Deep Hypergraph U-Net for Brain Graph Embedding and Classification
DDeep Hypergraph U-Net for Brain Graph Embeddingand Classification
Mert Lostar a , Islem Rekik a,b, ∗ , and for the Alzheimer’s DiseaseNeuroimaging Initiative ∗∗ a BASIRA lab, Faculty of Computer and Informatics, Istanbul Technical University,Istanbul, Turkey b School of Science and Engineering, Computing, University of Dundee, UK
Abstract-Background.
Network neuroscience examines the brain as a complexsystem represented by a network (or connectome), providing deeper insightsinto the brain morphology and function, allowing the identification of atypicalbrain connectivity alterations, which can be used as diagnostic markers ofneurological disorders. -Existing Methods.
Graph embedding methods which map data sam-ples (e.g., brain networks) into a low dimensional space have been widelyused to explore the relationship between samples for classification or predic-tion tasks. However, the majority of these works are based on modeling thepair-wise relationships between samples, failing to capture their higher-order ∗ Corresponding author; Dr Islem Rekik ([email protected]), http://basira-lab.com/ ,GitHub code: https://github.com/basiralab/HUNet ∗∗ Data used in preparation of this article were obtained from the Alzheimer’s DiseaseNeuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigatorswithin the ADNI contributed to the design and implementation of ADNI and/or provideddata but did not participate in analysis or writing of this report. A complete listing ofADNI investigators can be found at: http://adni.loni.usc.edu/wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf
Preprint submitted to Elsevier September 1, 2020 a r X i v : . [ q - b i o . N C ] A ug elationships. -New Method. In this paper, inspired by the nascent field of geometricdeep learning, we propose Hypergraph U-Net (HUNet), a novel data embed-ding framework leveraging the hypergraph structure to learn low-dimensionalembeddings of data samples while capturing their high-order relationships.Specifically, we generalize the U-Net architecture, naturally operating ongraphs, to hypergraphs by improving local feature aggregation and preserv-ing the high-order relationships present in the data. -Results.
We tested our method on small-scale and large-scale hetero-geneous brain connectomic datasets including morphological and functionalbrain networks of autistic and demented patients, respectively. -Conclusion.
Our HUNet outperformed state-of-the art geometric graphand hypergraph data embedding techniques with a gain of 4-14% in classifi-cation accuracy, demonstrating both scalability and generalizability.
Keywords:
Neurological disorder diagnosis, Machine Learning,Computer-Aided Diagnosis, Geometric Deep Learning, Hypergraph UNet
1. Introduction
Studying the connectivity of the human brain provides us with a deepunderstanding of how the brain operates as a highly complex interconnectedsystem. Network neuroscience, in particular, aims to chart the brain con-nectome by modeling it as a network, where each node represents a specificanatomical region of interest (ROI) and the weight of the edge connectingpairs of nodes encodes their relationship in function, structure or morphology(Fornito et al., 2015; Heuvel and Sporns, 2019). Studies of brain networks2rimarily investigated structural and functional connectivities derived fromdiffusion weighted and functional magnetic resonance imaging (MRI), respec-tively (Park and Friston, 2013; Heuvel and Sporns, 2019). On a methodolog-ical level, graph theory techniques have been widely used to analyze brainnetworks, giving new insights into atypical alterations of brain connectivitycaused by neurological or neuropsychiatric disorders (Fornito et al., 2015).Studies combining these techniques have uncovered that diseases such asschizophrenia (Fornito et al., 2012; Alexander-Bloch et al., 2012), Alzheimer’sDisease (AD) (Buckner et al., 2009; Mahjoub et al., 2018), autism spectrumdisorder (ASD) (Morris and Rekik, 2017; Soussia and Rekik, 2017) affect theconnectomics of the brain, implying that pinning down connectional changesin the brain could reveal clinically useful diagnostic markers.To this aim, investigating a population of brain connectomes using graph-based embedding techniques has become popular, given their capacity tomodel the one-to-one relationship between data samples (i.e. connectomes)and circumvent the curse of dimensionality in learning tasks such as brainconnectome classification or generation. Existing