# learnable hypergraph Laplacian for hypergraph learning

###### Abstract

HyperGraph Convolutional Neural Networks (HGCNNs) have demonstrated their
potential in modeling high-order relations preserved in graph structured data.
However, most existing convolution filters are localized and determined by the
pre-defined initial hypergraph topology, neglecting to explore implicit and
long-range relations in real-world data. In this paper, we propose the _first_
learning-based method tailored for constructing adaptive hypergraph structure,
termed HypERgrAph Laplacian aDaptor (HERALD), which serves as a generic plug-
in-play module for improving the representational power of HGCNNs.
Specifically, HERALD adaptively optimizes the adjacency relationship between
hypernodes and hyperedges in an end-to-end manner and thus the task-aware
hypergraph is learned. Furthermore, HERALD employs the self-attention
mechanism to capture the non-local paired-nodes relation. Extensive
experiments on various popular hypergraph datasets for node classification and
graph classification tasks demonstrate that our approach obtains consistent
and considerable performance enhancement, proving its effectiveness and
generalization ability.

Index Terms—  hypergraph convolutional neural network, adaptive hypergraph
structure, self-attention, non-local relation

##  1 Introduction

In recent years, Graph Convolutional Neural Networks (GCNNs) develop rapidly
[1, 2], due to their ability of processing graph structured data, such as
social networks [3], citation networks [4] and biomedical networks [5, 6].
GCNNs have shown superiority on graph representation learning compared with
traditional neural networks which are used to process regular data.

Many graph neural networks have been developed to model ordinary graph whose
edge connects exactly two vertices. At the meantime, more and more researchers
noted that the data structure in real practice could be beyond pair
connections, and intuitive pairwise connections among nodes are usually
insufficient for capturing higher-order relationships.

Consequently, a new research area aiming at establishing convolutional
networks on hypergraphs attracts a surge of attention recently. Hypergraph is
a generalization of the regular graph, whose edge could join any number of
vertices, and thus possess a more powerful capability of modeling complex
relationship preserved in the real-world data [7, 8, 9]. For example, in a co-
citation relationship [10], papers act as hypernodes, and citation
relationships become hyperedges. Existing representative works in literature
include HGNN [11], HyperGCN [10], HNHN [12], which have designed basic network
architecture for hypergraph learning. However, a major concern is that these
methods are built upon an intrinsic hypergraph and not capable of dynamically
optimizing the hypergraph topology. The topology plays an essential role in
the message passing across nodes, and its quality could make a significant
impact on the performance of the trained model [13]. Another barrier that
limits the representational power of HGCNNs is that they only aggregate
message of hypernodes in a localized range, while neglecting to explore
information about long-range relations.

Several works have made their attempts to remedy these issues. DHSL [14]
proposes to use the initial raw graph to update the hypergraph structure, but
it fails to capture high-order relations among features. Also, the
optimization algorithm in DHSL could be expensive cost and unable to
incorporate with convolutional networks as well. DHGNN [15] manages to capture
local and global feature, but the adopted K-NN method [16] leads the graph
structure to a k-uniform hypergraph and lost its flexibility. AGCN [13]
proposes a spectral graph convolution network that can optimize graph adjacent
matrix during training. Unfortunately, it can not be naturally extended to
hypergraph spectral learning for which is based on the incidence matrix.

In a hypergraph, incidence matrix represents its topology by recording the
connectioins and connection strength between hypernodes and hyperedges. As an
intuition, one could parameterize such a matrix and involve it into the end-
to-end training process of the network. To this end, we propose a noval
hypergraph laplacian adaptor (HERALD), the _first_ fully learnable module
designed for adaptively optimizing the hypergraph structure. Specifically,
HERALD takes the node features and the pre-defined hypergraph laplacian as
input, and then constructs the parameterized distance matrix between nodes and
hyperedges, which empowers the automated updating of the hypergraph laplacian
and thus the topology is adapted for the down-stream task. Notably, HERALD
employs the self-attention mechanism to model the non-local paired-nodes
relation for embedding global property of the hypergraph into the learned
topology. In the experiments, to evaluate the performance of the proposed
module, we have conducted experiments for hypergraph classification tasks on
node-level and graph-level. Our main contributions are summarized as below:

  1. 1.

We propose a generic and plug-in-play module, termed HypERgrAph Laplacian
aDaptor (HERALD), for automated adapting the hypergraph topology to the
specific down-stream task. It is the first learnable module that can update
hypergraph structure dynamically.

  2. 2.

HERALD adopts the self-attention mechanism to capture global information on
the hypergraph and the parameterized distance matrix is built, which empowers
the learning of the topology in an end-to-end manner.

  3. 3.

We have conducted extensive experiments on node classification and graph
classification tasks, and the results show that consistent and considerable
performance improvement is obtained, which verifies the effectiveness of the
proposed approach.

The rest of this paper is organized as follows. Section 2 introduces the basic
formulation of the spectral convolution of hypergraph and some common
notations. Section 3 gives details about the proposed module HERALD. Section 4
shows the experiment results and Section 5 concludes this paper.

##  2 learning of hypergraph Laplacian

Notation. Let $\mathcal{G}=(\mathcal{V,E})$ represents the input hypergraph
with hypernode set of $\mathcal{V}$ and hyperedge set of $\mathcal{E}$.
Hyperedge weights are assigned by a diagonal matrix
$\mathbf{W}\in\mathbb{R}^{|\mathcal{E}|\times|\mathcal{E}|}$. The structure of
hypergraph $\mathcal{G}$ can be denoted by a incidence matrix
$\mathbf{H}\in\mathbb{R}^{|\mathcal{V}|\times|\mathcal{E}|}$ with each entry
of $h(v,e)$, which equals 1 when $e$ is incident with $v$ and 0 otherwise. The
degree of hypernode and hyperedge are defined as
$d(v)=\sum_{e\in\mathcal{E}}w(e)h(v,e)$ and
$\delta(e)=\sum_{v\in\mathcal{V}}h(v,e)$ which can be denoted by diagonal
matrixes $\mathbf{D}_{v}\in\mathbb{R}^{|\mathcal{V}|\times|\mathcal{V}|}$ and
$\mathbf{D}_{e}\in\mathbb{R}^{|\mathcal{E}|\times|\mathcal{E}|}$.

