{k}}, d + 1\} \in \pi_{i}$  with length  $\tau_{k} + 1$  is  $PE\{\pi_{i}^{(k)}\} = b_{i, l_{1}} \cdots b_{l_{\tau_{k}}, (d + 1)}$ , by the path method, where  $b_{i,j}$  is the weight of the edge  $g_{i}(\mathbf{Z}) \to g_{j}(\mathbf{Z})$  if it exists, and  $b_{i,j} = 0$  otherwise, for  $i, j \in \{1, \dots, d\}$ , and  $b_{l_{\tau_{k}}, (d + 1)} = \theta_{l_{\tau_{k}}}$  as the direct edge from  $g_{l_{\tau_{k}}}(\mathbf{Z})$  to  $Y$ . Thus,
$$
T E _ {i} (\boldsymbol {B}; g) = \sum_ {k = 1} ^ {m _ {i}} P E \{\pi_ {i} ^ {(k)} \}.
$$
Both  $TE_{i}$  and  $DE_{i}$  can be explicitly calculated given a matrix  $B$  under a selector  $g$ . We denote their estimates as  $\widehat{TE}_{i}$  and  $\widehat{DE}_{i}$  given the estimated matrix  $\widehat{B}$  and  $g$ .
5.2 Learning Algorithm based on Causal Effects
The primary algorithm based on causal effects comprises three steps that quantify two sources of loss and learn the causal graph, specifically: loss of causal structural learning, loss of discovering causal features, and minimizing the overall loss to learn NSCG based on data  $\{o^{(j)} = (z^{(j)},y^{(j)})\}_{1\leq j\leq n}$ .
Step 1: Form the loss from causal structural learning. To estimate the matrix  $B$  in (1), we adopt the acyclicity constraint [44; 46] as  $h_1(B) \equiv \operatorname{tr}\left[(I_{d+1} + tB \circ B)^{d+1}\right] - (d+1) = 0$ , where  $I_{d+1}$  is a  $d+1$ -dimensional identity matrix, and  $\operatorname{tr}(\cdot)$  is the trace of a matrix and  $t$  is a hyperparameter that depends on the estimated largest eigenvalue of  $B$ . The first loss by the augmented Lagrangian is
$$
L _ {1} (\boldsymbol {B}, g, \theta , \lambda_ {1} | \{\boldsymbol {o} ^ {(j)} \}) = f (\boldsymbol {B}, g, \theta | \{\boldsymbol {o} ^ {(j)} \}) + \lambda_ {1} h _ {1} (\boldsymbol {B}), \tag {2}
$$
where  $f(\pmb{B}, g, \theta | \{\pmb{o}^{(j)}\})$  is some loss such as the least square error in NOTEARS [46] or the Kullback-Leibler divergence in DAG-GNN [44] with parameters  $\theta$ , and  $\lambda_1$  is the Lagrange multiplier. Other causal structural leaning algorithms [see e.g., 35; 7; 34; 14; 4; 27; 48] can also be applied by formulating the corresponding score or loss function.
Step 2: Constraints for causal relevance and causal identifiability. We next measure the causal relevance of the selection function  $g$  by the natural causal effects. We convert the outcome  $Y$  to be nonnegative. According to Thm. 4.6, we can avoid the estimation of POCs by using the related causal effects in Def. 4.5 with their explicit expressions. This part of loss thus becomes
$$
L _ {2} ^ {C E} (\boldsymbol {B}, g, \gamma | \{\boldsymbol {o} ^ {(j)} \}) = - \sum_ {i = 1} ^ {d} \left| \widehat {C E} _ {i} (\boldsymbol {B}; g) \right| + \sum_ {i = 1} ^ {d + 1} \left| b _ {i, d + 1} \right| + \gamma | g |. \tag {3}
$$
Here,  $\widehat{CE}_i$  can either take the estimated  $DE_i$  or the estimated  $TE_i$  given the matrix  $B$  and  $g$ , as detailed in § 5.1. The second term corresponds to the causal identification constraint on the last column of  $B$ , requiring all elements to be zeros as in (1). This constraint restricts the causal structural learning to a smaller class of DAGs. Finally,  $|g|$  denotes the number of selected nodes in  $g$ , with a penalty  $\gamma$  to control the complexity of the selector.
