SlowGuess/NeurIPS_2023_OCR huggingface dataset:
When there is only a single domain,  $\Pi = \{\Pi^1\}$ , the first constraint reduces to standard d-separation on a causal diagram. The second condition is a generalization of the  $\Psi$ -Markov property characterization [6], extending conditional invariances to multiple domains. We illustrate the definition with the following two examples.
Example 1. Consider the selection diagram in Figure 1(a) with two domains  $\Pi = \{\Pi^1, \Pi^2\}$ . Let  $\mathbf{P} = \langle P_1^1, P_2^1, P_1^2 \rangle$  be the result of the interventions  $\Psi^{\Pi} = \langle \{\}^1, \{X\}^1, \{\}^2 \rangle$ ,  $\mathbf{S} = \{S^{1,2}\}$  be the set of S-nodes, and  $\mathcal{K} = [1,0,1]$ . First, we have  $(Y \angle X | S^{1,2})_{G_S}$  so the first constraint of Def. 3.2 is not applicable for any distribution. Second, we compare  $P_1^1(y|x)$  and  $P_2^1(y|x)$ , where  $S^{1,1} = \emptyset$  by convention and  $\mathbf{K} = \{X\}$ . We have  $(Y \angle X)_{G_{S_X}}$  and the invariance is not required. For  $P_1^1(y|x)$  and  $P_1^2(y|x)$ , we have  $(Y \angle S | X)_{G_S}$ . Also, for  $P_2^1(y|x)$  and  $P_1^2(y|x)$ , we have  $(Y \angle \{X, S\})_{G_{S_X}}$ . Hence, no invariances are required between those pairs of distributions. A similar argument can be made when comparing other probability terms across distributions. Therefore,  $\mathbf{P}$  satisfies the S-Markov property with respect to  $\langle G_S, \Psi \rangle$ .
Example 2. Consider the setup from Ex. 1. We check if  $\mathbf{P}$  satisfies the S-Markov property relative to  $\langle G_S, \Psi' \rangle$  where  $\Psi'^{\Pi} = \langle \{\}^1, \{Y\}^1 \{\}^2 \rangle$ . We compare  $P_1^1(X)$  and  $P_2^1(X)$  and we have  $K = (\{\}^1 \Delta \{Y\}^1) \cup \emptyset = \{Y\}^1$ . The separation  $(X \perp Y | S^{1,2})_{G_{\overline{Y}}}$  holds true which implies the invariance  $P_1^1(X) = P_2^1(X)$ , but  $P_2^1$  was generated from an intervention on  $X$  so the invariance is not satisfied. Hence,  $\mathbf{P}$  does not satisfy the S-Markov property with respect to  $\langle G_S, \Psi' \rangle$ .
Next, we use Def. 3.2 to define S-Markov equivalence as follows. In words, two pairs of selection diagrams and their corresponding sets of intervention targets  $\langle G_S, \Psi \rangle$  and  $\langle G_S', \Psi' \rangle$  are S-Markov equivalent if they can induce the same set of distribution tuples.
Definition 3.3 (S-Markov Equivalence). Let  $\Pi$  and  $\mathcal{K}$  denote fixed sets of domains and indices of known intervention targets, respectively. Given selection diagrams  $G_{S}, G_{S}^{\prime}$  defined over  $\mathbf{V} \cup \mathbf{S}$  and the corresponding intervention targets  $\Psi, \Psi^{\prime}$ , the pairs  $\langle G_{S}, \Psi \rangle$  and  $\langle G_{S}^{\prime}, \Psi^{\prime} \rangle$  are said to be S-Markov equivalent if  $S_{\mathcal{K}}^{\Pi}(G_{S}, \Psi) = S_{\mathcal{K}}^{\Pi}(G_{S}^{\prime}, \Psi^{\prime})$ .
3.3 Multi-domain observational data
S-nodes introduced through the lens of selection diagrams are augmentations of the causal graph to represent different domains and changes in distributions that may occur [7, 15, 54, 57]. As part of this augmented graph, S-nodes are graphically similar to F-nodes, which have been successfully used to represent interventions [6, 7, 54]. F-nodes are utility nodes where each one is a parent to (each element
in) a symmetrical difference set, and they are used to represent invariances between interventional distributions. The significance of these F-nodes will be further emphasized in Section 3.4; more specifically, by Definition 3.5 and Proposition 1. Despite the similarity between F-nodes and S-nodes, it is worthy to distinguish S-nodes since many causal inference tasks, such as in transportability, rely on knowing the S-node structure [15, 53]. Before deriving the graphical characterization for the S-Markov equivalence class, we first focus on the setting where there is only observational data across different domains. We highlight that S-nodes can be exactly viewed as F-nodes constructed from interventions with unknown targets when there is only observational data to consider [6].