Hypergraph Spectral Convolution. The normalized hypergraph laplacian matrix
$\mathbf{L}$ is given by:

$\mathbf{L}=\mathbf{I}-\mathbf{D}_{v}^{-1/2}\mathbf{HW}\mathbf{D}_{e}^{-1}\mathbf{H}^{\top}\mathbf{D}_{v}^{-1/2},$
(1)

which represents the connection strength of every pair of nodes. Since
$\mathbf{L}$ is a symmetric positive semi-definite matrix, then its spectral
decomposition is $\mathbf{L}=\mathbf{U}\Lambda\mathbf{U}^{\top}$, where
$\mathbf{U}=\\{u_{1},u_{2},...,u_{n}\\}$ is the set of eigenvectors and
$\Lambda=diag(\lambda_{1},\lambda_{2},...,\lambda_{n})$ contains the
corresponding eigenvalues. The Fourier transform is then defined as
$\hat{x}=\mathbf{U}x$. The spectral convolution of signal $\mathbf{x}$ and
filter $g$ can be denoted as:

$\mathbf{g}\star\mathbf{x}=\mathbf{U}g(\Lambda)\mathbf{U}^{\top}\mathbf{x},$
(2)

here, $g(\Lambda)=diag(g(\lambda_{1}),...,g(\lambda_{n}))$ is a function of
the Fourier coefficients. Following spectral graph convolution[2], we can get
the polynomial parametrization of filter $g$:

$g_{\theta}(\Lambda)=\sum\nolimits_{k=0}^{K-1}\theta_{k}\Lambda^{k},$ (3)

Thus, the convolution calculation for hypergraph results in:

$\mathbf{g}\star\mathbf{x}=\mathbf{U}\sum\nolimits_{k=0}^{K-1}\theta_{k}\Lambda^{k}\mathbf{U}^{\top}\mathbf{x}=\sum\nolimits_{k=0}^{K-1}\theta_{k}\mathbf{L}^{k}\mathbf{x}.$
(4)

##  3 Methodology

Although we can get the spectral convolution in Eq.(4), it is noticed that the
convolution kernel Eq.(3) is only a K-localized kernel, which aggregates K-hop
nodes to the farthest per iteration and thus restricts the flexibility of
kernel. Furthermore, the initial pre-defined hypergraph structure is not
always the optimal one for the specific down-stream learning task. Notably,
it’s somehow proved that GCNs which are K-localized and topology-fixed
actually simulate a polynomial filter with fixed coefficients [17, 18]. As a
result, existing techniques [11, 10, 12] might neglect the modeling of non-
local information and fail in obtaining high-quality hypergraph embeddings as
well.

In order to improve the representaional power of HyperGraph convolutional
Neural Networks (HGCNNs), we propose the HypERgrAph Laplacian aDaptor (HERALD)
to dynamically optimize the hypergraph structure (a.k.a. adapt the hypergraph
topology to the specific down-stream task). The HERALD focuses the directly
learning of the filter $g$ to make a more flexible spectral convolution kernel
and captures the global information that is beneficial to the adapting of the
hypergraph topology. Based on Eq.(3), the new filter is defined as:

$g_{\theta}(\Lambda)=\sum\nolimits_{k=0}^{K-1}(\mathcal{T}(\mathbf{L,X,\Theta}))^{k},$
(5)

where $\mathbf{L}$ and $\mathbf{X}$ are the original Laplacian matrix and node
features respectively. $\mathbf{\Theta}$ is the parameter and $\mathcal{T}$ is
the learnable function that outputs the spectrum of updated Laplacian
$\mathbf{\tilde{L}}$. Then, the spectral convolution will become:

$\mathbf{Y}=\mathbf{g}\star\mathbf{x}=\mathbf{U}\sum\nolimits_{k=0}^{K-1}(\mathcal{T}(\mathbf{L,X,\Theta}))^{k}\mathbf{U}^{\top}\mathbf{x}.$
(6)

Learnable Incidence Matrix. In order to learn a suitable hypergraph structure,
HERALD takes the node features and the pre-defined togology to construct a
parametrized incidence matrix. To be specific, given original incidence matrix
$\mathbf{H}\in\mathbb{R}^{|\mathcal{V}|\times|\mathcal{E}|}$ and node features
$\mathbf{X}=\\{x_{1};...;x_{|\mathcal{V}|}\\}\in\mathbb{R}^{|\mathcal{V}|\times
d}$ , we first get the hyperedge features by averaging the features of
incident hypernodes.

$x_{e_{i}}=\frac{1}{|e_{i}|}\sum_{v\in e_{i}}x_{v},$ (7)

where $|e_{i}|$ denotes the number of nodes in hyperedge $e_{i}$. Then we use
a linear transformation to obtain the transformed hyperedge feature:

$x_{e_{i}}=\mathbf{W}_{e}^{T}x_{e_{i}},$ (8)

where $\mathbf{W}_{e}\in\mathbb{R}^{d\times h}$ is the learnable parameter.
Next, in order to enhance the representational power of the convolution
kernel, HERALD adopts the self-attention mechanism [19] to encode the non-
local relations between paired nodes into the updated node features
$\mathbf{X}$. That is to say, the enhanced node features are formulated as:

$x_{i}=\sum_{1\leq j\leq k}\alpha_{ij}\mathbf{W}_{v}^{T}x_{j},$ (9)

where $\mathbf{W}_{v}$ is set to have the same dimension as $\mathbf{W}_{e}$,
and the attention weights are calculated by:

$\gamma_{ij}=(\mathbf{W}_{v}^{T}x_{i})^{\top}(\mathbf{W}_{v}^{T}x_{j}),\;1\leq
i,j\leq|\mathcal{V}|,$ (10) $\alpha_{i,j}=\frac{exp(\gamma_{ij})}{\sum_{1\leq
k\leq|\mathcal{V}|}exp(\gamma_{ik})}$ (11)

With the generated node features $\\{x_{1},...,x_{|\mathcal{V}|}\\}$ and
hyperedge features $\\{x_{e_{1}},...,x_{e_{|\mathcal{E}|}}\\}$, we calculate
the Hardamard power (element-wise power) [20] of each pair of hypernode and
hyperedge. And then we obtain the pseudo-euclidean distance matrix of
hyperedge and hypernode after a linear transformation:

$\displaystyle d_{ij}=\mathbf{W}_{s}^{T}(x_{i}-x_{e_{j}})^{\circ 2},\;1\leq
i\leq|\mathcal{V}|,1\leq j\leq|\mathcal{E}|,$ (12)

where $\mathbf{W}_{s}\in\mathbb{R}^{h\times 1}$ and $(\cdot)^{{\circ 2}}$
denotes the Hardamard power. Finally, the soft assignment matrix H is
constructed by further parameterizing the generated distance matrix with a
Gaussian kernel, in which each element represents the probability that the
paired node-hyperedge is connected:

$\tilde{\mathbf{H}}_{i,j}=exp(-d_{ij}/2\sigma^{2}),$ (13)

where the hyper-parameter of $\sigma$ is fixed to 20 in our experiments. Based
on the parametrized $\mathbf{\tilde{H}}$ and Eq.(1), HERALD outputs the the
hypergraph Laplacian $\mathbf{\tilde{L}}$.