Step 3: Necessary and sufficient causal structural learning. Combining two sources of loss functions in (2) with (3), leads to the objective as
$$
\min  _ {\boldsymbol {B}, g} \left[ L _ {1} (\boldsymbol {B}, g, \theta , \lambda_ {1} | \{\boldsymbol {o} ^ {(j)} \}) + \alpha L _ {2} ^ {C E} (\boldsymbol {B}, g, \gamma | \{\boldsymbol {o} ^ {(j)} \}) \right], \tag {4}
$$
where  $\alpha$  can be reviewed as a trade-off parameter between two loss functions. Next, we provide a solution for (4) without tuning  $\alpha$ . Specifically, based on no unmeasured confounders in (A2), we can calculate the highest causal effects that could be achieved given all variables without any penalty. Denote the estimated highest absolute causal effects in data as  $\delta^{*}$ . The goal is to find a subset of  $Z$  such that  $g(Z)$  achieves a similar level of necessity and sufficiency, i.e., the resulting score is close to  $\delta^{*}$ . Hence, we set the second loss as a constraint by comparing the overall causal effects of the selected nodes with the highest reference over the entire observed feature space, i.e.,
$$
h _ {2} (\boldsymbol {B}; g) = \delta^ {*} - \sum_ {i = 1} ^ {d} | \widehat {C E} _ {i} (\boldsymbol {B}; g) | + \sum_ {i = 1} ^ {d + 1} | b _ {i, d + 1} |,
$$
should be approaching 0 given a good selector  $g$ . This yields a new objective function as
$$
\min  _ {\boldsymbol {B}, g} f (\boldsymbol {B}, g, \theta | \{\boldsymbol {o} ^ {(j)} \}) + \lambda_ {1} h _ {1} (\boldsymbol {B}) + \lambda_ {2} h _ {2} (\boldsymbol {B}; g) + c | h _ {1} (\boldsymbol {B}) | ^ {2} + d | h _ {2} (\boldsymbol {B}; g) | ^ {2} + \gamma | g |, \tag {5}
$$
where  $\lambda_{2}$  is the Lagrange multiplier for the new constraint, and  $c$  and  $d$  are penalty terms. To minimize the loss in (5) and satisfy both  $h_1(B)\to 0$  and  $h_2(B;g)\to 0$ , we simultaneously update  $\lambda_{1}$  and  $\lambda_{2}$  and increase  $c$  and  $d$  to infinity, by modifying the updating technique [see e.g., 46; 44] for multiple constraints, with the class of functions  $g$  specified as the subset of  $\mathbf{Z}$  and its penalty  $\gamma$  as the size of selected nodes in  $g$ . Here, the minimization can be solved using a black-box stochastic optimization such as 'Adam' [15]. Denote the estimated matrix as  $\widehat{\pmb{B}}$ , based on which we can obtain the estimated causal graph as  $\widehat{\mathcal{G}}_{\mathbf{V}}$  consisting of nodes  $\widehat{g} (\mathbf{Z})$ . Finally, we name this proposed algorithm as necessary and sufficient causal structural learning (NSCSL). The computational complexity of NSCSL is provided in App. B.3. We next establish the consistency of estimated causal graphs below.
Theorem 5.1. Assume Model (1) holds with independent Gaussian error and equal variance. Suppose the topological ordering of the true bounded matrix  $\mathbf{B}$  is consistently estimated. Then the estimated matrix  $\hat{\mathbf{B}}$  minimizing the loss in (5) converges to  $\mathbf{B}$  with the probability going to 1 as  $n \to \infty$ .
Table 2: Comparison results across S1 to S3 under different sample sizes  $(n)$ . Methods are evaluated by FDR, TPR, and SHD, with the standard error (SE) reported for each metric, over 50 replications.