graph embedding tech-niques can be broken down into three main categories: (1) matrix factor-ization based, (2) deep learning methods based on random walks and (3)neural network based methods. Matrix factorization focuses on factorizing ahigh dimensional data matrix into lower dimensional matrices while preserv-ing the topological properties of the data to factorize. Such methods firstencode relationships between nodes into an affinity matrix, which is then fac-torized to generate the embedding. These vary depending on the propertiesof the matrix. For instance, while graph factorization (GF) technique uses3he adjacency matrix (Ahmed et al., 2013), GraRep (Cao et al., 2015) uses k -step transition probability matrices. However, matrix factorization basedmethods usually consider the first order proximity and some of these meth-ods which consider high-order proximities such as the GraRep suffer fromscalability issues (Goyal and Ferrara, 2018).Unlike matrix factorization methods, random-walk based deep learningapproaches such as DeepWalk (Perozzi et al., 2014) and node2vec (Groverand Leskovec, 2016) focus on optimizing embeddings to encode the statisticsof random walks rather than trying to come up with a deterministic nodesimilarity measure. These approaches use random walks in graphs to gener-ate node sequences in order to learn node representations. Given a startingnode in a graph, these methods select one of the neighbors and then re-peat the process after moving onto the neighboring node to generate nodesequences. Random walks have had different uses in approximating differ-ent properties in graphs including node similarity (Fouss et al., 2007) andcentrality (Newman, 2005). They are especially helpful when a graph is toolarge to consider in its entirety or when a graph is only partially observ-able (Goyal and Ferrara, 2018). DeepWalk (Perozzi et al., 2014), one of theinitial works using this approach, performs truncated random walks graphs.Instead, node2vec (Grover and Leskovec, 2016) uses a biased random walkprocedure which uses depth first sampling and breadth first sampling to-gether. However, since these approaches use local windows to operate theyfail to characterize the global structure of the graph (Cai et al., 2018).More recently, there has been a surge of interest in deep graph neural net-works (GNN) (Kipf and Welling, 2017; Gao and Ji, 2019; Wang et al., 2016),4iven their remarkable capacity to model the deeply nonlinear relationshipbetween data samples (i.e. connectomes) rooted in message passing, aggre-gation, and composition rules between node connectomic features (Ktenaet al., 2017; Bessadok et al., 2019b,a; Banka and Rekik, 2019). Graph em-bedding also witnessed the introduction of different neural networks such asautoencoders (Wang et al., 2016), the multilayer perceptron (Tang et al.,2015), graph convolutional network (GCN) (Kipf and Welling, 2017) andgenerative adversarial network (GAN) (Wang et al., 2017). For example,structural deep network embedding (SDNE) (Wang et al., 2016) leverageddeep autoencoders to conserve the information from the first and second or-der proximities by jointly optimizing both proximities. Their method applieshighly non-linear functions in order to create an embedding that captures thenon-linearity of the graph. However this approach can be computationallyexpensive to operate on large sparse graphs (Goyal and Ferrara, 2018). GCNhandles this issue by defining a convolution operation for graphs with theaim of iteratively aggregating the embedding neighbors of nodes to updatethe embedding. Considering only the local neighborhood makes the methodscalable while multiple iterations allow for the characterization of the globalfeatures. Graph U-Net (GUNet) (Gao and Ji, 2019) is an encoder-decoderarchitecture leveraging graph convolution and it improves on GCN by gener-alizing the seminal U-Net (Ronneberger et al., 2015) designed for Euclideanspaces (e.g., images) to non-Euclidean spaces (e.g., graphs), allowing high-level feature encoding and receptive field enlargement through the samplingof important nodes in the graph. 5 igure 1: A) A simple undirected un-weighted graph, where an edge connects apair of nodes. B) An unweighted hyper-graph, where a hyperedge connects morethan two nodes. While an edge capture thelow-order interaction between graph nodes,a hyperedge capture a high-order betweennodes as a set. Hence, the learned nodeembeddings (gray vertical bars) in a hyper-graph better capture complex and represen-tative node interactions.