Residual connect. In fact, if we learn the hypergraph topology from scratch,
it may spend expensive cost for optimization converge due to the lacking prior
knowledge about a proper initialization on the parameters. To this end, we
reuse the intrinsic graph structure to accelerate the training and increase
the training stability. To simplify the notation, we denote
$\mathbf{D}_{v}^{-1/2}\mathbf{HW}\mathbf{D}_{e}^{-1}\mathbf{H}^{\top}\mathbf{D}_{v}^{-1/2}$
as $\mathbf{N}$ and thus the normalized hypergraph laplacian matrix
$\mathbf{L}=\mathbf{I}-\mathbf{N}$. Formally, we assume that the optimal
$\hat{\mathbf{N}}$ is slightly shifted away from the original $\mathbf{N}$:

$\hat{\mathbf{N}}=(1-a)\mathbf{N}+a\mathbf{N}_{res},$ (14)

where $a$ is the hyper-paremeter that controls the updating strength of the
topology. From this respect, the HERALD module actually learns the residual
$\mathbf{N}_{\text{res}}$ rather than $\hat{\mathbf{N}}$.

To sum up, the learning complexity of HERALD module is $\mathcal{O}(d_{i}h)$
($d_{i}$ is the dimension of the input features at layer $i$) with introduced
parameters $\\{\mathbf{W}_{v},\mathbf{W}_{e},\mathbf{W}_{s}\\}$, independent
of the input hypergraph size and node degrees. Finally, the algorithm of
HERALD is summarized as follow:

Algorithm 1 HERALD module Input: Hypergraph Features
$\mathbf{X}=\\{\mathbf{x_{i}}\\},\mathbf{N}=\mathbf{I-L}$;
$\mathbf{Parameter}~{}a,\mathbf{W_{v},W_{e},W_{s}}$ 1:
$\tilde{\mathbf{H}}\leftarrow$ Eq.(7-13) 2:
$\mathbf{N}_{res}=\tilde{\mathbf{D}}_{v}^{-1/2}\tilde{\mathbf{H}}\tilde{\mathbf{W}}\tilde{\mathbf{D}}_{e}^{-1}\tilde{\mathbf{H}}^{\top}\tilde{\mathbf{D}}_{v}^{-1/2}$
3: $\hat{\mathbf{N}}=(1-a)\mathbf{N}+a\mathbf{N}_{res}$ 4:
$\mathbf{\tilde{L}=I-\mathbf{\hat{N}}}$ 5: Return $\mathbf{\tilde{L}}$

##  4 experiments

Node classification and graph classification are two natural settings of
hypergraph learning. Most of hypernode classification problems are semi-
supervised learning, where the goal is to assign labels to unlabelled nodes in
hypergraph and only a little amount labelled data is accessible [21, 11].
While many spectral convolution approaches have been proposed to solve node
classification problems [11, 10, 12], however, there exists no previous works
conducting on the hypergraph classification to the best of our knowledge. In
the experiments, we evaluate the proposed HERALD on both these tasks, and we
select the representative model of HGNN [11] to work as the evaluation
backbone. We have also conducted many experiments to compare our method with
the vanilla GCN [1].

Table 1: The results on the task of hypergraph classification. We report the
average test accuracy and its standard deviation under the 10-fold cross-
validation.

Datasets | MUTAG | PTC | IMDB-B | PROTEINS | NCI1 | COLLAB
---|---|---|---|---|---|---
# layers | 4 | 2 | 3 | 3 | 2 | 3
GCN [1] | 68.60 $\pm$ 5.6 | 65.41 $\pm$ 3.7 | 53.00 $\pm$ 1.9 | 68.29 $\pm$ 3.7 | 58.08 $\pm$ 1.1 | 52.69 $\pm$ 0.5
HGNN [11] | 69.12 $\pm$ 6.2 | 66.56 $\pm$ 4.7 | 55.20 $\pm$ 3.7 | 68.38 $\pm$ 3.8 | 58.32 $\pm$ 1.3 | 55.61 $\pm$ 2.5
HGNN + HERALD | 71.23 $\pm$ 9.0 | 67.75 $\pm$ 5.9 | 58.20 $\pm$ 5.5 | 68.64 $\pm$ 3.4 | 58.37 $\pm$ 1.4 | 55.74 $\pm$ 2.3

###  4.1 Hypernode Classification

Datasets. In this experiment, we employ two hypergraph datasets [10], one is
the co-citation relationship of Cora [4], the other is the co-authorship of
it. The co-authorship data consists of a collection of the papers with their
authors and the co-citation data consists of a collection of the papers and
their citation relationship. The details of the datasets are shown in Table 2.

Table 2: The brief introduction about the node classification datasets used in
our work.

Dataset | Cora | Cora
---|---|---
| (co-citation) | (co-authorship)
# hypernodes, $|\mathcal{V}|$ | 2708 | 2708
# hyperedges, $|\mathcal{E}|$ | 1579 | 1072
# classes | 7 | 7

Experimental Setup. We use the backbone of a $3$-layer HGNN as the evaluation
baseline. Since different datasets have different hypergraph structure, we
perform random search on the hyper-parameters and report the case giving the
best accuracy on validation set. To evaluate the proposed module, we add the
HERALD module to the latter two layers in HGNN. That is to say, in the $l$-th
layer (where $2\leq l\leq 3$), HERALD generates $\mathbf{\hat{N}}^{(l)}$, and
the feature updating function is given by
$\mathbf{X}^{(l+1)}=\beta(\mathbf{\hat{N}}^{(l)}\mathbf{X}^{(l)}\mathbf{W})$,
where $\beta$ is the non-linear activation function. For the hyper-parameter
of $a$, we set $a=1-0.9*(\text{cos}(\pi(l-1)/10)+1)/2$ for gradually increase
the updating strength of the task-specific adapted topology. We also add a
loss regularizer of $||\mathbf{N}-\mathbf{N}_{res}||_{2}$ to make the training
more stable, of which the loss weight is fixed to $0.1$.