Scenario	Method	FDR±SE
n1 (small)	n2 (large)	TPR±SE
n1 (small)	n2 (large)	SHD±SE
n1 (small)	n2 (large)
S1	NSCSL-TE	0.09±0.03	0.02±0.01	0.95±0.02	1.00±0.00	0.64±0.20	0.14±0.06
p=5	NSCSL-DE	0.08±0.03	0.02±0.01	0.95±0.03	1.00±0.00	0.72±0.20	0.14±0.06
n1=30	NOTEARS	0.39±0.01	0.34±0.00	0.96±0.02	1.00±0.00	2.56±0.14	2.02±0.01
n2=100	PC	0.53±0.02	0.53±0.01	0.47±0.05	0.41±0.01	3.12±0.11	3.20±0.07
ER Model	LiNGAM	0.31±0.02	0.33±0.00	0.78±0.02	0.98±0.01	2.30±0.10	2.00±0.00
S2	NSCSL-TE	0.10±0.03	0.02±0.01	1.00±0.00	0.99±0.01	0.38±0.14	0.12±0.06
p=5	NSCSL-DE	0.12±0.04	0.01±0.01	0.67±0.04	0.50±0.00	1.00±0.12	1.02±0.01
n1=30	NOTEARS	0.63±0.01	0.60±0.00	1.00±0.00	1.00±0.00	3.46±0.13	3.02±0.01
n2=100	PC	0.73±0.02	0.79±0.00	0.50±0.00	0.50±0.00	3.62±0.15	3.88±0.03
ER Model	LiNGAM	0.62±0.01	0.60±0.00	0.98±0.02	1.00±0.00	3.18±0.09	3.00±0.00
S3	NSCSL-TE	0.10±0.02	0.01±0.00	0.94±0.02	1.00±0.00	0.84±0.16	0.06±0.02
p=5	NSCSL-DE	0.08±0.02	0.01±0.00	0.84±0.03	0.81±0.00	1.30±0.14	1.00±0.00
n1=30	NOTEARS	0.11±0.02	0.01±0.00	0.96±0.02	1.00±0.00	0.76±0.16	0.06±0.02
n2=100	PC	0.00±0.00	0.00±0.00	0.66±0.02	0.80±0.00	1.68±0.12	1.00±0.00
ER Model	LiNGAM	0.03±0.01	0.00±0.00	0.98±0.01	1.00±0.00	0.24±0.09	0.02±0.01
The proof and detailed conditions for Thm. 5.1 are provided in App. D, which align with those commonly imposed in causal structural learning [e.g., 33]. Our proof follows similar strategies but accounts for the extra penalty term from causal effects. Notice that the explicit forms of causal effects under LSEM are linear combinations of elements of  $B$ . This implies our new regulation can similarly vanish away as  $n$  goes to infinity.
Experiment design. Scenarios are generated as follows. We consider the dimension of variables in the graph as  $p = 5$  in Scenarios 1 to 3 (S1 to S3),  $p = 20$  in Scenario 4 (S4), and  $p = 50$  in Scenario 5 (S5), to examine the scalability of NSCSL. For each scenario, the DAG that characterizes the causal relationship among variables  $\mathcal{O} = (\mathbf{Z}, Y)$  is generated from the Erdős-Rényi (ER) model with an expected degree as 2 for S1 to S3 and degree as 5 for S4 to S5. We also generate the graph from the scale-free (SF) model for S5 with the degree of 5 to examine the robustness of the proposed method under diverse synthetic graphs. Each edge is assigned positive weights. We set the last variable as the outcome of interest  $Y$ , and generate the data based on LSEM by  $Z = B^{\top} Z + \epsilon$ , where  $\epsilon$  is a random vector of jointly independent binary variables with equal probability taking value one or zero. Thus, the outcome is nonnegative and discrete. In addition, we also include a nonlinear structural equation model for S4 where  $O_{i} := \psi_{i}\{\mathrm{PA}_{O_{i}}(\mathcal{G})\} + e_{O_{i}}$  and  $\psi_{i}(x) = \lfloor 2\log (x + 1)\rfloor$  where  $\lfloor x\rfloor$  rounds to nearest integer for  $x$ . In S1, the true causal graph contains one spurious node (indexed by 0) and three non-spurious nodes (indexed by 1, 2, and 3), as shown in sub-figures (a) of Fig. E.7, with the associated true NSCG in sub-figures (b) of Fig. E.7. Moreover, we design a balanced setting with half spurious variables and half non-spurious variables in S3, as depicted in Fig. E.8, and a baseline setting without any spurious variables in S3, shown in Fig. E.9. Finally, S4 contains 2 non-spurious variables with 17 spurious variables in Fig. E.10. The experiments are conducted on a Google Cloud Platform virtual machine with 8 processor cores and 32GB memory.
