# Improving Out-of-Distribution Generalization in Graphs
via Hierarchical Semantic Environments

Yinhua Piao1, Sangseon Lee2, Yijingxiu Lu1, Sun Kim1,3,4

Department of Computer Science and Engineering, Seoul National University1
Institute of Computer Technology, Seoul National University2, AIGENDRUG Co.,
Ltd.3
Interdisciplinary Program in Artificial Intelligence, Seoul National
University4
{2018-27910, sangseon486, solanoon0113<EMAIL_ADDRESS>
###### Abstract

Out-of-distribution (OOD) generalization in the graph domain is challenging
due to complex distribution shifts and a lack of environmental contexts.
Recent methods attempt to enhance graph OOD generalization by generating flat
environments. However, such flat environments come with inherent limitations
to capture more complex data distributions. Considering the DrugOOD dataset,
which contains diverse training environments (e.g., scaffold, size, etc.),
flat contexts cannot sufficiently address its high heterogeneity. Thus, a new
challenge is posed to generate more semantically enriched environments to
enhance graph invariant learning for handling distribution shifts. In this
paper, we propose a novel approach to generate hierarchical semantic
environments for each graph. Firstly, given an input graph, we explicitly
extract variant subgraphs from the input graph to generate proxy predictions
on local environments. Then, stochastic attention mechanisms are employed to
re-extract the subgraphs for regenerating global environments in a
hierarchical manner. In addition, we introduce a new learning objective that
guides our model to learn the diversity of environments within the same
hierarchy while maintaining consistency across different hierarchies. This
approach enables our model to consider the relationships between environments
and facilitates robust graph invariant learning. Extensive experiments on
real-world graph data have demonstrated the effectiveness of our framework.
Particularly, in the challenging dataset DrugOOD, our method achieves up to
1.29% and 2.83% improvement over the best baselines on IC50 and EC50
prediction tasks, respectively.

##  1 Introduction

Figure 1:  (a) Results on IC50-sca dataset from DrugOOD [20]. (b) Flat
environments from existing approaches. (c) Hierarchical environments from our
methods. For visualization, we set #real environments as $10$.

Graph-structured data is ubiquitous in real-world applications, from social
networks to biological networks and chemical molecules [21, 14, 16, 48]. One
notable advancement in this area is the emergence of Graph Neural Networks
(GNNs). GNN-based models have pioneered end-to-end learning strategies to
extract valuable information from graphs and have demonstrated remarkable
success across various applications [24, 42, 13]. However, the success of GNNs
encounters challenges in out-of-distribution (OOD) scenarios primarily due to
the intricate nature of graph distribution shifts [9, 3, 28]. Graph data,
characterized by nodes, edges, and potential attributes, poses additional
challenges compared to other domains such as natural language processing (NLP)
or computer vision (CV). Unlike those domains where context is often provided
by sentences, paragraphs, or spatial information in images [15, 4, 36], graph
lacks a built-in contextual framework, making it inherently challenging to
discern the relevance and context of individual graph elements in OOD
scenarios [49, 12].

Invariant Risk Minimization (IRM) [2] is a widely used strategy in the
Euclidean domain, relying on the assumption that training data is sourced from
distinct environments with varied data distributions. Motivated by the success
of invariant learning in the Euclidean domain [2, 1, 26, 8, 17, 29, 31, 32],
invariant learning methods for non-Euclidean graphs [43, 27, 6, 7, 10, 27, 30]
extend the concept to address the unique challenges posed by graph-structured
data. Despite commendable progress in addressing OOD scenarios within the
graph domain, existing studies have predominantly adhered to the assumption of
a flat environmental situation [5, 33]. For example, in the DrugOOD dataset,
there are nearly 7000 diverse training environments in ic50-sca subset,
including a wide variety of distinct substructures. Fig. 1 illustrates three
existing approaches to acquire environments when performing invariant learning
on this data: “Real” means directly utilizing provided environments [27],
“Rand#2” means randomly splitting samples into two environments [6, 2], and
“Infer#2” means inferring samples into two environments [8, 17]. As shown in
Fig. 1, all existing approaches exhibit poor or comparable performance to
empirical risk minimization (ERM), which is consistent with the findings of
Yang [43]. These phenomena indicate the shortcomings of flat environments: (1)
Limited consideration of local environment similarity degrades performance in
numerous environments (e.g., ‘Real’ case in Fig. 1) (2) Inferring from a small
number of environments may fail to capture global environment similarities and
interrelationships. (e.g., ‘Infer#2’ case in Fig. 1 )

Considering that graphs exhibit a hierarchical structure, with semantic
information organized across various levels. This inherent hierarchy is
fundamental to understanding structures and properties within graphs [44]. In
addition, inspired by the recent works [29, 17, 1], which show that the
diversity of environments is recognized as the key to effectively handling
graph OOD scenarios , we aim to bridge the gap of flat environments while
leveraging the inherent hierarchy of graphs. In this paper, we propose a
hierarchical approach to generate semantic environments of each graph for
effective graph invariant learning. Initially, we extract variant subgraphs
from the input graph, enabling the generation of proxy predictions on local
environments. Employing stochastic attention mechanisms, we iteratively re-
extract subgraphs, building global environments hierarchically. To guide the
robustness of hierarchical environment inference, we introduce a hierarchical
environment diversification loss which encourages our model to diversify
environments within the same hierarchy while maintaining consistency across
different hierarchical levels. This approach not only aids our model in
considering relationships between environments but also strengthens graph
invariant learning robustness. By modeling relationships between different
environments within a hierarchical framework, our approach acquires the rich
source of environmental information embedded in the hierarchical structure of
graphs. Extensive experiments demonstrate our approach to graph OOD
classification datasets. Our contributions can be summarized as follows:

  * •

We propose a hierarchical approach to generate semantic environments for
effective graph invariant learning. To the best of our knowledge, our proposed
method is the first attempt to generate the environments in a hierarchical way
in graph OOD generalization.

  * •

We introduce a new learning objective that guides our model to learn the
diversity of environments within the same hierarchy while maintaining
consistency across different hierarchies.

  * •

Extensive experiments have demonstrated our model yields significant
improvements over various domains. In molecule graph benchmarks DrugOOD, our
method achieves up to 1.29% and 2.83% higher ROC-AUC compared to SOTA graph
invariant learning approaches.

##  2 Related Works

##### Out-of-Distribution Generalization

In the Euclidean domain, such as computer vision, OOD generalization has been
studied extensively using a variety of strategies, such as Invariant Risk
Minimization (IRM) [2], data augmentation [47], and domain adaptation methods
[39]. Unlike deep neural networks with ERM, which suffer performance
degradation under data distributional shifts, IRM-based studies [26, 1] have
shown powerful and robust performance when providing environmental
information. Recently, a two-step optimization framework EIIL [8] has been
proposed to train the invariant learning model without explicit environment
labels. EIIL trains the environment inference model to learn and infer the
environment labels in the first step, and in the second step, EIIL uses
inferred environments to conduct invariant learning. Other recent approaches,
such as ZIN [29], HRM [31], KerHRM[32], and EDNIL [17] improve the OOD
generalization by augmenting the environment to address its high
heterogeneity. These augmentation strategies, such as the assistant model,
clustering-based model, or effective diversification objective function all
enhance the learning stage to infer diverse environments for effective
invariant learning.

##### Graph Invariant Learning

While general invariant learning methods are effective for the Euclidean
domain, challenges arise when applying them to non-Euclidean domains like
graph structures. In the realm of graph invariant learning, since graph data
have an inherent nature of complex distribution shifts, often lacks explicit
environment labels. To address this issue, methodologies have been widely
developed and implemented. Approaches such as EERM [41], LiSA [45], and GREA
[30] proposed to generate new environments, offering effective strategies to
discover invariant patterns without environment label knowledge. While methods
like MoleOOD [43] and GIL[27] focus on the inference of environments,
providing solutions to deal with challenging or unexplored environments.
Another dimension of exploration involves learning with auxiliary assistance
to ensure sufficient variation, with typical techniques found in CIGA [6],
GALA[7]. Collectively, these innovative methods offer detailed strategies to
navigate the challenges inherent in graph invariant learning.

##  3 Problem Definition and Preliminaries

In Sec. 3.1, we introduce notations about out-of-distribution generalization
and the background knowledge of graph invariant learning. In Sec. 3.2, we
introduce environment inference to explain how recent research works on the
graph dataset without environmental information.

###  3.1 Problem Definition

Following general settings of OOD generalization, we assume that the training
datasets are collected from multiple training environments. Let
$\mathcal{G}_{tr}=\\{G^{e}\\}_{e\in\texttt{supp}(\mathcal{E}_{tr})}$ with
different graphs $G^{e}=\\{(G_{i}^{e},y_{i}^{e})\,|\,1\leq i\leq N^{e}\\}$ be
the training datasets, where $G^{e}$ represents the graphs drawn from
environment $e$, and $\mathcal{E}_{tr}$ indicates the environment labels in
the training datasets. We denote the environments $\mathcal{E}_{all}$ as all
possible environment labels in the real world. Then, the test datasets contain
graphs $\mathcal{G}_{test}=\\{G^{e^{\prime}}\\}$ with unknown environment
labels $e^{\prime}$, where
$e^{\prime}\in\texttt{supp}(\mathcal{E}_{all})\setminus\texttt{supp}(\mathcal{E}_{tr})$.
Our approach addresses out-of-distribution challenges in graph-level
classification by incorporating invariant learning within environments derived
from complex heterogeneous data hierarchically.

###  3.2 Preliminaries

##### Graph Invariant Learning.

Recently, several studies have identified the cause of performance
deterioration in out-of-distribution (OOD) graphs, attributing it to the
introduction of spurious features. It is widely acknowledged that spurious
features exhibit sensitivity to environmental changes, often referred to as
environmental factors. The incorporation of spurious features diminishes the
significance of invariant features, resulting in a decline in OOD performance.
As a result, recent research [6, 7, 27, 45] often involves graph decomposition
to extract both variant subgraphs $G_{v}$ and invariant subgraphs $G_{inv}$
from the original graphs $G$, thereby facilitating the invariant GNN encoder
$f$ [24] to learn relationships between invariant features $h_{inv}$ and
labels $y$. Following the invariant risk minimization [2], the encoder $f$ is
trained with a regularization term enforcing jointly optimize the encoder $f$
in training environments $e$, which can generalize to unseen testing
environments $e^{\prime}$:

$\sum_{e\in{\texttt{supp}\\{\mathcal{E}_{tr}\\}}}{\mathcal{R}^{e}(f)+\lambda||\nabla{\mathcal{R}^{e}(f)}||^{2}},$
(1)

where $\mathcal{R}^{e}(f)=\mathbb{E}^{e}_{G,y}[\mathcal{L}(f(G),y)]$ denotes
the risk of $f$ in environment $e$, and $\mathcal{L}(\cdot,\cdot)$ denotes the
loss function.