Fast-HERALD. For constructing a more cost-friendly method, we also propose a
variant of the usage of HERALD, named Fast-HERALD. It constructs
$\hat{\mathbf{N}}$ at the beginning HGNN layer and reuses it in the process of
feature updating for all the rest layers. Since $\hat{\mathbf{N}}$ is shared
on each layer, it can reduce the number of parameters and increase the
training speed.

Table 3: The results on the task of node classification. We report the average
test accuracy and its standard deviation of test accuracy under 10 runs with
different random seeds.

Method | Cora | Cora
---|---|---
| (co-citation) | (co-authorship)
HGNN [11] | 48.23 $\pm$ 0.2 | 69.21 $\pm$ 0.3
HGNN + HERALD | 57.31 $\pm$ 0.2 | 70.05 $\pm$ 0.3
HGNN + FastHERALD | 57.27 $\pm$ 0.3 | 70.16 $\pm$ 0.4

Results and Disscussion. The results of experiments for hypernode
classification are given in Table 3. It is observed that HERALD module
consistently improves the testing accuracy for all the cases. It gains
0.84$\%$ improvement on Co-authorship Cora dataset while achieving a
remarkable 9.08$\%$ increase on Co-citation Cora dataset. We also notice the
FastHERALD get the best result on co-authorship Cora. The results show that
our proposed module can significantly improve the performance of hypergraph
convolutional network by adapting the topology to the downstream task.

###  4.2 Hypergraph Classification

Datasets. The task in this experiment is graph classification. The datasets we
use include six graph classification benchmarks: four bioinformatics datasets
(MUTAG, PTC, NCI1, PROTEINS) and two social network datasets (IMDB-BINARY,
COLLAB) [3]. However, all those datasets are ordinary graph. So we employ the
method proposed by Feng _et al._[11] to generate hypergrph, i.e. each node as
the centroid and its connected nodes form a hypergraph including the centroid
itself.

Experimental Setup. We also use HGNN as the evaluation backbone and employ the
same hyper-parameter search process as in the previous experiment. And we
conduct controlled experiments with the difference of with and without the
plugging of HERALD module. The hyper-parameters are used the same settings as
the illustration stated before. For obtaining the hypergraph embedding, we add
a summation operator as the permutation invariant layer at the end of the
backbone to readout the node embeddings.

Results and Disscussion. The results of experiments for hypergraph
classification are shown in Table 1. Comparing with the results of vanilla GCN
and HGNN, it can be ovserved that the performance of HGNN is better than that
of GCN, which demonstrates the meaning of using hypergraph to work as a more
powerful tool on modeling complex irregular relationship. One can also see
that the proposed HERALD module consistently improves the model performance
for all the cases and gains 1.12$\%$ test accuracy improvement on average,
which further verifies the effectiveness and generalization of the approach.

##  5 conclusion

We have presented a generic plug-in-play module of HypERgrAph Laplacian
aDaptor (HERALD) for improving the representational power of HGCNNs. The
module is tailored design for constructing task-aware hypergraph topology. To
this end, HERALD generates the parameterized hypergrapph laplacian and
involves it into the end-to-end training process of HGCNNs. The experiments
have shown our method gained remarkable performance on both hypernode and
hypergraph tasks, which verifies the effectiveness of the method.

## References

  * [1] Thomas N Kipf and Max Welling,  “Semi-supervised classification with graph convolutional networks,”  in Proceedings of the International Conference on Learning Representations, 2017.
  * [2] Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst,  “Convolutional neural networks on graphs with fast localized spectral filtering,”  in Advances in neural information processing systems, 2016, pp. 3844–3852.
  * [3] Pinar Yanardag and SVN Vishwanathan,  “Deep graph kernels,”  in Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2015, pp. 1365–1374.
  * [4] Prithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise Getoor, Brian Galligher, and Tina Eliassi-Rad,  “Collective classification in network data,”  AI magazine, vol. 29, no. 3, pp. 93–93, 2008.
  * [5] Xiang Yue, Zhen Wang, Jingong Huang, Srinivasan Parthasarathy, Soheil Moosavinasab, Yungui Huang, Simon M Lin, Wen Zhang, Ping Zhang, and Huan Sun,  “Graph embedding on biomedical networks: methods, applications and evaluations,”  Bioinformatics, vol. 36, no. 4, pp. 1241–1251, 2020.
  * [6] Wen Torng and Russ B Altman,  “Graph convolutional neural networks for predicting drug-target interactions,”  Journal of Chemical Information and Modeling, vol. 59, no. 10, pp. 4131–4149, 2019.
  * [7] Elena V Konstantinova and Vladimir A Skorobogatov,  “Application of hypergraph theory in chemistry,”  Discrete Mathematics, vol. 235, no. 1-3, pp. 365–383, 2001.
  * [8] Osamu Maruyama, Takayoshi Shoudai, Emiko Furuichi, Satoru Kuhara, and Satoru Miyano,  “Learning conformation rules,”  in International Conference on Discovery Science. Springer, 2001, pp. 243–257.
  * [9] Ke Tu, Peng Cui, Xiao Wang, Fei Wang, and Wenwu Zhu,  “Structural deep embedding for hyper-networks,”  in Proceedings of the 23rd AAAI Conference on Artificial Intelligence, 2018.
  * [10] Naganand Yadati, Madhav Nimishakavi, Prateek Yadav, Vikram Nitin, Anand Louis, and Partha Talukdar,  “Hypergcn: A new method for training graph convolutional networks on hypergraphs,”  in Advances in Neural Information Processing Systems, 2019, pp. 1511–1522.
  * [11] Yifan Feng, Haoxuan You, Zizhao Zhang, Rongrong Ji, and Yue Gao,  “Hypergraph neural networks,”  in Proceedings of the AAAI Conference on Artificial Intelligence, 2019, vol. 33, pp. 3558–3565.
  * [12] Yihe Dong, Will Sawin, and Yoshua Bengio,  “Hnhn: Hypergraph networks with hyperedge neurons,”  arXiv preprint arXiv:2006.12278, 2020.
  * [13] Ruoyu Li, Sheng Wang, Feiyun Zhu, and Junzhou Huang,  “Adaptive graph convolutional neural networks,”  arXiv preprint arXiv:1801.03226, 2018.
  * [14] Zizhao Zhang, Haojie Lin, Yue Gao, and KLISS BNRist,  “Dynamic hypergraph structure learning.,”  in IJCAI, 2018, pp. 3162–3169.
  * [15] Jianwen Jiang, Yuxuan Wei, Yifan Feng, Jingxuan Cao, and Yue Gao,  “Dynamic hypergraph neural networks.,”  in IJCAI, 2019, pp. 2635–2641.
  * [16] Naomi S Altman,  “An introduction to kernel and nearest-neighbor nonparametric regression,”  The American Statistician, vol. 46, no. 3, pp. 175–185, 1992.
  * [17] Qimai Li, Zhichao Han, and Xiao-Ming Wu,  “Deeper insights into graph convolutional networks for semi-supervised learning,”  arXiv preprint arXiv:1801.07606, 2018.
  * [18] Ming Chen, Zhewei Wei, Zengfeng Huang, Bolin Ding, and Yaliang Li,  “Simple and deep graph convolutional networks,”  arXiv preprint arXiv:2007.02133, 2020.
  * [19] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin,  “Attention is all you need,”  in Advances in neural information processing systems, 2017, pp. 5998–6008.
  * [20] Ruochi Zhang, Yuesong Zou, and Jian Ma,  “Hyper-sagnn: a self-attention based graph neural network for hypergraphs,”  arXiv preprint arXiv:1911.02613, 2019.
  * [21] Xiaojin Zhu and Andrew B Goldberg,  “Introduction to semi-supervised learning,”  Synthesis lectures on artificial intelligence and machine learning, vol. 3, no. 1, pp. 1–130, 2009.