Definition 3.4 (Corresponding Intervention Set). Consider the Multi-domain setup. For a selection diagram  $G_{S}$  over  $\mathbf{N}$  domains.  $\langle \mathbf{V}_{S^{i,j}}\rangle = \langle \mathbf{V}_{S^{1,2}},\mathbf{V}_{S^{1,3}},\dots,\mathbf{V}_{S^{N - 1,N}},\rangle \forall i\neq j\in [N]$  is an ordered tuple of the children of each S-node. The corresponding intervention set for  $\mathbf{V}_{\mathbf{S}}$  is  $\langle \mathbf{I}^1,\mathbf{I}^2,\dots,\mathbf{I}^N\rangle$ , such that  $\mathbf{I}^i\Delta \mathbf{I}^j = \mathbf{V}_{S^{i,j}}$  for all  $i\neq j$ .
The corresponding intervention set is a set that simplifies our presentation of the following theorem.
Theorem 1 (Equivalence of  $\Psi$  and S Markov property given multi-domain observational distributions). Consider the Multi-domain setup. Let  $G_{S}$  be a selection diagram among N domains and G be the corresponding causal diagram without S-nodes. Let  $\Psi^{\Pi} = \langle \{\}^{1},\dots,\{\}^{N}\rangle$  and  $\mathcal{K} = [1,1,\dots,1]$ , such that for each of the N domains, there is only observational data. Let  $\mathbf{I}_S$  be the corresponding intervention set for  $\mathbf{V}_{\mathbf{S}}$ . Let  $\mathbf{P}^{\Pi}$  be an arbitrary set of distributions generated by the corresponding interventions.  $\mathbf{P}^{\Pi}$  satisfies the S-Markov property with respect to  $\langle G_S,\Psi \rangle$  if and only if it satisfies the  $\Psi$ -Markov property with respect to  $\langle G,\mathbf{I}_S\rangle$ .
When given observations collected from multiple domains, it is equivalent to collecting distributions with unknown-target interventions. This coincides with other works, which treat different domains and interventions as the same [10, 13]. In this setting, S-nodes have a correspondence to the augmented graph's F-nodes in [6]. In some sense, the change-in-domain can be viewed as "nature's" intervention on the causal system. However, this simplification is not warranted when we consider interventions that occur in different domains.
3.4 Mixture of multi-domain observational and interventional data
Next, we analyze the general setting with multi-domain observational and interventional data. Def. 3.2 and 3.3 may be quite challenging to evaluate in practice since it involves surgically altering the selection diagram. One can leverage a graphical approach that encodes the symmetric differences of interventions using F-nodes [7].
Definition 3.5 (Augmented selection diagram). Consider the Multi-domain setup. Let the multiset  $\mathcal{I}$  be defined as  $\mathcal{I} = \{\mathbf{K}_1, \mathbf{K}_2, \dots, \mathbf{K}_k\} = \{\mathbf{K} | \Psi_m^i, \Psi_l^j \in \Psi \land \Psi_m^i \Delta \Psi_l^j = \mathbf{K}\}$ . The augmented graph of  $G_S$  with respect to  $\Psi$  is denoted as  $Aug_{\Psi}(G_S)$  and constructed as follows:  $Aug_{\Psi}(G_S) = (\mathbf{V} \cup \mathbf{L} \cup \mathbf{S} \cup \mathcal{F}, \mathbf{E} \cup \mathbf{E}_{\mathbf{S}} \cup \mathcal{E})$ , where  $\mathcal{F} = \{F_i^{j,k}\}_{i=1}^{N}$  is the set of added F-nodes and  $\mathcal{E} = \{(F_i^{j,k}, l)\}_{l \in \mathbf{K}_i}$  is the set of added F-node edges.