##### Environment Inference for GIL.

In most cases, environmental information is difficult to obtain, especially in
the context of graph generalization. While some datasets may provide metadata
that could serve as environmental information, it is not reliable to fully use
such information as the ground truth of the environment is not guaranteed.
Therefore, existing methods [8, 17, 29, 31, 32] often infer the environmental
labels $\mathcal{E}_{infer}$ by using variant subgraphs $G_{v}$ that are
related to the environment. A variant GNN encoder $f^{e}$ is used to learn the
variant features $h_{v}$ from $G_{v}$ and infer the label $\hat{e}$ of the
environment $e$. This approach allows joint optimization of environment
inference and graph invariant learning through a learning process.

##  4 Methods

Figure 2: Our Framework consists of (a) Hierarchical Stochastic Subgraph
Generation in Sec. 4.1, (b) Hierarchical Semantic Environments in Sec. 4.2,
(c) Robust GIL with Hierarchical Semantic Environments in Sec. 4.3.

Our method consists of three components: hierarchical stochastic subgraph
generation Sec. 4.1, hierarchical semantic environment inference Sec. 4.2, and
robust graph invariant learning (GIL) with inferred environments Sec. 4.3. In
Sec. 4.1, we generate invariant and variant subgraphs hierarchically for
hierarchical semantic environment inference. In Sec. 4.2, we learn
hierarchical semantic environments using the proposed hierarchical loss.
Finally, in Sec. 4.3, we leverage the inferred hierarchical semantic
environments to facilitate robust graph invariant learning, enabling us to
uncover the invariant relationships between input graphs and their
corresponding labels. These three steps together form a comprehensive approach
for enhanced graph analysis and understanding.

###  4.1 Hierarchical Stochastic Subgraph Generation

Given a graph $G=(V,E)$, where $V$ represents the set of nodes and $E$
represents the set of edges, the goal is to generate the invariant subgraph
$G^{k}_{v}$ and variant subgraph $G^{k}_{inv}$ from the original graph $G$ at
each hierarchy $k\in K$, where $K$ denotes the number of environmental
hierarchies. At each hierarchy $k$, we employ a graph neural network denoted
as $\text{GNN}_{k}$ to update the hidden embedding
$h^{k}_{i}=\text{GNN}_{k}(h^{k-1}_{i},A)$ for each node $v_{i}\in V$. We use
graph neural networks as $\text{GNN}_{k}$ [42] to aggregate information from
neighboring nodes for node embedding updating. In the process of learning
variant and invariant subgraphs at each hierarchy, we define a probability
distribution function $S^{k}(\cdot)$ for edge selection as follows. We omit
the hierarchy notations, while the computation of probability distribution is
conducted within each hierarchy:

$s_{ij}=S(h_{ij})=\sigma(\text{MLP}([h_{i},h_{j}]))$ (2)

Here, $s_{ij}$ is the probability of selecting edge $e_{ij}$, $\sigma$ is the
sigmoid function, and $\text{MLP}(h_{i},h_{j})$ is a multi-layer perceptron
(MLP) that takes node embeddings $v_{i}$ and $v_{j}$ as inputs.

To introduce stochasticity in edge selection, we generate a sampler
$\mathbf{p}_{ij}\in\\{0,1\\}$ from the Bernoulli distribution
$\mathbf{p}_{ij}\sim\\{\pi_{1}:=s_{ij},\pi_{0}:=1-s_{ij}\\}$. Gumbel-Softmax
[19] is then applied to obtain differentiable edge selection probabilities for
each edge:

$\hat{\mathbf{p}}_{ij}=\frac{\exp\left((\log{\pi_{1}}+\mathbf{g}_{1})/\tau\right)}{\sum_{i\in\\{0,1\\}}{\exp\left((\log{\pi_{i}}+\mathbf{g}_{i})/\tau\right)}}$
(3)

Here, $\mathbf{g}_{1}$ and $\mathbf{g}_{0}$ are i.i.d variables sampled from
the Gumbel distribution, and $\tau$ is a temperature parameter that controls
the smoothness of the sampling process. A lower $\tau$ leads to more
categorical (hard) selections, while a higher $\tau$ results in smoother
(soft) probabilities. As $\tau$ approaches 0, $\hat{\mathbf{p}}_{ij}$ can be
annealed to a categorical distribution.

To ensure hierarchical neighborhood consistency [38] in the generation of
stochastic subgraphs, we introduce an additional hierarchical neighbor masking
function denoted as $N^{k}$. This masking function compares the neighbor
matrix from the previous hierarchy ($k-1$) with the currently selected
neighbors, ensuring that nodes within the hierarchical neighborhood are
consistently either included or excluded. The hierarchical neighbor mask
$N^{k}_{ij}$ can be defined as:

$N^{k}_{ij}=N^{k-1}_{ij}+\boldsymbol{1}\\{N^{k-1}_{ij}=0\text{ and
}\mathbf{p}^{k}_{ij}>T\\}$ (4)

where $T$ denotes the threshold for selecting the edges based on
$\mathbf{p}^{(k)}_{ij}$, and $\boldsymbol{1}$ denotes indicator function. Then
we obtain the variant and invariant subgraph at each hierarchy $k$ as:

$A^{k}_{v}\leftarrow A\odot N^{k},A^{k}_{inv}\leftarrow A-A^{k}_{v}$ (5)

where the adjacency matrix of the variant subgraph, denoted as $A_{v}^{k}$, is
derived by performing element-wise multiplication between the original edge
matrix $A$ and the hierarchical neighbor mask matrix $N^{k}$. Conversely, the
adjacency matrix of the invariant subgraph, denoted as $A_{inv}^{k}$, is
obtained by subtracting the edges included in the variant subgraph $A_{v}^{k}$
from the original edge matrix $A$ [27, 5, 7]. By following this process
iteratively across hierarchies, we construct a series of variant and invariant
subgraphs that capture the hierarchical and structural nuances within the
original graph, enabling a multi-level analysis of downstream tasks and
analysis.

###  4.2 Hierarchical Semantic Environments

Since the existing environmental information is not reliable or available, we
design an environment inference model to assign the graphs to relatively
reliable environments. Given prior $p(e|G)$, we maximize the log-likelihood of
$p(y|G)$ to obtain the posterior $p(e|G,y)$. As there is no solution to the
true posterior, existing methods use clustering-based or variational
distribution to approximate and infer the posterior. Inspired by a multi-class
classification problem, we estimate the posterior using the softmax function
as:

$P(\hat{e}\in\mathcal{E}_{tr}|X,Y)=\frac{\exp{(-l(f^{e}(\Phi(X),Y)))}}{\sum_{e^{\prime}\in\texttt{supp}(\mathcal{E}_{tr})}{\exp(-l({f^{e^{\prime}}(\Phi(X),Y)))}}}$

where $f^{e}$ denotes neural network with $y^{e}$-class classifier and $\Phi$
indicates the variant subgraph generator.

####  4.2.1 Intra-Hierarchy Environment Diversification

In the previous section, we discussed the extraction of variant subgraphs
$\\{G^{k}_{v}|k\in K\\}$ utilizing GNN encoders in conjunction with
corresponding stochastic masking $\\{p_{ij}^{k}|k\in K\\}$ at each hierarchy
$k$. To guide the learning process effectively, we incorporate an intra-
hierarchy environment diversification loss with variant subgraph and estimated
posteriors, denoted as $L_{ED}$:

$L^{k}_{ED}=-\sum_{i}\max_{e^{k}}\log{P}(e^{k}|f^{k}({G^{k}_{v}}_{i},y_{i}))$
(6)

Specifically, $L_{ED}$ is designed to guide the network $f^{k}$ to maximize
the dependency between $e^{k}$ and $Y$ given the variant subgraph $G^{k}_{v}$
at each hierarchy $k$. By optimizing this loss, the network $f^{k}$ is ensured
to distinguish relationships between the generated environments, fostering
diversity within hierarchies.

####  4.2.2 Inter-Hierarchy Environment Augmentation

The goal of inter-hierarchy environment augmentation is to maximize the intra-
class mutual information of the estimated invariant subgraphs while maximizing
the intra-environment mutual information of the estimated variant subgraphs.
Before introducing our objective function, a standard InfoNCE loss [37] can
maximize the similarity between positive pairs and minimize the similarities
between some randomly sampled negative pairs, which is defined as:

$\mathcal{L}_{\text{InfoNCE}}(z,\mathcal{N}_{p},\tau)=-\log\left(\frac{\exp(z\cdot\mathcal{N}_{p}(z)/\tau)}{\sum_{n\in\mathcal{N}(z)}\exp(z\cdot
n/\tau)}\right)$

where $\mathcal{N}_{p}(z)$ denotes positive samples for $z$, and
$\mathcal{N}(z)$ denotes the batch samples for $z$, and $\tau$ denotes a
temperature parameter. To ensure the reliability and diversification of
learned hierarchical environments, we introduce two inter-hierarchy semantic
invariants: label-invariant and neighborhood-invariant. Inspired by [5, 7], we
introduce two contrastive learning losses: (1) Environment-based Contrastive
learning loss $\mathcal{L}_{\text{EnvCon}}$ and (2) Label-based Contrastive
learning loss $\mathcal{L}_{\text{LabelCon}}$. For environment-based
contrastive learning, we define an environment-based neighborhood function
$\mathcal{N}^{k}_{p_{e}}$ that constructs the positive set for the anchor
sample, where samples have the same inferred environment. To preserve
neighborhood consistency across hierarchies, the hierarchical neighborhood
function $\mathcal{N}^{k}_{p_{e}}$ can be defined as :

$\mathcal{N}^{k}_{p_{e}}(z)=\mathcal{N}^{k-1}_{p_{e}}(z)\cup\\{z_{i}\mid
e^{k}_{i}=e^{k}_{z},z_{i}\in\mathcal{N}(z)\\}$ (7)

Using $\mathcal{N}^{k}_{p_{e}}$, we formulate the environment consistency loss
$\mathcal{L}_{\text{EnvCon}}$ as follows:

$\mathcal{L}^{k}_{\text{EnvCon}}=\mathbb{E}_{G^{k}_{v}\sim{p_{d}}}\left[\mathcal{L}_{\text{InfoNCE}}(z^{k}_{v},\mathcal{N}^{k}_{p_{e}}(z^{k}_{v}),\tau)\right]$
(8)

where $p_{d}$ denotes the batch of data, and $z^{k}_{v}$ denotes the embedding
of variant subgraph $G^{k}_{v}$ from a multi-layer perception MLP at $k$-th
hierarchy. Environment consistency loss $\mathcal{L}^{k}_{\text{EnvCon}}$
attracts the variant subgraphs with the same environment labels and pushes the
variant subgraphs with different environments, which can augment the power of
learning the sufficiency of the environment.