# The JWST Resolved Stellar Populations Early Release Science Program III:
Photometric Star-Galaxy Separations for NIRCam

Jack T. Warfield Department of Astronomy, The University of Virginia, 530
McCormick Road, Charlottesville, VA 22904, USA  Hannah Richstein Department of
Astronomy, The University of Virginia, 530 McCormick Road, Charlottesville, VA
22904, USA  Nitya Kallivayalil Department of Astronomy, The University of
Virginia, 530 McCormick Road, Charlottesville, VA 22904, USA  Roger E. Cohen
Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen
Road, Piscataway, NJ 08854, USA  Alessandro Savino Department of Astronomy,
University of California, Berkeley, Berkeley, CA 94720, USA  Martha L. Boyer
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218,
USA  Christopher T. Garling Department of Astronomy, The University of
Virginia, 530 McCormick Road, Charlottesville, VA 22904, USA  Mario Gennaro
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218,
USA  The William H. Miller III Department of Physics & Astronomy, Bloomberg
Center for Physics and Astronomy, Johns Hopkins University, 3400 N. Charles
Street, Baltimore, MD 21218, USA  Kristen B. W. McQuinn Department of Physics
and Astronomy, Rutgers University, 136 Frelinghuysen Road, Piscataway, NJ
08854, USA  Max J. B. Newman Department of Physics and Astronomy, Rutgers
University, 136 Frelinghuysen Road, Piscataway, NJ 08854, USA  Jay Anderson
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218,
USA  Andrew A. Cole School of Natural Sciences, University of Tasmania,
Private Bag 37, Hobart, Tasmania 7001, Australia  Matteo Correnti Space
Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
Andrew E. Dolphin Raytheon Technologies, 1151 E. Hermans Road, Tucson, AZ
85756, USA  Steward Observatory, University of Arizona, 933 N. Cherry Avenue,
Tucson, AZ 85719, USA  Marla C. Geha Department of Astronomy, Yale University,
New Haven, CT 06520, USA  Karin M. Sandstrom Center for Astrophysics and Space
Sciences, Department of Physics, University of California San Diego, 9500
Gilman Drive, La Jolla, CA 92093, USA  Daniel R. Weisz Department of
Astronomy, University of California, Berkeley, Berkeley, CA 94720, USA
Benjamin F. Williams Department of Astronomy, University of Washington, Box
351580, U.W., Seattle, WA 98195-1580, USA

###### Abstract

We present criteria for separately classifying stars and unresolved background
galaxies in photometric catalogs generated with the point spread function
(PSF) fitting photometry software DOLPHOT from images taken of Draco II, WLM,
and M92 with the Near Infrared Camera (NIRCam) on JWST. Photometric quality
metrics from DOLPHOT in one or two filters can recover a pure sample of stars.
Conversely, colors formed between short-wavelength (SW) and long-wavelength
(LW) filters can be used to effectively identify pure samples of galaxies. Our
results highlight that the existing DOLPHOT output parameters can be used to
reliably classify stars in our NIRCam data without the need to resort to
external tools or more complex heuristics.

Multi-color photometry (1077) — Photometry (1234) — Stellar populations (1622)
— Two-color diagrams (1724) — Classification (1907) — JWST (2291)

††journal: RNAAS

##  1 Introduction

The JWST Resolved Stellar Populations ERS program (PID 1334; Weisz et al.
2023, hereafter W23) was approved to develop data reduction methods and tools
to investigate strategies for future observations of resolved stars in nearby
galaxies and star clusters. In this note, we present photometric cuts in color
and in the DOLPHOT (Dolphin, 2000, 2016) parameters ROUND, SHARP, CROWD, FLAG,
and OBJECT_TYPE that can be used to recover pure samples of stars or
background galaxies from our NIRCam observations of Draco II, WLM, and M92.
Separating out likely stars from the background is necessary for accurately
analyzing a target’s true stellar population—particularly important for cases
such as modeling a color-magnitude diagram (CMD) or measuring proper motions.

The DOLPHOT parameters presented herein can be used to recover star-like
objects from the photometric catalog produced for each target. Additionally,
for fields without significant crowding, colors formed through the combination
of SW and LW filters can be used to separate galaxies from stars in color-
color and CMD space, especially at high signal-to-noise ratios (SNR). These
proof-of-concept criteria demonstrate how long-baseline colors and photometric
morphology parameters can be employed to accurately determine object types,
and are offered as starting points for refining photometric catalogs from any
JWST NIRCam observations.

##  2 Observations and Photometry

The design of the ERS observational program is given in W23. For the SW
channel, all three targets have images in the F090W and F150W filters. The LW
filters for Draco II were F360M and F480M. The integration time was 11,810
seconds per filter for F090W and F277W and 5883 seconds for F150W and F444W.
For WLM, the LW filters were F250M and F430M. The integration times were
30,492 seconds in F090W and F430M and 23,706 seconds in F150W and F250M. For
M92, LW observations were taken in F227W and F444W. The integration time
chosen for all 4 filters was 1245 seconds.