Methods and benchmark specification. We apply the proposed NSCSL based on TE and DE as the criteria of necessity and sufficiency, respectively, to capture the marginal and conditional causal effect of the confounders. Note that we consider fully identifiable models in the experiments so that it is meaningful to evaluate causal effects from the estimated graph. The underlying causal structure learning algorithm is set to NOTEARS [46]. We also compare the proposed method against other methods, including PC [35] and LiNGAM [34] for S1 to S5; DAG-GNN [44], GES with generalized score [GSGES, 11], fast causal inference [FCI, 36], and the causal additive model [CAM, 4], for all high-dimensional settings in S4 and S5. Here, we use a graph threshold of 0.3 (commonly used in the literature [46; 44; 48; 5]) to prune the noise edges for a fair comparison. The training details are provided in Table E.1. The true and estimated graphs with the associated matrix under different approaches are illustrated in Figs. E.2 to E.6 and Figs. E.7 to E.11 in App. E for S1 to S4. The comparison results across different sample sizes  $(n)$  are presented in Table 2 for S1 to S3, in Table 3 for S4 to S5 with linear ER model, in Table 4 for S4 with nonlinear ER model, and in Table 5 for S5 with linear SF model. All the results are evaluated by false discovery rate (FDR), true positive
Table 3: Comparison studies under S4 to S5 under different sample sizes  $(n)$  and dimensions  $(p)$  with the Erdős-Rényi (ER) model. Methods are evaluated by FDR, TPR, SHD, and runtime (seconds), with standard errors (SE) reported for each metric, over 50 replications.
Scenario	Method	FDR±SE
n1 (small)	n2 (large)	TPR±SE
n1 (small)	n2 (large)	SHD±SE
n1 (small)	n2 (large)	Time±SE
n1 (small)	n2 (large)
S4	NSCSL-TE	0.11±0.02	0.00±0.01	1.00±0.00	1.00±0.00	0.86±0.24	0.04±0.01	10.8±0.3	56.2±1.1
p=20	NSCSL-DE	0.11±0.02	0.00±0.01	0.69±0.01	0.67±0.00	1.34±0.08	1.00±0.01	12.4±0.8	54.5±1.3
n1=100	NOTEARS	0.93±0.00	0.92±0.00	1.00±0.00	1.00±0.00	40.80±0.20	36.40±0.04	22.9±6.4	69.8±8.7
n2=1000	PC	0.92±0.00	0.95±0.00	0.51±0.02	0.67±0.00	19.08±0.17	33.34±0.03	6.9±0.5	16.3±0.8
ER Model	LiNGAM	0.92±0.00	0.93±0.00	0.99±0.01	1.00±0.00	33.00±0.20	37.10±0.02	8.1±0.5	24.6±0.6
Degree=5	DAGGNN	0.93±0.00	0.93±0.00	0.97±0.01	0.97±0.00	41.34±0.19	40.10±0.06	28.32±4.3	39.12±7.2
GSGES	0.98±0.01	0.98±0.00	0.22±0.02	0.20±0.01	43.20±0.26	45.70±0.34	14.92±7.5	26.32±9.1
FCI	0.98±0.01	0.99±0.00	0.10±0.02	0.06±0.01	22.70±0.17	32.90±0.02	6.6±0.3	12.7±0.2
CAM	0.93±0.00	0.94±0.00	0.63±0.02	1.00±0.00	29.80±0.38	40.30±0.07	19.62±18.3	25.62±23.2
S5	NSCSL-TE	0.03±0.01	0.02±0.01	0.86±0.03	0.93±0.01	2.18±0.13	1.58±0.07	110.1±3.9	21.12±12.1
p=50	NSCSL-DE	0.02±0.02	0.01±0.01	0.29±0.02	0.28±0.01	10.08±0.21	9.76±0.13	119.0±5.5	23.52±10.9
n1=1000	NOTEARS	0.86±0.04	0.85±0.01	0.93±0.03	0.92±0.01	79.20±1.40	77.12±0.53	128.3±8.2	26.92±15.1
n2=3000	PC	0.96±0.03	0.97±0.02	0.07±0.02	0.06±0.01	82.12±1.21	88.28±1.63	20.9±0.5	35.9±3.2
ER Model	LiNGAM	0.86±0.02	0.86±0.01	0.97±0.02	0.99±0.01	86.12±1.20	85.70±0.84	43.1±6.3	145.3±7.9