The F-nodes graphically encode the symmetrical difference sets between every pair of intervention targets in  $\Psi^{\mathrm{II}}$  (i.e.  $\Psi_{m}^{i}\Delta \Psi_{l}^{j}$ ) within and across the different domains in  $\Pi$ .  $F_{k}^{i,i} = F_{k}^{i}$  denotes an F-node representing the kth symmetric difference of intervention targets within domain  $i$  and  $F_{k}^{i,j}$  denotes an F-node from comparing intervention targets between domains  $i$  and  $j$ . The result is an augmented selection diagram with the original causal structure augmented with S-nodes, F-nodes, and their additional edges. For example, Figure 1(c) shows the augmented diagram of the selection diagram in Figure 1(a) with respect to  $\Psi^{\mathrm{II}} = \langle \{\}^{1},\{X\}^{1},\{\}^{2}\rangle$ . The significance of this construction follows from Proposition 1 where separation statements in the S-Markov definition are tied (shown to be equivalent, formally speaking) to ones in the augmented selection diagram, with no need to perform any graphical manipulations.
Proposition 1 (Graphical S-Markov Property). Consider the Multi-domain setup. Let  $Aug_{\Psi}(G_S)$  be the augmented graph of  $G_S$  with respect to  $\Psi$ . Let  $\mathbf{K}_i^{j,k} = \mathbf{K}_i \cup \{S^{j,k}\}$  be the union of the set of nodes adjacent to  $F_i^{j,k}$  and the corresponding S-node  $S^{j,k}$ . The following equivalence relations hold
Due to space constraints, all the proofs are provided in the Appendix.
for disjoint  $\mathbf{Y},\mathbf{Z},\mathbf{W}\subseteq \mathbf{V}$  , where  $\mathbf{W}_i = \mathbf{W}\cap \mathbf{K}_i$  and  $\mathbf{R} = \mathbf{K}_i\backslash \mathbf{W}_i$
$$
(\mathbf {Y} \perp \mathbf {Z} | \mathbf {W}, \mathbf {S}) _ {G _ {S}} \iff (\mathbf {Y} \perp \mathbf {Z} | \mathbf {W}, \mathbf {S}, \mathcal {F}) _ {A u g _ {\Psi} (G _ {S})} \tag {2}
$$
$$
\left(\mathbf {Y} \perp \mathbf {K} _ {i} ^ {j, k} | \mathbf {W} \backslash \mathbf {W} _ {i}, \mathbf {S} \backslash \{S ^ {j, k} \}\right) _ {G _ {S} \underline {{\mathbf {W}}} _ {i}, \overline {{\mathbf {R} (\mathbf {W})}}} \Longleftrightarrow (\mathbf {Y} \perp \left\{F _ {i} ^ {j, k}, S ^ {j, k} \right\} | \mathbf {W}, F _ {[ k ] \backslash \{i \}}, \mathbf {S} \backslash \{S ^ {j, k} \}) _ {A u g _ {\Psi} (G _ {S})} \tag {3}
$$
The result in the above proposition is illustrated in the following example.
Example 3. Consider the selection diagram in Fig. 1(b) with intervention targets  $\Psi = \langle \emptyset^1, \{Z\}^1, \emptyset^2 \rangle$ . By Prop. 1, we can evaluate the S-Markov property in the corresponding augmented diagram in Fig. 1(d) without manipulating it. For example,  $(Y \perp Z)_{G_S \overline{Z}}$  can be tested in Fig. 1(d) by  $(Y \perp F_z^1 | \mathbf{S}, F_z^{1,2})_{Aug_\Psi(G_S)}$  to determine if the invariance  $P^1(Y) = P_Z^1(Y)$  should hold. In addition, we can test if across-domain distributional invariances should hold. We have  $(Y \not\ll \{F_z^{1,2}, S^{1,2}\} | \{X, Z\}, F_z^1)_{Aug_\Psi(G_S)}$ , then the invariance  $P_Z^1(Y|X,Z) = P^2(Y|X,Z)$  is not required.
Maximal Ancestral Graphs (MAGs) provide a compact and convenient representation capable of preserving all the tested constraints in augmented selection diagrams which are represented by d-separations [58]; see also [59, p. 6]. This is formalized in Definition 3.6 and the construct is referred to as an S-MAG. The following example is provided for illustration.