Observations were reduced with the new NIRCam module of
DOLPHOT111http://americano.dolphinsim.com/dolphot (Dolphin, 2000, 2016), whose
specific functionality is discussed further in W23. DOLPHOT is a PSF
photometry package capable of identifying point sources and measuring their
positions and magnitudes from individual exposures. This module includes a
preprocessing routine to apply the appropriate masks and pixel area maps
provided by the Space Telescope Science Institute (STScI) pipeline for NIRCam,
along with functions for using image metadata and camera-specific PSF models.
Our initial reduction parameter choices were based on those used by the
Panchromatic Hubble Andromeda Treasury for target imaging (Williams et al.,
2014). For our stellar classification, we only consider sources for which
${\rm SNR}>10$ in the F090W and F150W filters. For our galaxy classification,
we only consider sources for which ${\rm SNR}>10$ in all filters.

##  3 Cuts for Stars

Figure 1: Blue stars represent objects classified as stars and orange pluses
as galaxies. Gray circles are objects that fit our general SNR cuts, but fail
either classification. The top row of plots shows CMDs for each field in the
sample. The bottom row shows examples of color-color plots for each field.

We find that a general cut for a purity-driven sample of stars can be made
with:

        F090W_SHARP^2 & F150W_SHARP^2 <= 0.01,
        F090W_CROWD & F150W_CROWD <= 0.5,
        F090W_FLAG & F150W_FLAG <= 2, and
        OBJECT_TYPE <= 2.

Here, we only consider the F090W and F150W filters. This choice is favorable
due to the sensitivities of the individual filters for probing stellar
populations, as well as the usefulness of the color baseline for experiments
such as deriving star formation histories. However, in principle, this set of
cuts can be done with single-filter photometry.

##  4 Cuts for Galaxies

Even in the NIR, distant galaxies tend to appear redder than stellar point
sources. While we see in Figure 1 (top) that the stellar sequence is entwined
with the background galaxies, it is possible to disentangle these objects from
each other by constructing long-baseline colors. Replacing the $x$-axis color
with both possible ${\rm F090W-LW}$ color combinations in each target’s CMD,
and drawing cuts for objects redder than the edge of each field’s stellar
sequence, we define cuts of $({\rm F090W-F360M}>2.1)$ and $({\rm
F090W-F480M}>2.1)$ for Draco II, $({\rm F090W-F250M}>1.75)$ and $({\rm
F090W-F430M}>2)$ for WLM, and $({\rm F090W-F277W}>2)$ and $({\rm
F090W-F444W}>2.3)$ for M92. Additionally, we defined cuts on SHARP and ROUND
to select extended sources:

        F150W_SHARP^2 > 0.05 & F090W_SHARP^2 > 0.05,
        |F090W_SHARP^2 - F150W_SHARP^2| < 0.15,
        F090W_ROUND > 0.05 & F150W_ROUND > 0.03, and
        |F090W_ROUND - F150W_ROUND| < 0.2.

These morphological cuts are effective at further purifying our selection; of
objects that satisfy the color cuts for Draco II, WLM, and M92, these
morphological cuts remove about 50%, 25%, and 10% of those objects,
respectively.

The color-color panels in Figure 1 (bottom) show the varying separation
between each field’s probable stars and galaxies. This separation is
especially apparent for Draco II, where we also see the selection effect of
only detecting the brightest stars at this SNR. In the case of WLM, we see
greater overlap between the two classifications in this space. For both Draco
II and WLM, we have visually verified that most of these sources appear likely
galaxies based on their extended morphologies in the images and found that
many of the spurious classifications correspond with diffraction artifacts
from saturated stellar sources.

For M92, the extreme crowding in this field leads to a much smaller sample of
detected background galaxies. Furthermore, it is difficult to confirm these
classifications from the images since the foreground visually obscures sources
tagged as galaxies.

##  5 Conclusions

From our DOLPHOT analysis of NIRCam observations of Draco II, WLM, and M92, we
conclude that:

  1. 1.

DOLPHOT output parameters, even from single-filter photometry, are effective
at cutting a catalog down to a pure sample of stars.

  2. 2.

Colors made by combining SW and LW filters can separate background galaxies
from stars. These baselines should be maximized for experiments where this
separation is desired.

  3. 3.

DOLPHOT output parameters can alone be used to distinguish background
galaxies, and greatly augment the sample purity when used in tandem with color
cuts.

  4. 4.

For crowded fields, diffraction artifacts from bright stars may significantly
affect one’s ability to recover background galaxies from NIRCam catalogs.

This work uses observations made with the NASA/ESA/CSA JWST. The data were
obtained from MAST at STScI, operated by the Association of Universities for
Research in Astronomy, Inc. (AURA), under NASA contract NAS 5-03127 for JWST.
These observations are associated with program DD-ERS-1334. This program also
benefits from recent DOLPHOT development work using observations made with the
NASA/ESA HST obtained from STScI (HST-GO-15902), and operated by AURA under
NASA contract NAS 5–26555. Facilities: JWST (NIRCam), MAST Data: NIRCam
observations http://dx.doi.org/10.17909/qnej-tt71 (catalog 10.17909/qnej-tt71)
Software: DOLPHOT (Dolphin, 2000, 2016)

## References

  * Dolphin (2016) Dolphin, A. 2016, DOLPHOT: Stellar photometry, Astrophysics Source Code Library, record ascl:1608.013.  http://ascl.net/1608.013
  * Dolphin (2000) Dolphin, A. E. 2000, PASP, 112, 1383, doi: 10.1086/316630
  * Weisz et al. (2023) Weisz, D. R., McQuinn, K. B. W., Savino, A., et al. 2023, arXiv e-prints, arXiv:2301.04659.  https://arxiv.org/abs/2301.04659
  * Williams et al. (2014) Williams, B. F., Lang, D., Dalcanton, J. J., et al. 2014, ApJS, 215, 9, doi: 10.1088/0067-0049/215/1/9


# About non-classical gravitational states

Federico Piazza  Aix Marseille Univ, Université de Toulon, CNRS, CPT,
Marseille, France

###### Abstract

Take the gravitational field, a number of macroscopic observers and put them
in a “highly-quantum” state, made by a coherent superposition of different
classical configurations. How would that look like? I try to study this system
by introducing a spatial distance operator among the observers and looking at
the properties of its expectation values.

##  1 Introduction

By a _classical state_ here I mean a state whose wave-function is peaked
around one solution of the classical equations of motion. The situation is
famously illustrated by coherent states, centered around a classical
trajectory and just “furbished” with the minimal uncertainty allowed by the
Heisenberg principle.