For label-based contrastive loss, we define a label-based neighborhood
function that represents the positive samples given ground truth label $y$,
which can be defined as $\mathcal{N}_{p_{y}}(z)=\\{z_{i}\mid
y_{i}=y_{z},z_{i}\in\mathcal{N}(z)\\}$. To measure the label consistency loss,
we formulate $\mathcal{L}_{\text{LabelCon}}$ as follows:

$\mathcal{L}^{k}_{\text{LabelCon}}=\mathbb{E}_{G^{k}_{\text{inv}}\sim{p_{d}}}\left[\mathcal{L}_{\text{InfoNCE}}(z^{k}_{\text{inv}},\mathcal{N}_{p_{y}}(z^{k}_{\text{inv}}),\tau)\right]$
(9)

where $z_{\text{inv}}$ denotes the embedding of invariant subgraph
$G^{k}_{\text{inv}}$ from MLP at $k$-th hierarchy.

####  4.2.3 Overall Objective Function

Considering the diversity of intra-hierarchy and consistency of inter-
hierarchy, we define a total loss of each hierarchy $k$ as follows:

$\mathcal{L}_{\text{hier}_{k}}=\mathcal{L}^{k}_{\text{ED}}+\alpha\cdot\mathcal{L}^{k}_{\text{EnvCon}}+\beta\cdot\mathcal{L}^{k}_{\text{LabelCon}}$
(10)

where $\alpha$ and $\beta$ denote coefficient parameters. And the overall
objective loss can be defined as
$\mathcal{L}_{\text{HEI}}=\sum_{k=1}^{K}\mathcal{L}_{\text{hier}_{k}}$ This
formulation ensures a unified optimization objective, allowing the model to
learn invariant subgraphs that capture both label and environment information
effectively.

###  4.3 Robust GIL with Hierarchical Semantic Environments

After the hierarchical learning stage of environment generation, we extract
the generated environment $e^{K-1}$ from the last hierarchy $K-1$. Our
objective is to minimize the invariant risk given by Eq. 1, employing the
hierarchically generated semantic environments $e^{K-1}$. The objective
function to calculate $\mathcal{L}_{inv}$ is as follows:

$\min_{f}\mathcal{L}_{cls}(f)+\nabla_{\hat{e}}(\mathcal{L}_{cls}(f))\text{
s.t. }\hat{e}=\arg\min_{e}\mathcal{L}_{\text{HEI}}$ (11)

This equation represents an optimization problem where the goal is to find a
function $f$ that minimizes the sum of two terms. The first term
$\mathcal{L}_{cls}(f)$ corresponds to the cross entropy loss of the
classification task. The second term $\nabla_{\hat{e}}(\mathcal{L}_{cls}(f))$
represents the gradient with respect to $\hat{e}$, which is a specific
environment. In essence, the objective is to discover a function $f$ that
performs well on a classification task while maintaining stability in
performance across different environments, adhering to the specified
environmental minimization condition.

methods | ic50-assay | ic50-sca | ic50-size | ec50-assay | ec50-sca | ec50-size |
---|---|---|---|---|---|---|---
#env | 311 | 6881 | 190 | 47 | 850 | 167 |
erm[40] | 71.79$\pm$0.27 | 68.85$\pm$0.62 | 66.70$\pm$1.08 | 76.42$\pm$1.59 | 64.56$\pm$1.25 | 62.79$\pm$1.15 |
irm[2] | 72.12$\pm$0.49 | 68.69$\pm$0.65 | 66.54$\pm$0.42 | 76.51$\pm$1.89 | 64.82$\pm$0.55 | 63.23$\pm$0.56 |
v-rex[26] | 72.05$\pm$1.25 | 68.92$\pm$0.98 | 66.33$\pm$0.74 | 76.73$\pm$2.26 | 62.83$\pm$1.20 | 59.27$\pm$1.65 |
eiil[8] | 72.60$\pm$0.47 | 68.45$\pm$0.53 | 66.38$\pm$0.66 | 76.96$\pm$0.25 | 64.95$\pm$1.12 | 62.65$\pm$1.88 |
ib-irm[1] | 72.50$\pm$0.49 | 68.50$\pm$0.40 | 66.64$\pm$0.28 | 76.72$\pm$0.98 | 64.43$\pm$0.85 | 64.10$\pm$0.61 |
grea [30] | 72.77$\pm$1.25 | 68.33$\pm$0.32 | 66.16$\pm$0.46 | 72.44$\pm$2.55 | 67.98$\pm$1.00 | 63.93$\pm$3.01 |
cigav1 [6] | 72.71$\pm$0.52 | 69.04$\pm$0.86 | 67.24$\pm$0.88 | 78.46$\pm$0.45 | 66.05$\pm$1.29 | 66.01$\pm$0.84 |
cigav2 [6] | 73.17$\pm$0.39 | 69.70$\pm$0.27 | 67.78$\pm$0.76 | - | - | - |
moleood [43] | 71.38$\pm$0.68 | 68.02$\pm$0.55 | 66.51$\pm$0.55 | 73.25$\pm$1.24 | 66.69$\pm$0.34 | 65.09$\pm$0.90 |
gala [7] | - | - | - | 79.24$\pm$1.36 | 66.00$\pm$1.86 | 66.01$\pm$0.84 |
ours | 74.01$\pm$0.11 | 70.72$\pm$0.30 | 68.64$\pm$0.23 | 80.82$\pm$0.21 | 69.73$\pm$0.21 | 66.87$\pm$0.38 |
Table 1: Test ROC-AUC of various models on DrugOOD benchmark datasets. The
mean $\pm$ standard deviation of all models is reported as an average of 5
executions of each model. The best methods are highlighted in bold and the
second best methods are underlined.

##  5 Experiments

##### Datasets.

Extensive experiments were conducted on real-world graph data from diverse
domains.

  * •

CMNIST-75sp. The task is to classify each graph that is converted from an
image in the ColoredMNIST dataset [2] into the corresponding handwritten digit
using the superpixels algorithm [25]. Distribution shifts exist on node
attributes by adding random noises in the testing set.

  * •

Graph-SST datasets. We utilize sentiment graph data from SST5 and SST-Twitter
datasets from [46]. For the Graph-SST5 dataset, graphs are split into
different sets based on averaged node degrees to create distribution shifts.
For the Graph-Twitter dataset, we invert the split order to assess the out-of-
distribution generalization capability of GNNs trained on large-degree graphs
to smaller ones.

  * •

DrugOOD. We use six datasets in DrugOOD [20] that are provided with manually
specified environment labels. DrugOOD provides more diverse splitting
indicators, including assay, scaffold, and size. To comprehensively evaluate
the performance of our method under different environment definitions, we
adopt these three different splitting schemes on categories IC50 and EC50
provided in DrugOOD. Then we obtain six datasets, EC50-* and IC50-*, where the
suffix * specifies the splitting scheme i.e. IC50/EC50-assay/scaffold/size.

##### Baselines.

We comprehensively compare our methods with the following categories of
baselines: (1) ERM denotes supervised learning with empirical risk
minimization [40]. (2) Euclidean OOD methods: We compare with SOTA invariant
learning methods from the Euclidean regime, including IRM [2], V-REX [26],
EIIL [8], IB-IRM [1]. (3) Graph OOD methods: We also compare with SOTA
invariant learning methods from the graph regime. Graph OOD methods can be
further split into three groups: environment generation-based baseline methods
including GREA [30] and LiSA [45], environment augmentation-based baseline
methods including DisC [10] and CIGA [6] and environment inference-based
baseline methods including GIL [27], MoleOOD [43], and GALA [7].

##### Environmental Setup.

All methods use the same GIN backbone [42] and the same optimization protocol
for fair comparisons. Each of the methods is configured using the same
parameters reported in the original paper or selected by grid search. For a
fair comparison, we use the same embedding size for all methods. We tune the
hyperparameters following the common practice. All details are given in
Appendix. We report the ROC-AUC score in the DrugOOD dataset and the accuracy
score for the rest of the datasets.

###  5.1 Performance Comparison

We first provide a detailed report on the DrugOOD benchmark dataset in Tab. 1.
We conduct IC50 and EC50 predictions with different split settings. As shown
in Tab. 1 and mentioned in Sec. 1, the DrugOOD dataset includes molecular
datasets with complex distributions, such as assay, scaffold, and size split
with various numbers of environments. Our approach consistently outperforms
existing methods, demonstrating the importance of hierarchical environment
learning in addressing complex drug OOD generalization applications. As shown
in Tab. 1, compared to Empirical Risk Minimization (ERM), Euclidean-based
invariant learning methods, such as IRM [2], V-REX [26], EIIL [8], etc, show
comparable or even degraded performance in most experimental settings. This
suggests that directly applying general invariant learning struggles to handle
shifts in graph distributions. In contrast, graph-based OOD methods exhibit
better performance, as they learn invariant subgraph patterns for graph OOD
generalization.

In addition, in the performance comparison of graph-based OOD methods, the
MoleOOD [43] pointed out that simple ERM sometimes outperforms several
existing methods when faced with a large number of provided environments in
graph OOD datasets, which is aligned with the results of Tab. 1. For example,
ic50-sca dataset is provided with 6881 environments, where 6881 scaffolds or
substructures construct a vast amount of environmental information for the
dataset. In ic50-sca dataset, GREA and MoleOOD show poor performance compared
to ERM, as shown in Tab. 1. We analyze this phenomenon for two potential
reasons: (1) The first reason is that the provided environments are overly
fine-grained, thereby similar substructure environments in the training and
testing sets. ERM can capture similar substructures to learn similar data
distributions, showing better performance. (2) The second reason is that GREA
and MoleOOD only generate a small number of environments, which overly
simplifies the environment and neglects the relationships between
environments. This limits their ability to learn heterogeneous and
interrelated environments, resulting in noticeable performance degradation.

Our method performs a hierarchical approach to learn relationships between
redundant environments and maximize the diversity of environments. Leveraging
the high-level semantic environments from the last hierarchy, we minimize
invariant prediction risk, thereby achieving better graph OOD generalization.
In Appendix, we report performance comparisons showing that our performance is
better or comparable to existing methods in general domains.