More interestingly, such graphembedding architectures allow tocircumvent the curse of dimen-sionality in learning based taskssuch as brain connectome classi-fication or generation (Bessadoket al., 2019b,a) by learning low-dimensional embeddings of node at-tributes such as connectome fea-tures while preserving their similar-ities. These learned embeddings canthen be used as inputs for machinelearning methods for tasks such asnode classification (Bessadok et al.,2019a; Banka and Rekik, 2019) andlink prediction (Liu et al., 2017).However, a major limitation of cur-rent deep graph embedding architectures is that they are unable to capture many-to-many (i.e., high-order) relationships between samples, hence thelearned feature embeddings only consider node-to-node edges in the popula-tion graph.Hypergraph Neural Network (HGNN) (Feng et al., 2019) addresses thisproblem through the use of hypergraph structure for data modeling. Themain difference between a traditional graph and a hypergraph, as illustratedin (
Fig. hyperedge connecting a subset of nodes .Even tough the traditional hypergraph learning approach usually suffers fromhigh computational costs, HGNN manages to eliminate this challenge by de-vising a hyperedge convolution operation. However, HGNN only uses thedevised hypergraph convolution operation for learning the hypernode em-beddings.In this paper we propose the Hypergraph U-Net (HUNet) architecturefor high-order data embedding by generalizing the graph U-Net (Gao andJi, 2019) to hypergraphs. HUNet, unlike HGNN takes advantage of the U-Net architecture to improve the local feature aggregation through poolingand unpooling operations while still leveraging the hypergraph convolutionoperation to learn more representative feature embeddings. It enables theinferring and aggregation of node embeddings while exploring the global high-order structure present between subsets of data samples, becoming moregnostic of real data complexity.We evaluate HUNet on two different types of brain connectomic datasetsfor neurological disorder diagnosis and show that HUNet achieves a largegain in classification accuracy compared to state-of-the-art methods on bothdatasets, demonstrating scalability and generalizability as we perturb train-ing and test sets and vary their sizes.7 igure 2:
Hypergraph U-Net (HUNet) architecture. A) Each brain network S i of subject i is encoded in a connectivity matrix, which is vectorized by extracting C in connectivityweights stored in its off-diagonal upper triangular part. We stack all N samples into X with C in rows. B) Using X , we generate the normalized hypergraph connectivitymatrix H ∈ R N × N . H and X are used as inputs for the proposed HUNet architecture,stacking our proposed hypergraph pooling (hPool) and unpooling (hUnpool) layers withhypergraph convolution layers. Connectivity information of the removed nodes at hPoollayer at HUNet level d are transferred to the hUnpool layer at the same level to be usedwhen up-sampling X and restoring H . Outputs of the HUNet are the learned featureembeddings ˜ X , which can be used for the target learning task.