Degree=5	DAGGNN	0.87±0.02	0.88±0.01	0.93±0.02	0.94±0.01	87.50±1.10	85.62±0.96	49.32±7.5	81.12±87.6
GSGES	0.89±0.03	0.93±0.01	0.19±0.03	0.12±0.01	93.54±1.46	95.70±0.79	31.22±10.1	45.32±18.1
FCI	0.96±0.02	0.97±0.01	0.08±0.01	0.07±0.01	84.00±0.80	88.50±0.60	14.3±0.8	17.7±0.5
CAM	0.93±0.04	0.95±0.02	0.66±0.03	0.67±0.02	126.00±3.64	127.80±2.06	28.42±31.7	75.62±63.9
rate (TPR), and the structural Hamming distance (SHD) to the true causal graph, with their standard errors, over 50 replications. The average running time of these methods is also reported in Table 2 to Table 5 to reflect the computational complexities. In addition, the sensitivity analyses concerning all hyperparameters in Table E.1 are conducted using S4, as presented in Fig. 2.
Results and conclusion. NSCSL performs the best in discovering NSCG in S1, S2, S4, and S5, and shows comparably best results in S3 (a setting without any spurious variables). To be specific, the benchmark methods for causal structural learning aim to reveal causal relationships in the whole graph (i.e., sub-figures (a) in Figs. E.2 to E.6), which contains spurious effects on the target outcome  $Y$ . The proposed algorithm overcomes this drawback by purely identifying the true important causal relationships (i.e., sub-figures (b) in Figs. E.2 to E.6). By comparing the sub-figures (c) and (d) in Figs. E.2 to E.6 as well as Table 2 to Table 5, NSCSL-TE detects all necessary and sufficient causal paths towards the outcome, while NSCSL-DE only extracts direct causal relationships between the features and the outcome, resulting in a slightly lower TPR and slightly higher SHD than NSCSL-TE. Furthermore, from Table 2 to Table 5, the results under the proposed algorithm approach the truth more closely as the sample size increases in all scenarios, regardless of the underlying graph models and data-generating process. In contrast, the benchmark methods all fail to discover NSCG in the high-dimensional setting, exhibiting extremely high FDRs and SHDs. In addition, Table 3 to Table 5 reveal that NSCSL is as fast as the quickest benchmarks such as PC, LiNGAM, and FCI, and significantly faster than others like DAGGNN, GSGES, and CAM. Our method's integration of treatment effects into the optimization adds efficiency and restricts the searching space, making it practical and even beating NOTEARS in computation. The sensitivity analyses in Fig. 2 indicate that our method remains robust to these parameters set within a reasonable range.
Real data analysis - Sachs et al. [30]. We conduct real data analysis using the benchmark data from Sachs et al. [30]. To validate our method's capacity to find the NSCG and align with Def. 3.2, we designated the protein Akt as the target outcome. This designation ensures that NSCG exists (see Fig. 3) and that finding an NSCG is meaningful. Our method (NSCSL-TE) and seven baseline methods were applied and evaluated against the true NSCG associated with the protein Akt. Table 6 shows that our method achieves the best performance in finding the NSCG concerning the protein Akt.