###  5.2 Effect of Hierarchical Semantic Environments

In this section, similar to existing environment inference-based works for OOD
generalization [7, 8], we first analyze the role of environment inference, and
then we discuss the effects of hierarchical environments learned by our model,
showing the necessity of learning the hierarchical environments for graph
invariant learning.

methods | ic50-sca | ic50-size
---|---|---
w/ $\text{env}_{\text{\\#non-infer(rand)=2}}$ | 68.54$\pm$0.64 | 67.63$\pm$0.33
w/ $\text{env}_{\text{\\#non-infer=real}}$ | 68.77$\pm$0.72 | 67.60$\pm$0.32
w/ $\text{env}_{\\#\text{infer}=\text{2}}$ | 69.14$\pm$0.80 | 67.55$\pm$0.34
w/ $\text{env}_{\\#\text{infer}=\text{real}}$ | 69.08$\pm$0.64 | 67.74$\pm$0.13
w/ $\text{env}_{\\#\text{hier-infer}}$ (ours) | 70.72$\pm$0.30 | 68.64$\pm$0.23
Table 2: The ablation study on ic50-sca and ic50-size datasets. Configurations | ic50-sca | ic50-size
---|---|---
#real = [#$e_{p}$] | 69.08$\pm$0.64 | 67.60$\pm$0.32
#env = [5] | 69.35$\pm$0.67 | 67.70$\pm$0.50
#env = [2] | 69.14$\pm$0.80 | 67.73$\pm$0.64
#env=[#$e_{p}$ $\to$ #$e_{p}/2$ $\to$ 5] | 70.12$\pm$0.14 | 68.62$\pm$0.34
#env=[#$e_{p}$ $\to$ #$e_{p}/2$ $\to$ 2] | 70.72$\pm$0.30 | 68.64$\pm$0.23
Table 3: Sensitivity analysis on generated environments. #$e_{p}$ denotes the
number of provided environments in datasets.

##### Environment Inference.

As shown in Tab. 2, we analyze the role of environment inference by comparing
the direct usage of non-inferred and inferred environments. Direct usage of
the non-inferred environments can be divided into two scenarios: (1) When
environments are unavailable, existing methods randomly assign two
environments for graph invariant learning, which is shown as ‘w/
$\text{env}_{\text{\\#non-infer (rand)}}$’ from Tab. 2. (2) DrugOOD dataset
provides environmental information, e.g., scaffold information in ic50-sca
dataset. As shown in Tab. 1, we can know the number of environments for both
dataset ic50-sca and dataset ic50-size. In such cases, we use environments for
invariant learning, as illustrated by the ‘w/ $\text{env}_{\text{\\#non-infer
(real)}}$’ in Tab. 2.

Secondly, we consider the usage of inferred environments for invariant
learning, as indicated in Tab. 2. Through the model of environment inference,
we directly predict the flat environment and use it in invariant learning, as
illustrated by ‘w/ $\text{env}_{\\#\text{infer=*}}$’. From Tab. 2, it can be
observed that the model with inferred environments exhibits better
performance, indicating that learned environments from data are more effective
than direct usage of non-inferred environments. Moreover, in dataset ic50-sca,
the effects of inference are more pronounced, emphasizing the necessity of
performing environment inference operations in situations with complex real-
world environments.

##### Hierarchical Environment Inference.

We further analyze the effects of hierarchical environments by comparing them
with flat environments. Our model adopts a hierarchical approach to learning
environmental information, regulating semantic environmental content between
hierarchies through both intra-hierarchy and inter-hierarchy mechanisms. Upon
comparing experimental results in ic50-sca and ic50-size datasets, as shown in
Tab. 2, our hierarchical model consistently outperforms invariant learning
models with flat environments. This indicates that in real-world datasets, the
environmental factors influencing label predictions are complex and
interdependent. Consequently, investigating graph invariant learning with
hierarchical environment inference is essential for attaining more
sophisticated and effective in graph OOD generalization.

###  5.3 Sensitive analysis on hierarchy

We also conduct sensitive analysis on the number of hierarchies and the number
of environments at each hierarchy. Specifically, in hierarchical settings, we
start from the number of environments as the number of environments provided
in the first hierarchy for a fair comparison. By comparing the results shown
in Tab. 3, we find inferred environments with hierarchical settings help OOD
generalization in the graph domain. Inferring flat environments is worse than
environment inference with three hierarchies. In addition, inferring a large
number of environments $\text{\\#}e_{p}$ in a single hierarchy shows poor
performance compared to inferring a small number of environments. This is
because the model with a large number of environments learns too local and
redundant to capture the relationships between environments. However, in a
hierarchical setting, even if inference starts from a large number of
environments to a small number of environments, different hierarchies can
capture the local relationships among a huge number of environments and
dynamically maximize the diversity of inferred high-level environments.
Therefore, despite the final number of inferred environments is still small,
these environments learn the semantic information from complex data
distribution hierarchically and can improve the Graph OOD generalization using
the high-level semantic environments.

###  5.4 Discussions

We discuss the diversity of environments generated by different methods as
shown in Fig. 3. we compare two environment assignment methods with our
method, which are random sampling-based method rand#2, and flat environment
inference-based method infer#2, respectively. For a fair comparison, we set
two environments for both methods and set the hierarchical environment as
$[6881\to 3440\to 2]$ for our method using the ic50-sca dataset from DrugOOD.

As mentioned in [8], the diversity of environments is important to obtain
effective IRM models, and large discrepancy of spurious correlations between
environments benefits the IRM model. We measure the diversity of environments
by calculating the distance of distributions among the acquired environments
for each method. More specifically, we measure the Kolomogorov-Smirnov (K-S)
statistic [34] as a distance between the two inferred scaffold distributions
for each method, respectively. The K-S test can capture the dissimilarity
between cumulative distribution functions, making it suitable for our
analysis. As shown in Fig. 3, the two environments generated by rand#2 methods
are almost the same since they assign the sample to two environments using
random sampling, showing the lowest diversity of environments. infer#2 shows
better diversity of generated environment distributions with significant
p-values ($p=1.86e-26$). Our method generates more significantly diverse
environment distributions ($p=4.52e-73$) compared to other methods, surpassing
them in inter-env-distance by a large margin. Moreover, the results are
aligned with Fig. 1, our methods outperform other methods in ic50-sca dataset,
indicating that diverse environments can give a clear indication of
distribution shifts, therefore IRM can easily identify and eliminate variant
features.

Figure 3: Discussions on the diversity of generated environments. We show
distributions of two generated environments $env_{0}$ and $env_{1}$ for (a)
random sampling methods, (b) flat environment inference methods, and (c) our
hierarchical environment inference methods. (d) We employ the Kolmogorov-
Smirnov test [34] to calculate the diversity of three methods.

##  6 Conclusion

In our research, we take on the formidable challenge of improving out-of-
distribution (OOD) generalization in the graph domain. We observe that recent
works often overlook the crucial aspect of studying hierarchical environments
in graph invariant learning. To address this gap, we introduce a novel method
that generates hierarchical dynamic environments for each graph. Our approach
involves hierarchical stochastic subgraph generation, hierarchical environment
inference, and a carefully designed learning objective. By incorporating graph
invariant learning with inferred high-level environments, our model not only
achieves meaningful and diverse environments within the same hierarchy but
also ensures consistency across different hierarchies. The effectiveness of
our method is particularly pronounced in the DrugOOD dataset, shedding light
on the potential for further exploration in hierarchical graph learning within
OOD scenarios.

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##  7 Implementation Details

Our proposed method contains three components: hierarchical stochastic
subgraph generation (Sec. 7.1), hierarchical semantic environments (Sec. 7.2),
and graph invariant learning. In this section, we extend the details of our
architectures. We provide a detailed description of our model in Algorithm 1
at the end of this section.

###  7.1 Hierarchical Stochastic Subgraph Generation

We employ graph isomorphism networks (GINs) [42] for graph encoding, utilizing
different GINs for different hierarchies. A two-layer multilayer perceptron
(MLP) with ReLU activation functions serves as the scoring function for edge
selection in each hierarchy, with different weight parameters. A sampler is
generated from a Bernoulli distribution based on the computed edge scores.
Edge selection uses the Gumbel-Softmax function with a threshold $T$ and
temperature $\tau$. During training, we set the temperature at 0.05 for edge
selection and we conduct a grid search for the threshold. Variant subgraphs
are extracted by the selected edges at each hierarchy, while invariant
subgraphs are obtained by directly subtracting the selected edges from the
original input graph. Notably, we retain the selected edges from previous
hierarchies for stable training.

###  7.2 Hierarchical Semantic Environments

####  7.2.1 Intra-Hierarchy Environment Diversification

To predict environment labels in each hierarchy, we employ a simple neural
network, denoted as $f^{e}$, for multi-classification, drawing inspiration
from the work [17]. Following Eq. 6, the input of the neural network $f^{e}$
is formed by concatenating the hidden embedding of the variant subgraph and
the one-hot encoding of the label. The output dimension of the neural network
$f^{e}$ corresponds to the number of environment labels at each hierarchy.
During inference, predicted environment labels at each hierarchy are obtained
by the neural network’s output.

Figure 4: Illustration of objective $\mathcal{L_{\text{EnvCon}}}$ in Inter-
Hierarchy Environment Augmentation. (a) We pull environment-based
neighborhoods $\mathcal{N}^{k}_{p_{e}}(z^{k}_{v})$ and
$\mathcal{N}^{k-1}_{p_{e}}(z^{k}_{v})$ toward anchor variant subgraph
embedding $z^{k}_{v}$. (b) We show a simple illustration of anchor variant
subgraph $z^{k-1}_{v}$ in the previous hierarchy $k-1$. (c) Notations of the
illustration figure.

####  7.2.2 Inter-Hierarchy Environment Augmentation

As emphasized by Huang [17], multi-class classification often outperforms
clustering-based methods. However, in hierarchical settings, we encounter
challenges in consistently assigning samples and their neighbors from previous
hierarchies to the same class. To address this issue, we propose a contrastive
objective that enhances environmental semantics in the latent embedding space
by preserving neighborhood consistencies across hierarchies. Taking
inspiration from prior studies [7, 6], we employ contrastive learning on two
dimensions: one based on the environment and the other on labels. In Fig. 4,
we illustrate the objective $\mathcal{L}_{\text{EnvCon}}$. For each anchor
variant subgraph, positive neighborhoods encompass graphs with the same
predicted environmental labels as the anchor graphs, including neighborhoods
from the previous hierarchy. The objective $\mathcal{L}_{\text{LabelCon}}$ is
computed similarly to $\mathcal{L}_{\text{EnvCon}}$, but the label-based
neighborhoods of anchor subgraphs consistently remain the same. Notably, to
distinguish embeddings between variant and invariant subgraphs, we employ two
MLP layers with distinct parameters to embed each subgraph.

Data: Dataset $\mathcal{G}^{train}=\\{(G_{i},y_{i})\\}_{i=1}^{N_{train}}$,
hierarchical subgraph generation module $s$, hierarchical environment
inference module $f$, graph invariant learning module $F$. Number of
hierarchies $K$, number of training epochs for hierarchical environment
inference module $E_{1}$, number of training epochs for the graph invariant
learning $E_{2}$, Batch size $B$.