2. Proposed method
In graph embedding the aim is learning how to project node features intoa low-dimensional space while preserving the structural relationships in the8 able 1:
Major mathematical notations used in this paper . Mathematical notation Definition S i brain network of subject iN number of nodes in the initial hypergraph X ∈ R N × C in input feature embeddings where C in is the input feature dimension E number of hyperedges in the initial hypergraph N d number of nodes at HUNet level d X d ∈ R N d × C d feature embeddings at HUNet level d , C d is the feature dimension Q ∈ { , } N × E hypergraph incidence matrix Q ( v, e ) = , ifv / ∈ e , ifv ∈ e Q entries where v is a vertex and e is a hyperedge d ( v ) = (cid:80) e ∈E Q ( v, e ) degree of a vertex d ( e ) = (cid:80) v ∈ V Q ( v, e ) degree of a hyperedge D v ∈ R N × N diagonal matrix of vertex degrees D ε ∈ R E × E diagonal matrix of hyperedge degrees W ∈ R E × E diagonal weight matrix of a hypergraph H = D v − QWD ε − Q T D v − ∈ R N × N normalized hypergraph connectivity based on Q incidence matrix Θ d ∈ R C d × C d +1 a learnable matrix where C d is the input feature sizeand C d +1 is the output feature size Z d ∈ R N d × C d +1 updated feature embeddings at HUNet level d Z d = H d X d Θ d hypergraph convolution operation at HUNet level d A wide variety of graph embedding models using different approaches havebeen proposed and applied to tasks such as node classification (Perozzi et al.,2014; Tang et al., 2015), node clustering (Tang et al., 2016), link prediction(Wang et al., 2016), graph alignment (Bessadok et al., 2019a), graph clas-sification (Dai et al., 2016; Banka and Rekik, 2019) and graph visualization(Cao et al., 2016) in the recent years. More recently, Graph U-Net (GUNet)(Gao and Ji, 2019) was proposed as an U-Net like architecture for graph data.By adapting Euclidean pooling and unpooling operations, which are criticalwhen building encoder-decoder architectures, to non-Euclidean graph datawith no spatial locality and order information. GUNet comprises Graph Con-volutional Networks (GCN) layers (Kipf and Welling, 2017) to learn deeplycomposed embeddings of the graph nodes by exploring their hierarchicaltopological neighbors via the ‘neighbor of a neighbor’ composition rule. Theyshow that this encoder-decoder architecture outperforms conventional GCNin learning well representative embeddings of the node features. However,GUNet is only able to learn from pair-wise relationships between the nodes,thereby ignoring the many-to-many high-order relationships present in manyreal-world data. 10 .1.2. Hypergraph learning
Very recently, hypergraphs, originally introduced in (Zhou et al., 2006),have started to gain momentum in geometric deep learning thanks to theirability to capture high-order relationship between data samples in varioustasks such as feature selection (Zhang et al., 2017), image classification (Yuet al., 2012), social network analysis (Fang et al., 2014) and sentiment predic-tion using multi-modal data (Ji et al., 2018). More recently, the inception ofhypergraph neural network (HGNN) (Feng et al., 2019) as the first geometricdeep learning model on hypergraph structure introduced hyperedge convolu-tion operations based on the hypergraph Laplacian encoding the hypergraphspectra. However, HGNN is restricted to using hypergraph convolution forlearning hypernode embeddings in a semi-supervised manner where train-ing node labels supervise the estimation of the node feature mapping fromlayer to layer. In the absence of node labels, HGNN cannot be trained. Be-sides, HUNet takes advantage of the U-Net architecture, aiming to learn wellrepresentative feature embeddings by improving the first-order local featureaggregation through pooling and unpooling operations while combining themwith hypergraph convolution operation for improving global feature aggre-gation compared to applying convolution alone for feature aggregation as inHGNN architecture (Feng et al., 2019).
In this section we explain our proposed HUNet architecture, shown in(
Fig.
2) and its components. The key idea behind the HUNet architecture isto learn a many-to-many node embedding with a high-order feature aggre-gation rule by leveraging the advantage of using hypergraphs to model the11igh-order relations between hypernodes compared to existing deep graph-based embedding methods. With this purpose, we propose the hypergraphpooling (hPool) and hypergraph unpooling (hUnpool) layers.