Definition 3.6 (S-MAG). Given a selection diagram  $G_{S}$  and a set of intervention targets  $\Psi$ , an S-MAG is the MAG constructed from  $Aug_{\Psi}(G_{S})$ . That is  $\mathrm{MAG}(Aug_{\Psi}(G_{S}))$ .
Example 4. Consider the selection diagram in Figure 1(a) and let  $\Psi = \langle \{\}^1, \{X\}^1, \{\}^2 \rangle$ . The corresponding augmented selection diagram  $A u g_{\Psi}(G_S)$  is shown in Fig. 1(c). Finally, the corresponding S-MAG is  $M A G(A u g_{\Psi}(G_S)) = \{X \leftarrow F_x^1 \rightarrow Y, X \leftarrow F_x^{1,2} \rightarrow Y, X \leftarrow S^{1,2} \rightarrow Y, X \rightarrow Y\}$ .
Finally, putting these results together, we derive a graphical characterization for two selection diagrams with corresponding sets of intervention targets to be S-Markov equivalent.
Theorem 2 (S-Markov Characterization). Let  $\Pi$  and  $\mathcal{K}$  denote fixed sets of domains and indices of known intervention targets, respectively. Given selection diagrams  $G_{S}, G_{S}^{\prime}$  defined over  $\mathbf{V} \cup \mathbf{S}$  and the corresponding intervention targets  $\Psi, \Psi^{\prime}$ , the pairs  $\langle G_{S}, \Psi \rangle$  and  $\langle G_{S}^{\prime}, \Psi^{\prime} \rangle$  are S-Markov equivalent if and only if for  $M = MAG(Aug_{\Psi}(G_{S}))$  and  $M^{\prime} = MAG(Aug_{\Psi^{\prime}}(G_{S}^{\prime}))$ :<sup>6</sup>
1.  $M$  and  $M^{\prime}$  have the same skeleton;
2.  $M$  and  $M^{\prime}$  have the same unshielded colliders; and,
3. If a path  $p$  is a discriminating path for a node  $Y$  in both  $M$  and  $M'$ , then  $Y$  is a collider on the path in one graph if and only if it is a collider on the path in the other.
Theorem 2 states that the pairs  $\langle G_S, \Psi \rangle$  and  $\langle G_S', \Psi' \rangle$  are S-Markov equivalent if their corresponding S-MAGs satisfy the corresponding three conditions, as illustrated in the example below.
Example 5. Consider the tuples  $\langle G_S, \Psi^{\Pi} \rangle$  from Example 1 and  $\langle G_S, \Psi'^{\Pi} \rangle$  from Example 2. S-MAGs  $M = MAG(Aug_{\Psi}(G_S))$  is shown in Ex. 4 and  $M' = MAG(Aug_{\Psi'}(G_S)) = \{F_y^1 \to Y, F_y^{1,2} \to Y, X \gets S^{1,2} \to Y, X \to Y\}$ . Therefore,  $M_1$  and  $M_2$  have differing skeletons and thus are not S-Markov equivalent.
Next, we leverage this characterization to devise an algorithm to learn the corresponding equivalence class of a true, underlying selection diagram.
4 Causal Discovery From Multiple Domains
We investigate in this section how to learn an EC of selection diagrams from a mixture of observational and interventional data that is generated from multiple domains. The graphical characterization of S-Markov equivalence in Theorem 2 and the significance of ancestral graphs (MAGs) in deriving this result motivate the following definition of S-PAG.
<sup>6</sup>We assume that the symmetrical difference sets are indexed for both diagrams in the same pattern such that the correspondence between F-nodes and S-nodes is the same in  $M$  and  $M'$ .
Figure 2: Example of S-FCI applied with  $\Psi = \langle \{\}^1, \{X\}^1, \{\}^2 \rangle$  and  $\mathcal{K} = [1,1,1]$ . The S-node representing domain-shift between domains 1 and 2 is the black square in (a).
Algorithm 1 S-FCI: Algorithm for Learning a S-PAG - SepSet the separating sets,  $\mathbf{S}$  is the S-node set,  $\mathcal{F}^{\Pi}$  the F-node set,  $\mathbf{H}$  maps each pair of known-targets symmetric diff.s., and  $\sigma$  maps each pair of distributions to a pair of domains.