Result: Final estimated model $F$

1 Initialize the models $s,f,F$;

2 for  _${p_{1}}\leftarrow 1$ to $E_{1}$_ do

3 Sample a batch of data $\mathcal{B}$ from $\mathcal{G}_{\text{train}}$ with
batch size $B$;

4 for  _$G_{j}$ , $y_{j}\in\mathcal{B}$_ do

5 for  _$k\leftarrow 1$ to $K$_ do

6 Obtain environment predictions $e^{k}_{j}$ from $f^{k}$; Find environment-
based neighbors $\mathcal{N}^{k}_{pe}(G_{j})$ with the same $e^{k}_{j}$ in $k$
and $k-1$-th layer;

7 Find label-based neighbors $\mathcal{N}_{py}(G_{j})$ with the same $y_{j}$;

8 Compute loss $\mathcal{L}^{k}_{\text{hier}}(s^{k},f^{k};G_{j})$ via Eq. 10;

9

10 Compute loss $\mathcal{L}_{\text{HEI}}$ across all $K$ hierarchies;

11 Back propagate and optimize $s,f$;

12

13

14Freeze the parameters of module $s,f$;

15 for  _${p_{2}}\leftarrow 1$ to $E_{2}$_ do

16 Sample a batch of data $B$ from $\mathcal{G}_{\text{train}}$ with batch
size $B$;

17 for  _$G_{j}$ , $y_{j}\in\mathcal{B}$_ do

18 Determine the environment of each sample $(G,y)$ in $G_{j}$ by
$\text{argmax}_{\hat{e}}f(\hat{e}|s(G),y)$ in last hierarchy $K-1$;

19 Compute loss $\mathcal{L}_{\text{inv}}(F;G_{j},\hat{e})$ according to Eq.
11;

20 Backpropagate and optimize $F$;

21

22

23Return the final estimated model $F$;

Algorithm 1 The training procedure. Table 4: Statistics of the datasets used in experiments. The number of nodes is averaged numbers among all datasets. Datasets | #Training | #Validation | #Testing | #Labels | #Envs | #Nodes | #Metrics
---|---|---|---|---|---|---|---
cminst-sp | 40,000 | 5,000 | 15,000 | 2 | N/A | 56.90 | ACC
graph-sst5 | 6,090 | 1,186 | 2,240 | 5 | N/A | 19.85 | ACC
graph-twitter | 3,238 | 694 | 1,509 | 3 | N/A | 21.10 | ACC
ic50-assay | 34,179 | 19,028 | 19,032 | 2 | 311 | 34.58 | ROC-AUC
ic50-sca | 21,519 | 19,041 | 19,048 | 2 | 6,881 | 39.38 | ROC-AUC
ic50-size | 36,597 | 17,660 | 16,415 | 2 | 190 | 37.99 | ROC-AUC
ec50-assay | 4,540 | 2,572 | 2,490 | 2 | 47 | 39.81 | ROC-AUC
ec50-sca | 2,570 | 2,532 | 2,533 | 2 | 850 | 56.84 | ROC-AUC
ec50-size | 4,684 | 2,313 | 2,398 | 2 | 167 | 48.40 | ROC-AUC

##  8 Data Statistics

##### DrugOOD datasets.

We present the data statistics for the DrugOOD dataset in Tab. 4. To evaluate
the performance of our model in realistic scenarios involving distribution
shifts, we have integrated three datasets from the DrugOOD benchmark [20].
This benchmark offers a comprehensive out-of-distribution (OOD) evaluation in
AI-driven drug discovery, specifically focusing on predicting drug-target
binding affinity for both protein targets and drug compounds. The molecular
data and annotations are precisely curated from the ChEMBL 29 database [35],
capturing various distribution shifts across various assays, scaffolds, and
molecule sizes. We specifically select datasets with three different splits,
assay, scaffold, and size, from the ligand-based affinity prediction task.
This task involves IC50 and EC50 measurements and encompasses annotation noise
at the core level.

##### General datasets.

In CMNIST-sp dataset, digits 0-4 are assigned to $y=0$, and digits 5-9 are
assigned to $y=1$. Following previous studies, the label $y$ will be flipped
with a probability of $0.25$. In addition, green and red colors will be
assigned to images with labels $0$ and $1$ on an average probability of $0.15$
for the training data, respectively. In Graph-SST datasets, the node features
are generated using BERT [22] and the edges are parsed by a Biaffine parser
[11]. We follow previous works [7, 6] and split the datasets according to the
averaged degrees of each graph. In the Graph-SST5 dataset, we partition graphs
into the training and test sets using a sorting approach that arranges them
from small to large degrees. In contrast, in the Twitter dataset, we adopt an
inverse ordering strategy. This means that during training, the model is
exposed to graphs with larger degrees, allowing us to assess its
generalization performance on graphs with smaller degrees during evaluation.

##  9 Baselines and Experimental Settings

##### GNN encoder.

For a fair comparison, we employ the same graph neural network (GNN)
architecture in all methods. Consistent with prior studies, we utilize a
multi-layer graph isomorphism network (GIN) [42] with Batch Normalization [18]
between layers and jumping knowledge (JK) residual connections at the last
layer. The hidden dimension of the GNNs is set to 128, and we explore the
number of GNN layers within the range of $\\{3,4,5\\}$.

##### Optimization and model selection.

We utilize the Adam optimizer [23] and perform a grid search for learning
rates in
$\\{1\mathrm{e}{-}3,1\mathrm{e}{-}4,4\mathrm{e}{-}5,1\mathrm{e}{-}5\\}$, as
well as batch sizes in $\\{32,64,128\\}$ across all models and datasets. Early
stopping based on validation performance is implemented with a patience of
$20$ epochs. A default dropout value of 0.5 is used for all datasets. The
final model is selected based on its performance on the validation set. All
experiments are conducted with five different random seeds (1, 2, 3, 4, 5),
and the results are reported as the mean and standard deviation from these
five runs.

##### Euclidean OOD methods.

Building on the implementation of Euclidean OOD methods (IRM [2], V-Rex [26],
IB-IRM [1], and EIIL [8]) in the previous study [7], we adopt the same
implementation setup to reproduce the results of these methods. As
environmental information is not provided, we follow the methodology of
previous works [7, 6] and randomly assign training data to two environments.
The weights for the regularization term are selected from
$\\{1.0,0.01,0.001\\}$ to calculate the IRM loss. Since the obtained results
closely align with those reported in [7], we directly use the reported results
from [7] for comparison.

methods | CMNIST-sp | Graph-SST5 | Twitter
---|---|---|---
erm | 21.56$\pm$5.38 | 42.62$\pm$2.54 | 59.34$\pm$1.13
grea | 18.64$\pm$6.44 | 43.29$\pm$0.85 | 59.92$\pm$1.48
disc | 54.07$\pm$15.3 | 40.67$\pm$1.19 | 57.89$\pm$2.02
cigav1 | 23.66$\pm$8.65 | 44.71$\pm$1.14 | 63.66$\pm$0.84
cigav2 | 44.91$\pm$4.31 | 45.25$\pm$1.27 | 64.45$\pm$1.99
gil | 18.04$\pm$4.39 | 43.30$\pm$1.24 | 61.78$\pm$1.66
moleood | 39.55$\pm$4.35 | 40.36$\pm$1.85 | 59.26$\pm$1.67
gala | 59.16$\pm$3.64 | 44.88$\pm$1.02 | 62.45$\pm$0.62
ours | 46.37$\pm$4.33 | 46.83$\pm$0.54 | 64.51$\pm$0.10
Table 5: Test accuracies of various models on CMNIST-sp, Graph-SST5, and
Twitter benchmark datasets. The mean $\pm$ standard deviation of all models is
reported as an average of 5 executions of each model. The best performance in
each dataset is highlighted in bolded and the second best methods are
underlined.

##### Graph OOD methods.

We reproduce the experiments for GREA, CIGAv1, CIGAv2, MoleOOD, and GALA using
the provided codes, if available. Although the previous work GALA [7]
implemented these baselines, our reproduced results differ. We adhere to the
author-recommended hyperparameters for training the baselines, and
consequently, some of the baselines exhibit superior results compared to the
reported GALA results. For the remaining baselines, including LiSA [45], DisC
[10], and GIL [27], we observe that these models struggle to fit the DrugOOD
dataset with the hyperparameters provided, as indicated in Tab. 6.

##### Our methods.

As previously mentioned, we employ the same GNN architecture to obtain graph
embeddings in our methods. In stochastic subgraph generation, we explore
threshold values in $\\{0.5,0.6,0.8\\}$ for different datasets. To address
potential over-smoothing issues in GNNs, we set the number of GNN layers in
the environment inference stage (first stage) to 1 and search the number of
GNN layers in graph invariant learning (second stage) in $\\{3,4,5\\}$. For
hierarchical settings, we search for the optimal number of hierarchy layers
(i.e., $K$) from $\\{1,2,3\\}$ for different datasets. As shown in Tab. 4,
some datasets, such as CMNIST-sp and Graph-SST, do not provide environmental
information. For general datasets, we set the number of environment labels in
the first hierarchy (i.e., $E_{1}$) to 100. For DrugOOD datasets, we set
$E_{1}$ as the provided number, e.g., $E_{1}$ of the ic50-sca dataset is 6881.
We search for the optimal number of environment labels at the last hierarchy
in $\\{2,5,10\\}$ for different datasets. For intermediate hierarchies, we set
the median as the number of environment labels, e.g., $K=3$ and
$(E_{1},E_{2},E_{3})=(100,50,2)$. We iteratively conduct hierarchical
environment inference to obtain more high-level semantic environments. In
graph variant learning with inferred environments, we select the weights for
the regularization term from $\\{1.0,0.01,0.001\\}$ to calculate the IRM loss.
In Fig. 5 (a), we conducted experiments for two inter-hierarchy contrastive
learning. The performance of the model shows an upward trend with varying
values of both $\alpha$ and $\beta$, demonstrating the effectiveness of
incorporating $L_{EnvCon}$ and $L_{LabelCon}$ in our model. In Fig. 5 (b), the
hierarchical inference of environments shows superiority over the flat
inference (= existing methods) and the selection of the starting number of
environments can have different performances.

Figure 5: Hyperparameter Selection of alpha, beta, #Envs, and #Hierarchies.