Even tough graphs are adequate for representing pair-wise relationshipsbetween different nodes, in many applications higher-order relationships,which graphs are unable to represent, are present between subsets of nodes.For such applications one can take advantage of the hypergraph structureencoding shared interactions between a subset of nodes by connecting themwith a single hyperedge (Zhou et al., 2006) (
Fig. G = { V, E , W } , where V is a node set, E is a hyperedge set, W ∈ R E × E is a diagonal weight matrix, where E is the number of hyper-edges, assigning weights to hyperedges. In our experiments W is initializedas an identity matrix meaning that all the hyperedges have the same weight.We then define a N × E hypergraph incidence matrix Q , where N is the num-ber of hypernodes in the hypergraph, with elements representing whether ahypernode is contained in a hyperedge or not as follows: Q ( v, e ) = , if v / ∈ e , if v ∈ e (1)where v ∈ V represents a hypernode and e ∈ E represents a hyperedge.We also define the hyperedge degree d ( e ) which represents the number ofhypernodes in a hyperedge e and node degree d ( v ) denoting the number ofhyperedges connected to a hypernode v as:12 ( e ) = (cid:88) v ∈ V Q ( v, e ) d ( v ) = (cid:88) e ∈E Q ( v, e ) (2)In order to adapt the hypergraph convolution operation to our encoder-decoder U-Net architecture, given a hypergraph with N hypernodes, we pro-duce the normalized hypergraph connectivity H ∈ R N × N as follows: H = D v − QWD ε − Q T D v − (3)where D v ∈ N N × N denotes the diagonal hypernode degree matrix and D ε ∈ N E × E represents the diagonal hyperedge degree matrix. H , constructedfrom the initial hyperpraph, is also pooled and unpooled along with the fea-ture embeddings in the pooling and unpooling layers but unlike the featureembeddings, it only changes in dimensionality. Normalized hypergraph con-nectivity H d ∈ R N d × N d at HUNet level d , where N d is the number of nodesin the hypergraph at level d , is pooled and restored by the respective poolingand unpooling layers at level d . The feature embedding X d ∈ R N d × C d , where C d is the feature dimension at level d , is taken as input from the previouspooling or unpooling layer along with H d . The hypergraph feature matrix X d is first transformed by the embedding function Θ d ∈ R C d × C d +1 learnedby the hypergraph convolution layer at level d to extract C d +1 dimensionalnode features, then diffused through H d to aggregate the embedded hyper-node features across hyperedges that contain them. The hypernode featureembedding matrix Z ∈ R N d × C d +1 is then passed on to the next HUNet leveland updated as follows: 13 d = H d X d Θ d (4) We propose a hypergraph pooling (hPool) layer to down-sample our hy-pergraph. Instead of passing the graph adjacency matrix to our pooling layeras in (Gao and Ji, 2019), we use the H described in the previous part in or-der to adapt the pooling operation to hypergraphs ( Fig. p ∈ R N d is used to map all hyper-node features to a real-valued score. This projection allows the use of top- k hypernode pooling for selecting the k most important hypernodes. The indi-vidual hypernode scores represent how much information is preserved afterthe projection onto the p vector. This means that selecting the top scor-ing k -hypernodes to form the new hypergraph would maximize informationpreservation. We define the layer-wise propagation rule for this pooling layeras follows: z pd = X d p || p || indexes = topk ( z pd , k )˜ z pd = sigmoid ( z pd [ indexes ])˜ X d = X d [ indexes, :] X d +1 = ˜ X d (cid:12) ˜ z pd H d +1 = H d [ indexes, indexes ] (5)14here X d ∈ R N d × C d and H d ∈ R N d × N d denote the feature embeddingand hypergraph connectivity, respectively, at depth d with C d denoting thefeature embedding dimension. z pd ∈ R N d is output of the projection of X d onto p . topk ( z pd , k ) operation returns the indexes of nodes with the largest scores in z pd . These indexes are then used to produce the pooled feature embeddings˜ X d ∈ R k × C d and the pooled hypergraph connectivity H d +1 ∈ R k × k . Weselect the largest entries from z pd and apply a sigmoid function to produce ˜ z pd .Next, we apply an element-wise multiplication, represented by (cid:12) to ˜ X d and˜ z pd , thereby generating the new feature embedding X d +1 to pass onto the nextHUNet level ( d +1) along with H d +1 . Note that only the dimensionality of thehypergraph connectivity changes from layer to layer during both encoding(i.e., pooling) and decoding (i.e., unpooling) steps in the HUNet architecture( Fig igure 3:
HUNet pooling layer (hPool).