Input: Tuple of distributions  $\mathbf{P}^{\mathrm{II}} = \langle P_1^1,\dots,P_m^N\rangle$ , vector of known intervention targets  $\kappa$  and  $\Psi^{\mathrm{II}}$ . Output: S-PAG,  $\mathcal{P}$
$\mathbf{S},\mathcal{F}\gets \phi ,k\gets 0,\sigma :\mathbb{N}\to \mathbb{N}\times \mathbb{N},\mathbf{H}\gets \phi$ $(\mathbf{S},\mathcal{F},\mathbf{H},\sigma)\gets$  CreateAugmentedNodes(  $\Psi^{\Pi},V)$  (see Alg. E.2)
Phase I: Learn skeleton
for all pairs  $X,Y\in \mathbf{V}\cup \mathcal{F}\cup \mathbf{S}$  do
$SepSet(X,Y), SepFlag \gets$  GeneralizedDoConstraints $(X,Y,\mathcal{F},\mathbf{S},\sigma, \Psi^{\Pi},\mathcal{K},\mathbf{V})$  (see Alg. E.4)
if SepFlag = True then
Remove edge between X and Y
Phase IIa: Orient unshielded colliders
For every unshielded triple  $\langle X,Y,Z\rangle$  in  $\mathcal{P}$  orient it as a collider iff  $Z\notin \operatorname{Set}(X,Y)$
Phase IIb: Apply logical orientation rules
R1-7: Apply 7 FCI rules from [39] and following two rules until none apply.
Rule 8': For  $F_{k}^{i,j} \in \mathcal{F}^{\Pi}$  and for  $S^{i,j} \in \mathbf{S}$ , orient adjacent edges out of  $F_{k}^{i,j}$  and  $S^{i,j}$ .
Rule 9': For  $F_{k}^{i,j} \in \mathcal{F}^{\Pi}$  with  $X \in H_{k}^{i,j}$ , that is adjacent to a node  $Y \notin H_{k}^{i,j}$ , if  $|H_{k}^{i,j}| = 1$ , then orient  $X \to Y$ .
Definition 4.1 (S-PAG). Consider the Multi-domain setup. Let  $M = MAG(Aug_{\Psi}(G_S))$  and let  $[M]$  be the set of S-MAGs corresponding to all the tuples  $\langle G_S', \Psi'^{\Pi} \rangle$  that are S-Markov equivalent to  $\langle G_S, \Psi^{\Pi} \rangle$ . The S-PAG for  $\langle G_S, \Psi^{\Pi} \rangle$ , denoted  $\mathcal{P}$  is a graph such that:
1.  $\mathcal{P}$  has the same adjacencies as M and any member of [M] does; and
2. every non-circle mark (tail or arrowhead) in  $\mathcal{P}$  is an invariant mark in [M] (i.e. present in all the S-MAGs in [M]).
S-PAGs generalize PAGs and  $\Psi$ -PAGs from the single-domain to the multiple-domain setting[6, 42]. The S-nodes and F-nodes are not so much "random variables" as they are graphical instruments that encode differences across domains and among interventional distributions in this equivalence class. Next, we introduce a generalization of c-faithfulness [6] that enables causal discovery from multi-domain data.
Definition 4.2 (S-faithfulness). Consider a selection diagram  $G_{S}$  over  $\mathbf{N}$  domains. A tuple of distributions  $\langle \mathbf{P}_{\mathbf{I}} \rangle_{\mathbf{I} \in \Psi^{\Pi}} \in S_{K}^{\Pi}(G_{S}, \Psi^{\Pi})$  is called s-faithful to  $G_{S}$  if the converse of each of the S-Markov conditions (Definition 3.2) holds.
The new algorithm, called S-FCI is shown in Alg. 1. Due to space constraints, we only include the high-level algorithm here. The algorithm proceeds by first constructing the augmented graph using Alg. E.2, by adding S-nodes and F-nodes to represent every pair of domains and interventions. Then it uses hypothesis testing to learn invariances in the skeleton (Alg. E.3) and finally applies orientation rules (Alg. E.5). S-FCI learns the skeleton by mapping pairs of distributions in  $\mathbf{P}^{\mathrm{II}}$  to F-nodes, or S-nodes by testing for the distributional invariances discussed in Section 3.1. Def. 3.2 and Prop. 1 connect these invariances to graphical criterion, which allow us to reconstruct the skeleton of the causal diagram. Interventional distributions across domains are used to learn F-node structure, and
(a)
Figure 3: Causal graphs related to example 6 - selection diagram with an intervention at Y, and S-node pointing to X (a), the graph after applying  $\mathcal{L}$ -FCI without considering domain-changes (b) and the S-PAG learned by S-FCI (c).
whereas observational distributions across domains are used to learn S-node structure. Besides the standard FCI rules that apply in the absence of selection bias, the algorithm also applies the following rules R8'-9'.