##  10 Additional Experiments

We report results on general datasets (CMNIST-SP, Graph-SST5, and Twitter) as
shown in Tab. 6. In particular, our methods consistently outperform baseline
approaches on Graph-SST datasets. We acknowledge that our methods did not
yield successful results on the CMNIST-SP dataset. We analyze the potential
reasons as follows: (1) Our model focuses on learning variant substructures
hierarchically, which can help infer more meaningful semantic environments for
graph invariant learning. (2) While CMNIST-SP contains shifts in node
attributes, Graph-SST encounters shifts in node degree (structure). Our method
demonstrates robustness in graphs featuring more complex distribution shifts,
particularly those influenced by structural variations, as indicated by
results on DrugOOD. Furthermore, in Tab. 7, our method outperforms baseline
performance when evaluated on three molecular property prediction datasets
from the Open Graph Benchmark under fair conditions. As shown in Tab. 8, our
model outperforms the additional three baselines on ec50-sca and molhiv
datasets.

methods | ec50-assay | ec50-sca | ec50-size
---|---|---|---
#env | 47 | 850 | 167
lisa [45] | 68.10$\pm$2.60 | 64.60$\pm$1.70 | 61.20$\pm$1.90
disc [10] | 65.40$\pm$5.34 | 54.97$\pm$3.86 | 56.97$\pm$2.56
gil [27] | 72.13$\pm$4.70 | 63.05$\pm$1.04 | 62.08$\pm$1.06
ours | 80.82$\pm$0.21 | 69.73$\pm$0.21 | 66.87$\pm$0.38
Table 6: Test ROC-AUC of rest models on DrugOOD benchmark datasets. The mean $\pm$ standard deviation of all models is reported as an average of 5 executions of each model. The best methods are highlighted in bold.  methods | molbace | molbbbp | molhiv
---|---|---|---
gin |  77.83$\pm$3.15 |  66.93$\pm$2.31 |  76.58$\pm$1.02
gin+moleood |  81.09$\pm$2.03 |  69.84$\pm$1.84 |  78.31$\pm$0.24
ours | 82.26$\pm$0.88 | 71.30$\pm$0.47 | 79.65$\pm$0.14
Table 7: Molecular property prediction from Open Graph Benchmark (OGB). datasets | dir | gsat | gil | ours
---|---|---|---|---
ec50-sca |  64.45$\pm$1.69 |  66.02$\pm$1.13 |  63.05$\pm$1.04 | 69.73$\pm$0.21
molhiv |  77.05$\pm$0.57 |  76.47$\pm$1.53 |  79.08$\pm$0.54 | 79.65$\pm$0.14
Table 8: More baseline comparisons in ec50-sca (DrugOOD) and molhiv (OGB).

##  11 Qualitative Analysis of Hierarchical Semantic Environments.

To evaluate the practical efficacy of our proposed hierarchical environmental
inference model in capturing complex environmental patterns, we utilize a
visualization-based approach to compare the assignment outcomes between a flat
environment model (Fig. 7) and our model (Fig. 8) for the same set of
molecules. We specifically focus on molecules labeled 0 from the ic50-sca
dataset, where our model exhibits accurate predictions while other flat
environment-inference models (such as the Infer#2 model) fail to predict
accurately. We then visualize the allocation of these molecules in each
environment, with particular emphasis on highlighting scaffold information.

In the Infer#2 model, the necessity of directly predicting assignments to two
environments in molecules with intricate scaffold structures may result in
overfitting complex structural features. In contrast, our model addresses the
challenge of overfitting to molecular structural features. Through the
combined learning of hierarchical environmental inference and graph invariant
learning, we effectively extract spurious features, facilitating higher-level
assignments of molecules to distinct environments.

In Fig. 7, the Infer#2 model may allocate seemingly similar molecules to the
same environment, but environments inferred from our model have a higher
inter-environment diversification score, which is described in the discussion
section (Sec.5.4). This indicates that the Infer#2 model fails to capture
spurious features, instead blending causal features and resulting in
inaccurate predictions. In summary, our hierarchical environmental assignment
model generates more sophisticated semantic environments by learning
hierarchical spurious features through graph invariant learning. This approach
mitigates the overfitting challenges observed in models with flat environment
inference, leading to a more accurate delineation of meaningful environments.

Figure 6: (a) Grey: Scaffold. (b) Red: Functional group (Chlorobenzene), and
structural alert COC(=O)C from PubChem. (c, d) Blue: Learned variant subgraphs
from the first and second hierarchies (the rest of the graph is considered
learned invariant).

(a) True label: 0; Predicted label: 1; Env: 0

(b) True label: 0; Predicted label: 1; Env: 2

Figure 7: Visualizing molecules with incorrect predictions in their
environments assigned by Infer#2 model in ic50-sca dataset. The highlighted
part with red color represents the scaffold information of each molecule.

(a) True label: 0; Predicted label: 0; Env: 0

(b) True label: 0; Predicted label: 0; Env: 1

Figure 8: Visualizing molecules with incorrect predictions in their
environments assigned by our model in ic50-sca dataset. The highlighted part
with red color represents the scaffold information of each molecule.

###  11.1 Specific Case Studies on Learned Invariant Graphs.

In the DrugOOD-SCA dataset, consisting of 6,881 diverse scaffolds
(environments), we computed Morgan-fingerprint Tanimoto similarity for all
pairwise scaffolds. The results show a long-tailed distribution with a mean of
0.13 and a standard deviation of 0.04. This reveals the importance of learning
the hierarchy of complex environments, given the presence of both similar and
dissimilar scaffolds. We performed a specific case study using an active
molecule from DrugOOD. Fig. 6 shows our model capturing scaffold-like variants
hierarchically while preserving invariant substructures. The final remaining
invariant subgraphs are shown to be aligned with the functional group and
structural alert.

###  11.2 Theoretical Analysis of Our Method.

Our method enhances out-of-distribution (OOD) performance by hierarchically
stacking intra-hierarchy and inter-hierarchy losses. Specifically, grounded in
the theorem of Structrual Causal Models [2], the intra-hierarchy loss $L_{ED}$
satisfies the condition $H(Y|X_{v})-H(Y|X_{v},E_{\text{learn}})>0$ from EDNIL
[17], thereby maximizing the diversity of inferred environments at each
hierarchy. Furthermore, the $L_{\text{EnvCon}}$ and $L_{\text{LabelCon}}$ in
the inter-hierarchy adjust the boundary of the embedding space for variant and
invariant subgraphs, resulting in both theoretically and empirically diverse
and reliable environments for graph invariant learning (Fig. 5).


#  $\theta$-splitting Densities and Reflection Positivity

Jobst Ziebell  0000-0002-9715-6356
Abbe Center of Photonics, Friedrich-Schiller-University, Jena, Germany

(2nd August 2023)

###### Abstract

A simple condition is given that is sufficient to determine whether a measure
that is absolutely continuous with respect to a Gaußian measure on the space
of distributions is reflection positive. It readily generalises conventional
lattice results to an abstract setting, enabling the construction of many
reflection positive measures that are not supported on lattices.

##  1 Introduction

Reflection positivity is one of the pillars of Euclidean quantum field
theories. It is readily established for wide sets of Gaußian measures but for
non-Gaußian measures, the author feels that - with the exception of measures
supported on lattices - there is no general framework that can be easily
applied. That is fixed in this article by introducing the set of
$\theta$-splitting functions, which can work as densities to directly
generalise the lattice methods used e.g. in [GJ87]. The result is very simple:
Given a $\theta$-invariant reflection positive Gaußian measure and applying a
measurable density to it that is $\theta$-splitting, the outcome is a
reflection positive measure.

##  2 Preliminaries

A locally convex space is a real topological vector space whose topology is
induced by some family of seminorms. The dual of a locally convex space $X$
equipped with the strong dual topology will be denoted by $X^{*}_{\beta}$.
Inner products denoted with round brackets $(\cdot,\cdot)$ are taken to be
$\mathbb{R}$-bilinear. Throughout this work, $d\in\mathbb{N}$ is fixed. We
shall work on the following spaces of real test functions with their canonical
LF topologies [Tre67, p. 131-133]:

$\mathcal{D}:=\mathcal{D}(\mathbb{R}^{d+1})\qquad\text{and}\qquad\mathcal{D}_{+}:=\mathcal{D}(\mathbb{R}_{>0}\times\mathbb{R}^{d})\,.$
(1)

Let us denote the corresponding continuous restriction map by
$\pi_{+}:\mathcal{D}^{*}_{\beta}\to(\mathcal{D}_{+})^{*}_{\beta}$ (see e.g.
[Tre67, p. 245-246]). $\mathcal{D}$ and $\mathcal{D}_{+}$ as well as their
strong duals $\mathcal{D}^{*}_{\beta}$ and $(\mathcal{D}_{+})^{*}_{\beta}$ are
complete [Tre67, Theorem 13.1], barrelled [Tre67, p. 347], nuclear spaces
[Tre67, p. 530] (hence, reflexive by [SW99, p. 147]) that are also Lusin
spaces [Sch73, p. 128] and thus in particular Souslin spaces.

######  Theorem 2.1 ([Bog07, Lemma 6.4.2.(ii), Lemma 6.6.4]).

Let $X$ and $Y$ be Souslin spaces. Then, the Borel $\sigma$-algebra of
$X\times Y$ coincides with the $\sigma$-algebra generated by all products of
Borel sets in $X$ and $Y$ respectively.

In this work, a measure is taken to be a countably additive nonnegative
function on a $\sigma$-algebra. A Borel measure is thus a measure on a Borel
$\sigma$-algebra and a Radon measure is a Borel measure that is inner regular
over compact sets. A centred Gaußian measure on a locally convex space $X$ is
a Borel probability measure with the property that the pushforward measures by
elements of $X^{*}$ are centred Gaußians or the Dirac delta measure
$\delta_{0}$ at the origin. One can in general consider non-Radon Gaußian
measures on locally convex spaces. However, every Borel measure on the spaces
$\mathcal{D}^{*}_{\beta},(\mathcal{D}_{+})^{*}_{\beta}$ and countable products
thereof is automatically Radon [Bog07, Theorem 7.4.3].

A subset $A\subseteq X$ is $\bm{\mu}$-measurable with respect to a measure
$\mu$ on some $\sigma$-algebra $\mathcal{A}$ on $X$ if it is in the Lebesgue
completion $\mathcal{A}_{\mu}$ of $\mathcal{A}$ with respect to $\mu$.
Similarly, a function $f:X\to[-\infty,\infty]$ is $\bm{\mu}$-measurable if the
preimage of every Borel subset of $[-\infty,\infty]$ is in
$\mathcal{A}_{\mu}$. Likewise, $f:X\to[-\infty,\infty]$ is
$\bm{\mu}$-integrable if $f$ is $\mu$-measurable and
$\int|f|\mathrm{d}\mu<\infty$. A subset $A\subseteq X$ is
$\bm{\mu}$-negligible if it is a subset of some $B\in\mathcal{A}$ with
$\mu(B)=0$.

The pushforward of a Borel measure $\mu$ on a Hausdorff space $X$ by a
continuous function $f:X\to Y$ to a Hausdorff space $Y$ will be denoted by
$f_{*}\mu$. It is automatically a Borel measure on $Y$ and if $\mu$ is Radon,
so is $f_{*}\mu$ [Bog07, Theorem 9.1.1.(i)]. The convolution of two Borel
measures $\mu$ and $\nu$ on a Souslin locally convex space $X$ is given by
$\mu*\nu=s_{*}(\mu\times\nu)$ where $s:X\times X\to X,(x,y)\mapsto x+y$. This
is well-defined by theorem 2.1.