The double-well potential offers an equally known example of _deviations_ from
classicality. Its ground state does not correspond to a classical
configuration sitting on either of the minima, but is a coherent superposition
of the two. As nicely explained by Coleman [1], this generalizes to periodic
potentials and, most notably, to gauge theories. The ground state of a non-
Abelian Yang-Mills theory _is not_ just $A^{\mu}=0$ “furbished” with the
minimal amount of quantum indeterminacy. The _$\theta$ -vacua_ of such
theories are coherent superpositions of topologically inequivalent
configurations. When time-dependent or transient, deviations from classicality
are normally associated with tunneling.

Hovering over the latter examples are _Euclidean_ path integrals. Broadly
speaking, when the relevant classical saddles live in Euclidean spacetime and
are non-trivial, in the sense that they interpolate different classical
configurations, this often signals a departure from classicality.

Recently, gravitational Euclidean methods have played an important role in
addressing aspects of the black hole information puzzle [2]. Non-trivial
Euclidean saddles connecting different boundaries account for the decrease of
entropy in the late-Hawking radiation, as expected if the whole system evolves
unitarily. It is tempting to wonder if _gravitational_ deviations from
classicality at low energy are at play in this context. Perhaps related to all
this, gravitational instantons/wormholes have been advocated as a non
perturbative mechanism to solve the cosmological constant problem [3]. But
what would it mean for the gravitational field to be in a coherent
superposition of different classical metrics? What would an observer living in
such a spacetime experience?

In this talk I report on recent little advances in understanding these issues
[4]. I will show that, perhaps counterintuitively, a highly quantum
gravitational state does not seem to imply anything exotic, _locally_.
However, anomalous geometrical effects can build up at large separations.

##  2 A fluid (or a solid) of observers

I am interested in low energy phenomena, governed by Einstein gravity. In the
standard canonical theory [5] the spatial metric $h_{ij}$ is the relevant
dynamical variable. Physical states satisfy Hamiltonian and momentum
constraints. The idea is then to consider a highly quantum such state
$\Psi[h_{ij}(x^{k})]$ and try to understand it “operationally”, i.e. in terms
of the mutual observations of a bunch of observers living therein. One should
thus introduce additional fields in the low energy theory to model the
observers and their mutual interactions.

My choice of observers is that of a continuum “fluid” of them. The effective
field theory description of a fluid or a solid [7] involves three scalar
fields $X^{I}$, $I=1,2,3$. Each triplet of values labels and “follows in time”
an infinitesimal volume of the medium. Equivalently, each triplet is the
_name_ of an observer. One can arrange the Lagrangian in such a way that
$X^{I}=const.$ is a geodesic on any classical solution. In this case we are
dealing with a non-relativistic fluid. However, for the moment, I will
maintain agnostic about the dynamics of the medium.

The three fields $X^{I}$ effectively serve as a reference system, to which I
will add a fourth “time” field $T$. Dynamical reference systems have a long
story as a mean to define gauge invariant observables [8] and dress local
operators [9] in quantum gravity. Ideally one would like to add also a light
field to model signal exchanges among the observers. A more economic strategy
is to focus directly on spacetime _distances_ as a proxy for all such mutual
relations.111E.g. if a light ray connects events $A$ and $B$ then $A$ and $B$
are at null Lorentzian distance. See also [6] on this.

In the canonical theory the fields introduced above are commuting, and thus
can be used as independent variables in the wave-function. It is known that
$\Psi[h_{ij}(x^{k}),\,X^{I}(x^{k}),\,T(x^{k})]$, despite depending of spatial
quantities, portraits a portion of space _time_. The point is that the
Hamiltonian constraint makes $\Psi$ invariant under time-diffeomorphisms.
However, my very limited task here is to define _spatial distances between the
observers along a given time slice_ , which can be made gauge-invariant with
the aid of the reference fields $X^{I}$ and $T$.

##  3 Spatial-distance operator

I will use the field $T$ to define the _conditional probability amplitude_
that all remaining fields be in some configuration _given that_ the value of
$T(x^{k})$ is a constant, $T_{0}$,

$\tilde{\Psi}[h_{ij}(x^{k}),\,X^{I}(x^{k});T_{0}]\ \equiv\ {\cal
N}(T_{0})\Psi[h_{ij}(x^{k}),\,X^{I}(x^{k}),\,T(x^{k})=T_{0}].\ $ (1)

This is a portrait of the system “at a given time”. The
$\tilde{\Psi}$-functionals only depend on $h_{ij}$ and $X^{I}$ and must
satisfy the momentum constraint. A convenient (overcomplete) basis for the
corresponding Hilbert space are the common eigenvectors of such commuting
variables, $|h_{ij}(x^{k}),\,X^{I}(x^{k})\rangle$. Each element of this basis
represents a very classical three-dimensional geometry with the positions of
the observers perfectly specified on it. On
$|h_{ij}(x^{k}),\,X^{I}(x^{k})\rangle$ it is thus possible to calculate the
distance between any chosen pair of observers.

So I define a spatial distance operator between $\vec{X}$ and $\vec{Y}$,
$\hat{d}(\vec{X},\vec{Y})$, by declaring that it is diagonal on
$|h_{ij}(x^{k}),\,X^{I}(x^{k})\rangle$,

$\hat{d}(\vec{X},\vec{Y})\ |h_{ij}(x^{k}),\,X^{I}(x^{k})\rangle\ =\
d(\vec{X},\vec{Y})_{\\{h,X^{I}\\}}\ |h_{ij}(x^{k}),\,X^{I}(x^{k})\rangle\,.$
(2)

In the above equation the two triplets $\vec{X}$ and $\vec{Y}$ are the names
of the two observers and the eigenvalue $d(\vec{X},\vec{Y})_{\\{h,X^{I}\\}}$
is their distance on the corresponding eigenstate. In practice, as usual, one
needs to find the geodesic between $\vec{X}$ and $\vec{Y}$ and integrate the
spatial line element along it. The subscript ${\\{h,X^{I}\\}}$ is there to
remind us that this procedure depends on the geometry and on the fields’
configuration on the eigenstate $|h_{ij}(x^{k}),\,X^{I}(x^{k})\rangle$.