Real data analysis - Brem & Kruglyak [2]. Furthermore, we apply NSCSL to gene expression traits in yeast [2] using a dataset of 104 yeast segregants with diverse genotypes. The goal is to identify genes, known as quantitative trait loci (QTLs), influencing the expression level (nonnegative) of the genetic variant YER124C, a daughter cell-specific protein involved in cell wall metabolism. Following a similar approach as in Chakraborty et al. [6], we include 492 QTLs by filtering out genes with missing or low variability in expression levels. The total sample size is 262. Given the high-dimensional QTLs, constructing an NSCG with only causally relevant variables for the outcome of interest is essential. We apply NSCSL-TE and compare it with all baseline methods. The summarized results in Table 7 highlight our method's ability to identify relevant genetic influencers
Table 4: Comparison studies under the nonlinear structural equation model for S4 over 50 replications.
Scenario	Method	FDR±SE	TPR±SE	SHD±SE	Time±SE
S4	NSCSL-TE	0.03±0.01	0.83±0.01	0.60±0.02	55.8±0.3
p=20	NSCSL-DE	0.03±0.01	0.50±0.01	1.60±0.02	56.8±0.2
n=1000	NOTEARS	0.91±0.01	0.83±0.01	35.90±0.04	56.5±0.7
ER Model	PC	0.99±0.01	0.12±0.01	44.12±0.03	15.7±0.2
Degree=5	LiNGAM	0.93±0.01	0.70±0.00	37.30±0.03	11.6±0.1
DAGGNN	0.94±0.01	0.86±0.01	34.80±0.06	41.32±9.8
GSGES	0.98±0.01	0.23±0.01	52.40±0.38	22.12±7.5
FCI	0.97±0.01	0.13±0.01	33.80±0.06	12.6±0.3
CAM	0.95±0.01	1.00±0.01	31.60±0.08	27.92±33.7
Table 5: Comparison studies under the scale-free (SF) model for S5 over 50 replications.
Scenario	Method	FDR±SE	TPR±SE	SHD±SE	Time±SE
S5	NSCSL-TE	0.02±0.02	0.78±0.03	5.08±0.11	135.7±5.6
p=50	NSCSL-DE	0.02±0.02	0.51±0.02	17.20±0.35	147.0±6.3
n=1000	NOTEARS	0.88±0.04	0.75±0.03	123.10±1.50	160.5±8.1
SF Model	PC	0.97±0.03	0.06±0.02	79.34±1.17	32.1±0.8
Degree=5	LiNGAM	0.91±0.02	0.98±0.01	212.00±5.12	117.7±6.3
DAGGNN	0.92±0.02	0.85±0.02	203.50±7.80	57.32±13.6
GSGES	0.96±0.03	0.10±0.03	98.34±2.15	35.92±13.5
FCI	0.97±0.02	0.07±0.01	81.70±1.40	15.7±1.1
CAM	0.98±0.04	0.24±0.03	218.00±8.15	37.12±53.6
Table 6: Real data results for the single-cell data by Sachs et al. [30], evaluated by total edges, correct edges, and SHD, based on the true NSCG with respect to the protein Akt.
Method	NSCSL	NOTEARS	PC	LiNGAM	DAGGNN	GSGES	FCI	CAM
Total Edges	8	20	25	6	33	28	24	7
Correct Edges	4	2	2	1	4	2	2	1
SHD	8	21	28	11	30	32	27	10
Figure 2: Sensitivity analyses.
Figure 3: The causal signaling network in Sachs et al. [30], where the blue-colored sub-graph is the true NSCG for protein Akt (purple).
Table 7: Real data results for the yeast gene data, evaluated by total edges, the identified parents/ancestors of the variant YER124C, and the edges towards YER124C.
Method	NSCSL	NOTEARS	PC	LiNGAM	DAGGNN	GSGES	FCI	CAM
Total Edges	11	25	22	15	35	27	22	33
# Parents/Ancestors of YER124C	8	8	6	4	8	7	5	7
# Edges towards YER124C	11	9	8	6	10	9	8	10