To every finite Borel measure $\mu$ on a locally convex space $X$ we associate
its characteristic function $\hat{\mu}:X^{*}\to\mathbb{C}$ with

$\phi\mapsto\int_{X}\exp\left[i\phi\left(x\right)\right]\mathrm{d}\mu\left(x\right)\,.$
(2)

It is well-known that two Radon measures on a locally convex space are equal
if and only if their characteristic functions are equal [Bog07, Lemma 7.13.5].
Moreover, if $\mu$ is a centred Gaußian measure on $X$, its characteristic
function is given by

$\hat{\mu}\left(\phi\right)=\exp\left[-\frac{1}{2}\left(\phi,\phi\right)_{L^{2}(\mu)}\right]$
(3)

for all $\phi\in X^{*}$ [Bog98, Theorem 2.2.4, Corollary 2.2.5].

######  Theorem 2.2.

Let $f:X\to Y$ be a continuous map from a Souslin space $X$ to a Hausdorff
space $Y$. Then, for every Borel set $B\subseteq X$, $f(B)$ is measurable by
any Radon measure on $Y$.

###### Proof.

Since every Borel subset of a Souslin space is Souslin [Sch73, p. 96 Theorem
3], this follows directly from [Bog98, Theorem A.3.15]. ∎

######  Corollary 2.3.

Let $p:X\to Y$ be a continuous map from a Souslin space $X$ to a Hausdorff
space $Y$ and $\mu$ a Radon measure on $X$. Then every function
$f:Y\to[-\infty,\infty]$ with the property that $f\circ p$ is $\mu$-measurable
is $(p_{*}\mu)$-measurable.

###### Proof.

First, note that $p(X)$ is $(p_{*}\mu)$-measurable by the preceeding theorem.
Now, letting $B\subset[-\infty,\infty]$ be a Borel set, we have

$p^{-1}\left(f^{-1}\left(B\right)\right)=A\cup N_{1}$ (4)

for some Borel subset $A\subseteq X$ and some $\mu$-negligible set
$N_{1}\subseteq X$. For brevity, let $N_{2}=Y\setminus p(X)$, which is clearly
$(p_{*}\mu)$-negligible. Then,

$\displaystyle f^{-1}\left(B\right)$
$\displaystyle=\left[f^{-1}\left(B\right)\cap
p(X)\right]\cup\left[f^{-1}\left(B\right)\cap N_{2}\right]$ (5)
$\displaystyle=p\left(p^{-1}\left(f^{-1}\left(B\right)\right)\right)\cup\left[f^{-1}\left(B\right)\cap
N_{2}\right]$ $\displaystyle=p\left(A\right)\cup
p\left(N_{1}\right)\cup\left[f^{-1}\left(B\right)\cap N_{2}\right]\,.$

$p(A)$ is $(p_{*}\mu)$-measurable by the preceeding theorem and $p(N_{1})$ as
well as $f^{-1}(B)\cap N_{2}$ are clearly $(p_{*}\mu)$-negligible. ∎

We close this section by a simple lemma on positive semidefinite matrices.

######  Lemma 2.4 ([Sch11, Satz VII]).

Let $N\in\mathbb{N}$ and $A,B$ be positive semidefinite $N\times N$ matrices
with respect to the standard inner product on $\mathbb{C}^{N}$. Then the
matrix $(A_{m,n}B_{m,n})_{m,n=1}^{N}$ given by component-wise multiplication
is positive semidefinite.

###### Proof.

Diagonalising $B$ by a unitary matrix $U$, we obtain

$B_{m,n}=\sum_{a=1}^{N}U_{m,a}^{*}\lambda_{a}U_{n,a}$ (6)

for some nonegative numbers $\lambda_{1},\dots,\lambda_{N}$. Hence, for any
$c\in\mathbb{C}^{N}$,

$\sum_{m,n,a,b=1}^{N}c_{m}^{*}A_{m,n}B_{m,n}c_{n}=\sum_{a=1}^{N}\lambda_{a}\sum_{m,n=1}^{N}\left(U_{m,a}c_{m}\right)^{*}A_{m,n}\left(U_{n,a}c_{n}\right)\geq
0\,.$ (7)

∎

##  3 Reflection Positivity

On $\mathbb{R}^{d+1}$ we define the operation of time reflection which we
shall denote by
$\theta:\mathbb{R}^{d+1}\to\mathbb{R}^{d+1},(x_{1},\dots,x_{d+1})\mapsto(-x_{1},\dots,x_{d+1})$.
By a slight abuse of notation, $\theta$ extends continuously and linearly to
$\mathcal{D}$ and $\mathcal{D}^{*}_{\beta}$ in the obvious way.

######  Definition 3.1 ([GJ87, p. 90]).

Let $\mu$ be a finite Borel measure on $\mathcal{D}^{*}_{\beta}$. Then $\mu$
is reflection positive if for every sequence $(\phi_{n})_{n\in\mathbb{N}}$ in
$\mathcal{D}_{+}$, every sequence $(c_{n})_{n\in\mathbb{N}}$ of complex
numbers and every $N\in\mathbb{N}$,

$\sum_{m,n=1}^{N}c_{m}^{*}\hat{\mu}\left(\phi_{m}-\theta\phi_{n}\right)c_{n}\geq
0\,.$ (8)

Furthermore, $\mu$ is $\bm{\theta}$-invariant if $\theta_{*}\mu=\mu$.

To begin with, let us recapitulate two of the most important (in the author’s
opinion) theorems on reflection positive measures along with their proofs.

######  Theorem 3.2 ([GJ87, Theorem 6.2.3]).

Let $\mu$ be a finite, reflection positive Borel measure on
$\mathcal{D}^{*}_{\beta}$ with the property that for every
$\phi\in\mathcal{D}_{+}$ the function
$\mathbb{R}\to\mathbb{C},t\mapsto\hat{\mu}(t\phi)$ has an analytic
continuation to some neighbourhood of zero in the complex plane. Then,
$(\phi,\theta\phi)_{L^{2}(\mu)}\geq 0$ for all $\phi\in\mathcal{D}_{+}$.

###### Proof.

For $\lambda>0$ let $\psi_{1}=\lambda\phi$, $\psi_{2}=0$, $c_{1}=\lambda^{-1}$
and $c_{2}=-\lambda^{-1}$. Since $\mu$ is reflection positive, we obtain

$\displaystyle 0$
$\displaystyle\leq\sum_{m,n=1}^{2}c_{m}^{*}\hat{\mu}\left(\psi_{m}-\theta\psi_{n}\right)c_{n}$
(9)
$\displaystyle=\frac{1}{\lambda^{2}}\int_{\mathcal{D}^{*}_{\beta}}\left(\exp\left[i\lambda
T\left(\phi-\theta\phi\right)\right]-\exp\left[-i\lambda
T\left(\phi\right)\right]-\exp\left[-i\lambda
T\left(\theta\phi\right)\right]+1\right)\mathrm{d}\mu\left(T\right)\,.$

By a classical theorem of Lukacs [LC70, p. 192], the moment-generating
functions of the pushforward measures $\phi_{*}\mu$, $(\theta\phi)_{*}\mu$ and
$(\phi-\theta\phi)_{*}\mu$ exist as integrals in some neighbourhood of zero.
Consequently, we can take $\lambda\to 0$ under the integral and obtain

$\lim_{\lambda\to
0}\sum_{m,n=1}^{2}c_{m}^{*}\hat{\mu}\left(\psi_{m}-\theta\psi_{n}\right)c_{n}=\int_{\mathcal{D}^{*}_{\beta}}T\left(\phi\right)T\left(\theta\phi\right)\mathrm{d}\mu\left(T\right)=\left<\phi,\theta\phi\right>_{L^{2}(\mu)}\geq
0\,.$ (10)

∎

######  Theorem 3.3 ([GJ87, Theorem 6.2.2]).

Let $\mu$ be a $\theta$-invariant Gaußian measure on
$\mathcal{D}^{*}_{\beta}$. Then $\mu$ is reflection positive if and only if
$(\phi,\theta\phi)_{L^{2}(\mu)}\geq 0$ for all $\phi\in\mathcal{D}_{+}$.

###### Proof.

$\Rightarrow$: This is clear by the preceeding theorem.
$\Leftarrow$: Let $(\cdot,\cdot)$ denote the inner product in $L^{2}(\mu)$ and
let $(\phi_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{D}_{+}$,
$(c_{n})_{n\in\mathbb{N}}$ a sequence of complex numbers and $N\in\mathbb{N}$.
Then, $\theta$-invariance implies

$\sum_{m,n=1}^{N}c_{m}^{*}\hat{\mu}\left(\psi_{m}-\theta\psi_{n}\right)c_{n}=\sum_{m,n=1}^{N}c_{m}^{*}\hat{\mu}\left(\phi_{m}\right)\exp\left[(\phi_{m},\theta\phi_{n})\right]\hat{\mu}\left(\phi_{n}\right)c_{n}\,.$
(11)

Since $\hat{\mu}$ is real, the statement follows if
$(\exp\left[(\phi_{m},\theta\phi_{n})\right])_{m,n=1}^{N}$ is a positive
semidefinite matrix. Since
$(\phi_{m},\theta\phi_{n})=(\theta\phi_{m},\phi_{n})$ by the
$\theta$-invariance of $\mu$, $\theta$ extends to a positive semidefinite
linear operator on the complexification of
$\mathrm{span}\\{\phi_{n}:n\in\mathbb{N}\\}$. Consequently,
$((\phi_{m},\theta\phi_{n}))_{m,n=1}^{N}$ is positive semidefinite. By
decomposing the exponential as a power series, the claim now follows from
lemma 2.4. ∎

The main theorem of this article depends on the following simple property of a
function with respect to $\theta$.

######  Definition 3.4.

A function $F:\mathcal{D}^{*}_{\beta}\to[-\infty,\infty]$ is called
$\bm{\theta}$-splitting if there exists a function
$G:(\mathcal{D}_{+})^{*}_{\beta}\to[-\infty,\infty]$ such that

$F=G\circ\pi_{+}+G\circ\pi_{+}\circ\theta\,.$ (12)

######  Theorem 3.5.

Let $\mu$ be a $\theta$-invariant reflection positive centred Gaußian measure
on $\mathcal{D}^{*}_{\beta}$. Then, for any $\mu$-measurable
$\theta$-splitting function $F:\mathcal{D}^{*}_{\beta}\to[-\infty,\infty]$
with $exp\circ F\in L^{1}(\mu)$, the finite Borel measure

$\omega=\exp\left[F\right]\cdot\mu$ (13)

is reflection positive.