##  4 Expectation values

On a generic state $\tilde{\Psi}$, expectation values of
$\hat{d}(\vec{X},\vec{Y})$ are calculated by expanding $\tilde{\Psi}$ on the
classical eigenbasis,

$\left\langle\hat{d}(\vec{X},\vec{Y})^{2}\right\rangle=\sum_{I}\int\widetilde{{\cal
D}h}\,{\cal D}X^{I}\left|\langle\tilde{\Psi}|h,X^{I}\rangle\right|^{2}\
\left(d(\vec{X},\vec{Y})_{\\{h,X^{I}\\}}\right)^{2}\,.$ (3)

Figure 1: Every classical configuration contributes to the distance
expectation value, weighted by the amplitude probability $\tilde{\Psi}$. On
each configuration the distance between $\vec{X}$ and $\vec{Y}$ is a well-
defined calculable quantity.

Because of gauge redundancy, different metric configurations represent the
same geometry. The tilde on ${\cal D}h$ means that we should pick just one
representative for each gauge orbit.

In (3) I have evaluated the average of $\hat{d}^{2}$ because it has better
analytic properties than that of $\hat{d}$ and is more naturally generalizable
to the Lorentzian case: Lorentzian distances can be either positive or
imaginary and averaging over them could be meaningless. The distances
_squared_ , instead, are always real and thus average to a positive, negative
or null value.

The observers’ labels $\vec{X}$ define a coordinate system on a three-
dimensional manifold which is very naturally equipped with the distance

$\overline{d(\vec{X},\vec{Y})}\equiv\sqrt{\left\langle\hat{d}(\vec{X},\vec{Y})^{2}\right\rangle}\,.$
(4)

Is (4) itself a _Riemannian_ distance? In other words, is there an “average
line element” on the manifold that reproduces (4) for each pair of points? It
will be sufficient to show a counterexample in order to give a negative answer
to this question.

##  5 Average distances are non-additive

Consider a superposition of two classical eigenvectors,
$|\tilde{\Psi}\rangle=(|\tilde{\Psi}_{1}\rangle+|\tilde{\Psi}_{2}\rangle)/\sqrt{2}\,.$
The system is one-dimensional and the classical states
$|\tilde{\Psi}_{1}\rangle$ and $|\tilde{\Psi}_{2}\rangle$ are depicted in the
figure below. Only four observers are picked out in the figure: $A$, $B$, $C$,
and $D$.

$\displaystyle\ |\tilde{\Psi}_{1}\rangle\ \
\qquad\quad\qquad\qquad\qquad\qquad\qquad|\tilde{\Psi}_{2}\rangle$

To first order in $\epsilon$,

$\displaystyle\overline{d(A,B)}=\overline{d(B,C)}=\overline{d(A,D)}=\overline{d(C,D)}=\pi/2\,,\quad\
\overline{d(B,D)}=\pi\,,\quad\ \ \overline{d(A,C)}=\pi-2\epsilon\,.$ (5)

It is clear from this example that average distances _are not_ Riemannian
distances. In particular, even in one dimension they fail to add up:
$\overline{d(A,B)}+\overline{d(B,C)}\neq\overline{d(A,C)}$.

This anomalous behavior, in general, builds up at large separation and is
reminiscent of some IR non-local effect found in double holography [10]. On
the opposite, in the neighbors of each observer nothing particularly exotic
happens. There is [4] an average metric ${\cal G}_{IJ}$ such that

$\overline{d(0,\vec{X})}\ =\ \sqrt{{\cal G}_{IJ}\ X^{I}X^{J}}\ +\ {\cal
O}(X)^{3}\,,$ (6)

where the labels $\vec{X}$ have been used directly as coordinates and one of
the two observers has been located at the origin. Overall, average distances
behave like _chord distances_ of embedded manifolds: they have a standard
locally flat limit but fail to be additive at large separation [4].

##  6 Conclusions

I have tried to study departures from classicality in gravity with the aid of
certain auxiliary scalar fields that serve as a reference frame of
“observers”. The expectation values of their mutual distances are a good
diagnostic of the quantum-ness of the state. Most strikingly, average
distances can grow in some non-additive way while maintaining a regular
behavior at small separations. The physical implications of this effect are
far from obvious—space-like distances do not have direct operational meaning
anyway. However, if some of this non-additivity leaks along the time
direction, this could have interesting implications e.g. on causality [4]
[10].

My considerations are purely kinematical in the sense that, from the onset, I
consider states that already deviate from classicality. There is no firm
indication, at present, about when and how a low-energy gravitational system
could find itself in such a highly quantum state. However, it is not difficult
to see that the non-additive effect just described is always going to be
there, to some extent. Perhaps it is just negligible in most realistic
situations. It would be interesting to quantify this effect, say, in a Hartle-
Hawking state. It would be nice also to extend this study to four-dimensional
spacetime distances, which have more direct operational meaning.

## References

## References

  * [1] S. Coleman, “Aspects of Symmetry: Selected Erice Lectures,” (1985), Chap. 7.
  * [2] See e.g. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, “The entropy of Hawking radiation,” Rev. Mod. Phys. 93 (2021) no.3, 035002 [arXiv:2006.06872 [hep-th]], and references therein.
  * [3] See e.g. A. Hebecker, T. Mikhail and P. Soler, “Euclidean wormholes, baby universes, and their impact on particle physics and cosmology,” Front. Astron. Space Sci. 5 (2018), 35 [arXiv:1807.00824 [hep-th]], and references therein.
  * [4] F. Piazza, “Glimmers of a post-geometric perspective,” [arXiv:2108.12362 [hep-th]].
  * [5] B. S. DeWitt, “Quantum Theory of Gravity. 1. The Canonical Theory,” Phys. Rev. 160 (1967), 1113-1148
  * [6] E. Witten, “Light Rays, Singularities, and All That,” Rev. Mod. Phys. 92 (2020) no.4, 045004 [arXiv:1901.03928 [hep-th]].
  * [7] S. Dubovsky, T. Gregoire, A. Nicolis and R. Rattazzi, “Null energy condition and superluminal propagation,” JHEP 03 (2006), 025 [arXiv:hep-th/0512260 [hep-th]]. S. Endlich, A. Nicolis, R. Rattazzi and J. Wang, “The Quantum mechanics of perfect fluids,” JHEP 04 (2011), 102 [arXiv:1011.6396 [hep-th]].
  * [8] C. Rovelli, “Quantum Reference Systems,” Class. Quant. Grav. 8 (1991), 317-332.
  * [9] D. Marolf, “Comments on Microcausality, Chaos, and Gravitational Observables,” Class. Quant. Grav. 32 (2015) no.24, 245003 [arXiv:1508.00939 [gr-qc]].
  * [10] H. Omiya and Z. Wei, “Causal Structures and Nonlocality in Double Holography,” [arXiv:2107.01219 [hep-th]].