The feature embedding X d ∈ R N d × C d andhypergraph connectivity matrix H d ∈ R N d × N d are passed to the hPool layer as inputswhere N d is the number of nodes, C d is the feature dimension at HUNet level d . In theprojection phase, the feature embedding is projected on the learnable vector p ∈ R N d .The output of this operation, represented by z pd ∈ R N d , is then used to determine theindices of the top- k nodes. This index information is what we use in order to pool ourinputs in the top- k node selection phase. This pooling operation produces ˜ X d ∈ R k × C d which represents the pooled feature embedding and H d +1 ∈ R k × k denoting the hypergraphconnectivity which we pass onto the next HUNet level. An unpooling operation is needed in order to up-sample the hypergraphdata that was previously pooled in the hypergraph encoding phase. To thisend, we propose hypergraph unpooling layer (hUnpool) that generalizes theunpooling operation proposed in (Gao and Ji, 2019) to hypergraphs by lever-aging the normalized hypergraph connectivity H . This layer takes the con-nectivity information about the removed nodes from the hPool layer at the16ame level of the HUNet to reconstruct the hypergraph and place back theremoved nodes ( Fig
Eq. H drives thehypergraph convolution in both encoding (top-down) and decoding phases(bottom-up) as illustrated in Fig
2. Blocks of hypergraph pooling and hy-pergraph unpooling layers are stacked in order to construct the HUNet archi-tecture. Hypergraph convolution follows every hypergraph pooling layer toupdate the features of the nodes using their first-order local neighbors as wellas every hypergraph unpooling layer to fill in the empty node features thatwere added back. The algorithm of our HUNet architecture with d -depth isdetailed in Algorithm.
3. Results
We evaluate our HUNet and comparison state-of-the-art methods on small-scale and large-scale connectomic datasets derived from different neuroimag-ing modalities (structural and functional MRI) to demonstrate the ability ofHUNet in better generalizing across data scales and handling heterogeneousdata distributions. 17 lgorithm 1
A Hypergraph U-Net of depth d Definitions: x s : array of feature embeddings (empty at initialization) h s : array of normalized hypergraph connectivities (empty at initialization) N : number of rows in X idx s : array of indices (empty at initialization) idx : selected node indices in top- k pooling hP ool (˙): hypergraph pooling layer r : pooling ratio to be used in the hPool layers hConv (): hypergraph convolution uact (): activation function that is used between HUNet levels zeros ( input ): creates a zero matrix in the shape of the input matrix oact (): output activation function INPUTS: X : feature embeddings; H : hypergraph connectivity; x s .append ( X ) h s .append ( H ) idx s .append ([1 , . . . N ]) for depth = 1, 2, . . . , d do X , H , idx = hP ool ( X , H , r ) X = uact ( hConv ( X , H )) idx s .append ( idx ) if depth < d then x s .append ( X ) H s .append ( H ) end if end for for depth = d - 1, d - 2, . . . , 0 do ret x = x s [ depth ] ret H = h s [ depth ] ret idx = idx s [ depth + 1] upsample = zeros ( ret x ) upsample [ ret idx ] = X [ ret idx ] X = ret x + upsample X = hConv ( X , ret H ) if depth > then X = uact ( X ) else X = oact ( X ) end if end for OUTPUT: learned hypernode feature embedding X mall-scale morphological data We use a subset of ADNI GO publicdataset, consisting of 77 subjects (41 AD and 36 Late Mild Cognitive Impair-ment), where each subject has a structural T1-w MR image (Mueller et al.,2005). Data used in the preperation of this article were obtained from theAlzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu).The ADNI was launched in 2003 as a public-private partnership, led byPrincipal Investigator Michael W. Weiner, MD. The primary goal of ADNIhas been to test whether serial magnetic resonance imaging (MRI), positronemission tomography (PET), other biological markers, and clinical and neu-ropsychological assessment can be combined to measure the progression ofmild cognitive impairment (MCI) and early Alzheimer’s disease (AD). Forpreprocessing, we follow the steps defined by (Mahjoub et al., 2018). In or-der to reconstruct left and right cortical hemispheres from T1-w MRI (Fischl,2004), FreeSurfer (Fischl, 2012) processing pipeline was used for each sub-ject. Next, each cortical hemisphere was divided into 35 cortical regions usingDesikan-Killany cortical atlas (Fischl, 2004). We then use cortical attributes:maximum principal curvature, cortical thickness, sulcal depth and averagecurvature to derive four 35 ×
35 morphological brain connectivity matrices.For each attribute, we extract a feature vector by retrieving the off-diagonalupper triangular part elements of each attribute-specific connectivity matrix.