Rule 8' (Augmented Node Edges) - We orient edges out of F-nodes.
Rule 9' (Identifiable Inducing Paths) - If  $F_{k}^{i,j} \in \mathcal{F}$  is adjacent to a  $Y \notin H_{k}^{i,j}$  known-target node and we know that the intervention target is node  $X$ , one can orient  $X \to Y$  because the  $F_{k}^{i,j} \to Y$  is only present due to an inducing path between  $X$  and  $Y$ .
In Figure 2, the different stages of the S-FCI algorithm are shown. Next we prove the proposed S-FCI algorithm is sound.
Theorem 3 (S-FCI Soundness). Given  $\mathcal{K}$ , let  $\mathbf{P}^{\Pi}$  be generated by some unknown tuple  $\langle G_S, \Psi^{\Pi} \rangle$  from domains  $\Pi$  with a corresponding selection diagram  $G_S$  and is s-faithful to the selection diagram  $G_S$ . S-FCI algorithm is sound (i.e. every adjacency and orientation in  $\mathcal{P}_{\mathrm{S - F C I}}$ , the S-PAG learned by S-FCI, is common for  $MAG(Aug_{\Psi}(G_S)))$ ).
Next, we illustrate some subtleties between the S-FCI and related algorithms that say pool observational and interventional distributions, ignoring the domain change. The example is motivated from biomedical sciences, where interventions are commonly performed in different domains and the goal is to leverage all datasets for learning. A group of scientists are trying to determine the causal structure of a set of proteins, but leverage data across the lab and hospital setting. Different experiments are run in each setting and combined into a single dataset [29]. We provide additional examples and commentary on the S-FCI subtleties in the Appendix.
Example 6. Let  $G_{S}$  be a selection diagram as shown in Figure 3(a). Let  $\Pi = \langle \Pi^{1}, \Pi^{2} \rangle$  be the set of domains representing the lab  $(\Pi^{1})$  and the hospital  $(\Pi^{2})$ . These are a tuple of distributions  $\mathbf{P} = \langle P_{1}^{1}, P_{1}^{2} \rangle$  with intervention targets  $\Psi^{\Pi} = \langle \{\}^{1}, \{Y\}^{1}, \{\}^{2} \rangle$  and  $\mathcal{K} = [1, 1, 1]$ , where  $X$  represents some protein in the dataset.
In this example, let  $G_{S}$  be the true selection diagram as shown in Figure 3(a). Given the interventional and observational data, we may be tempted to use the  $\mathcal{I}$ -FCI algorithm and simply pool the observational data, while ignoring the domain differences [7], but this would learn the graph in Figure 3(b) with an incorrect orientation (shown as the red edge). This I-PAG only contains one F-node because there is only two distributions: i) the pooled observational data and ii) the data resulting from intervention on Y. Applying R9 of the  $\mathcal{I}$ -FCI algorithm incorrectly orients the edge  $X \gets Y$ . Thus, R9 of the  $\mathcal{I}$ -FCI algorithm is not sound when the domains are ignored [7, 44].
Figure 3(c) contains what S-FCI would recover. Intuitively, one should learn (c) instead of (b) because even though there is a change in distribution among X and Y, one cannot ascertain whether there is an inducing path from  $F_{y}^{1}$  to X, or a change in distribution due to the domain.
In this paper, we introduced a generalized Markov property called S-Markov, which defines a new equivalence class (EC), the S-PAG, representing the constraints found across observational and experimental distributions collected from multiple domains. Building on this new characterization, we develop a causal discovery algorithm called S-FCI, which subsumes FCI,  $\mathcal{L}$ -FCI and  $\Psi$ -FCI, and accepts as input a mixture of observational and interventional data from multiple domains.
AL was supported by the NSF Computing Innovation Fellowship (#2127309). EB was supported in part by the NSF, ONR, AFOSR, DoE, Amazon, JP Morgan, and The Alfred P. Sloan Foundation.
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