###### Proof.

Define

$j:\mathcal{D}^{*}_{\beta}\to(\mathcal{D}_{+})^{*}_{\beta}\times(\mathcal{D}_{+})^{*}_{\beta}\qquad
T\mapsto\left(\pi_{+}T,\pi_{+}\theta T\right)\,.$ (14)

$j$ is clearly continuous such that the pushforward measure $j_{*}\mu$ is a
Radon measure $\nu$ on
$(\mathcal{D}_{+})^{*}_{\beta}\times(\mathcal{D}_{+})^{*}_{\beta}$. Now, let

$F_{2}:(\mathcal{D}_{+})^{*}_{\beta}\times(\mathcal{D}_{+})^{*}_{\beta}\to\mathbb{R}\qquad\left(T,K\right)\mapsto
G\left(T\right)+G\left(K\right)\,.$ (15)

Then, for every $T\in\mathcal{D}^{*}_{\beta}$,

$\left(F_{2}\circ
j\right)\left(T\right)=G\left(\pi_{+}T\right)+G\left(\pi_{+}\theta
T\right)=F\left(T\right)\,,$ (16)

such that $F_{2}$ is $\nu$-measurable by corollary 2.3. Turning to reflection
positivity, let $(\phi_{n})_{n\in\mathbb{N}}$ be a sequence in
$\mathcal{D}_{+}$ and note that

$\displaystyle\hat{\omega}\left(\phi_{m}-\theta\phi_{n}\right)$
$\displaystyle=\int_{\mathcal{D}^{*}_{\beta}}\exp\left[iT\left(\phi_{m}\right)-iT\left(\theta\phi_{n}\right)+F\left(T\right)\right]\mathrm{d}\mu\left(T\right)$
(17)
$\displaystyle=\int_{\mathcal{D}^{*}_{\beta}}\exp\left[i\;j\left(T\right)\left(\phi_{m},-\phi_{n}\right)+\left(F_{2}\circ
j\right)\left(T\right)\right]\mathrm{d}\mu\left(T\right)$
$\displaystyle=\int_{((\mathcal{D}_{+})^{*}_{\beta})^{2}}\exp\left[iT\left(\phi_{m}\right)-iK\left(\phi_{n}\right)+F_{2}\left(T,K\right)\right]\mathrm{d}\nu\left(T,K\right)$
$\displaystyle=\int_{((\mathcal{D}_{+})^{*}_{\beta})^{2}}\exp\left[iT\left(\phi_{m}\right)-iK\left(\phi_{n}\right)+G\left(T\right)+G\left(K\right)\right]\mathrm{d}\nu\left(T,K\right)\,.$

The above expression suggests to find a disintegration of $\nu$ that separates
the $T$ and $K$ variables. To that end, recall that $\mu$ is Gaußian such that
for any $\phi,\psi\in\mathcal{D}_{+}$, we have

$\hat{\nu}\left(\phi,\psi\right)=\int_{\mathcal{D}^{*}_{\beta}}\exp\left[iT\left(\phi\right)+iT\left(\theta\psi\right)\right]\mathrm{d}\mu\left(T\right)=\exp\left[-\frac{1}{2}\left\|\phi+\theta\psi\right\|_{L^{2}(\mu)}^{2}\right]\,.$
(18)

Furthermore, by theorem 3.2, Cauchy-Schwartz and the $\theta$-invariance of
$\mu$,

$0\leq\left<\phi,\theta\phi\right>_{L^{2}(\mu)}\leq\left<\phi,\phi\right>_{L^{2}(\mu)}\,.$
(19)

Moreover, since $(\mathcal{D}_{+})^{*}_{\beta}$ is a reflexive, nuclear,
barrelled space, there exist uniquely determined Radon Gaussian measures $P$
and $Q$ on $(\mathcal{D}_{+})^{*}_{\beta}$ with

$\displaystyle\hat{P}\left(\phi\right)$
$\displaystyle=\exp\left[-\frac{1}{2}\left<\phi,\phi\right>_{L^{2}(\mu)}+\frac{1}{2}\left<\phi,\theta\phi\right>_{L^{2}(\mu)}\right]\,,$
(20) $\displaystyle\hat{Q}\left(\phi\right)$
$\displaystyle=\exp\left[-\frac{1}{2}\left<\phi,\theta\phi\right>_{L^{2}(\mu)}\right]$
(21)

by Minlos theorem [Bog07, Theorem 7.13.9]. Defining the diagonal map

$\Delta:(\mathcal{D}_{+})^{*}_{\beta}\to(\mathcal{D}_{+})^{*}_{\beta}\times(\mathcal{D}_{+})^{*}_{\beta}\qquad
T\mapsto\left(T,T\right)$ (22)

it is clear that

$\hat{\nu}\left(\phi,\psi\right)=\hat{P}\left(\phi\right)\hat{P}\left(\psi\right)\hat{Q}\left(\phi+\psi\right)=\hat{P}\left(\phi\right)\hat{P}\left(\psi\right)\widehat{\Delta_{*}Q}\left(\phi,\psi\right)$
(23)

for all $\phi,\psi\in\mathcal{D}_{+}$. Equivalently, $\nu=(P\times
P)*(\Delta_{*}Q)$ by theorem 2.1. Hence, it is straightforward to verify that

$\displaystyle\hat{\omega}\left(\phi_{m}-\theta\phi_{n}\right)=\int_{((\mathcal{D}_{+})^{*}_{\beta})^{3}}\exp$
$\displaystyle\big{[}i\left(T+L\right)\left(\phi_{m}\right)-i\left(K+L\right)\left(\phi_{n}\right)$
(24)
$\displaystyle+G\left(T+L\right)+G\left(K+L\right)\big{]}\mathrm{d}\left(P\times
P\times Q\right)\left(T,K,L\right)\,.$

Now, the functions

$H_{m}\left(L\right)=\int_{(\mathcal{D}_{+})^{*}_{\beta}}\exp\left[-i\left(T+L\right)\left(\phi_{m}\right)+G\left(T+L\right)\right]\mathrm{d}P\left(T\right)$
(25)

for $m\in\mathbb{N}$ are well-defined $Q$-almost everywhere. Thus, using
Fubini, we arrive at

$\sum_{m,n=1}^{N}c_{m}^{*}\hat{\omega}\left(\phi_{m}-\theta\phi_{n}\right)c_{n}=\int_{(\mathcal{D}_{+})^{*}_{\beta}}\left|\sum_{n=1}^{N}c_{n}H_{n}\left(L\right)\right|^{2}\mathrm{d}Q\left(L\right)\geq
0$ (26)

for any $N\in\mathbb{N}$ and any sequence $(c_{n})_{n\in\mathbb{N}}$ of
complex numbers. ∎

This theorem is strikingly simple and can be applied very easily. Let us call
a locally convex space $X$ together with a continuous, linear map
$j:X\to\mathcal{D}^{*}_{\beta}$ a $\theta$-model space, if there is a
continuous, linear operator (slight abuse of terminology) $\theta:X\to X$ such
that $\theta\circ j=j\circ\theta$.

######  Example 3.6.

Examples of such $\theta$-model spaces are e.g. function spaces on
$\theta$-symmetric lattice subsets of $\mathbb{R}^{d+1}$, $\mathcal{D}$ or the
space of Schwartz functions on $\mathbb{R}^{d+1}$ together with their
respective usual injections into $\mathcal{D}^{*}_{\beta}$.

######  Remark 3.7.

The above examples cover most of what is used in literature on Euclidean
interacting quantum field theories and are also Souslin spaces.

We may now extend the definition of a $\bm{\theta}$-splitting function to
$\theta$-model spaces.

######  Definition 3.8.

A function $F:X\to[-\infty,\infty]$ on a $\theta$-model space $(X,j)$ is
called $\bm{\theta}$-splitting if there exists a function
$G:X\to[-\infty,\infty]$ such that

$F=G\circ\pi^{X}_{+}+G\circ\pi^{X}_{+}\circ\theta$ (27)

Here, $\pi^{X}_{+}:X\to X/j^{-1}(\ker\pi_{+})$ is the canonical quotient map.

######  Corollary 3.9.

Let $(X,j)$ be a Souslin $\theta$-model space. Furthermore, let $\mu$ be a
Gaußian measure on $X$ with the property that $j_{*}\mu$ is $\theta$-invariant
and reflection positive. Then, for any $\mu$-measurable $\theta$-splitting
function $F:X\to[-\infty,\infty]$ with $exp\circ F\in L^{1}(\mu)$, the finite
Borel measure

$\omega=j_{*}\left(\exp\left[F\right]\cdot\mu\right)$ (28)

is reflection positive.

###### Proof.

Let $G$ and $\pi^{X}_{+}$ be given as in definition 3.8 and define the
function $G_{2}:(\mathcal{D}_{+})^{*}_{\beta}\to[-\infty,\infty]$ given by

$T\mapsto\begin{cases}G(\pi^{X}_{+}x)&\text{if }\exists\,x\in X:T=\pi_{+}jx\\\
0&\text{else.}\end{cases}$ (29)

To see that $G_{2}$ is well-defined, note that if $\pi_{+}jx=\pi_{+}jy$ for
some $x,y\in X$, we have that there is some $T\in\ker\pi_{+}$ with $j(x-y)=T$,
i.e. $x-y\in j^{-1}(\ker\pi_{+})=\mathrm{ker}\,\pi^{X}_{+}$. Now, define the
function $F_{2}:\mathcal{D}^{*}_{\beta}\to[-\infty,\infty]$ given by

$T\mapsto G_{2}\left(\pi_{+}T\right)+G_{2}\left(\pi_{+}\theta T\right)\,.$
(30)

Clearly, $F_{2}\circ j=F$ such that $F_{2}$ is $(j_{*}\mu)$-measurable by
corollary 2.3. Consequently, $\omega=\exp[F_{2}]\cdot(j_{*}\mu)$ and theorem
3.5 applies. ∎

We finish this article by a simple example.

######  Example 3.10.

Let $\mathcal{S}$ denote the space of Schwartz functions on
$\mathbb{R}^{d+1}$. Define $j:\mathcal{S}\to\mathcal{D}^{*}_{\beta}$ by
$j(\phi)(\psi)=\int_{\mathbb{R}^{d+1}}\psi\phi$ for all $\phi\in\mathcal{S}$
and $\psi\in\mathcal{D}$. Moreover, let $\mu$ be a Gaußian measure on
$\mathcal{S}$ with the property that $j_{*}\mu$ is $\theta$-invariant and
reflection positive. Furthermore, let
$F:\mathcal{S}\to\mathbb{R},\phi\mapsto-\lambda\int_{\mathbb{R}^{d+1}}\phi^{4}$
for some $\lambda>0$. Then,

$F(\phi)=-\lambda\int_{\mathbb{R}_{>0}\times\mathbb{R}^{d}}\phi^{4}-\lambda\int_{\mathbb{R}_{>0}\times\mathbb{R}^{d}}\left(\theta\phi\right)^{4}$
(31)

provides a $\theta$-splitting of $F$.

## Acknowledgments

This work has been supported by the Deutsche Forschungsgemeinschaft (DFG)
under Grant No. 406116891 within the Research Training Group RTG 2522/1.

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