Large-scale functional network data
We also evaluate our method ona large scale functional network dataset, consisting of 517 subjects (245 ASDand 272 Control) from the ABIDE preprocessed dataset (Cameron et al., http://adni.loni.usc.edu http://preprocessed-connectomes-project.org/abide/ × × mm . In order to boost the signal to noise ratio, spatialsmoothing was then applied using a Gaussian kernel of 6 mm. Lastly, a band-pass filtering (0.01-0.1 Hz) was applied to the time series of each voxel (Priceet al., 2014; Huang et al., 2017). Detailed explanations for these steps can befound in http://preprocessed-connectomes-project.org/abide/ . Eachbrain rfMRI was partitioned into 116 ROIs to construct 116 ×
116 connectivitymatrices where we select the upper off-diagonal triangles as feature vectorsfor the individual subjects (hypernodes).
Performance measures
We evaluated the performance of the methodsfor node classification by using the results derived from the morphologicaland functional connectomic datasets in terms of accuracy, sensitivity andspecificity. For the ADNI dataset we also averaged the results on the fourmorphological attributes to calculate an overall result.
Parameter setting
The initial hypergraph was constructed from thefeatures using k -nearest neighbors algorithm with k = 2 for the morphologicaldataset and k = 4 for the functional dataset as it has more nodes. We used20 epth = 2 for both GUNet and HUNet and pooling ratio = 0 . pooling ratio = 0 . .
01 across architectures and datasets. For HGNN, we used 2 hypergraphconvolutional layers with a dropout layer in-between at 0 . Evaluation Table. ∼ ∼
4% as shown in
Table.
3. The resultsshow that HUNet achieves a classification accuracy gain of both ∼ Model Max principal curvature Cortical thickness Sulcal depth Average curvature Overall
ACC SEN SPEC ACC SEN SPEC ACC SEN SPEC ACC SEN SPEC ACC SEN SPECGUNet (Gao and Ji, 2019) . % % 89% 66 .
7% 62% % % 67% % . % % % 80% 75% . %HGNN (Feng et al., 2019) 80% 50% % 66 .
7% 75% 57% % 78% 83% . % % % 78 .
3% 72 .
7% 81 . HUNet (ours)
80% 67% 89% % % % . % % 83% . % % % . % % 82 . Table 2:
Classification results using different views from the morphological connectomicdata. ACC: accuracy. SEN: sensitivity. SPEC: specificity.
Model Accuracy Sensitivity Specificity
GUNet (Gao and Ji, 2019) 65% 65% %HGNN (Feng et al., 2019) 66% % 43% HUNet (ours) 69 % 86% 53%
Table 3:
Classification results using the ABIDE functional connectomic data. . Discussion We have presented HUNet, a hypergraph embedding architecture for learn-ing high-order representative data embeddings that surpasses state-of-the artnetwork embedding frameworks. We proposed our embedding architecture,designed to avoid the inability of existing deep graph embedding architec-tures to learn from the many-to-many relationships between different nodes(i.e., data samples). With our proposed framework, we treated the braingraph of each patient as a node in a hypergraph structure and learned afeature embedding which recapitulates the higher-order relationships preva-lent between different subjects and used this feature embedding to classifynodes (i.e., subjects) into different brain states. Finally we demonstrated theoutperformance of our method on two different types of brain connectomicdatasets for neurological disorder diagnosis.Our experimental results showed that HUNet was able to improve onGUNet architecture by an average of 3 .
3% on the small-scale morphologicaldataset, achieving up to a 13 .
3% difference in terms of classification accuracyin the individual views as shown in (
Table.
Table. .
3% in the individual views in terms of classification accuracy(
Table.
Table.
Limitations and recommendations for future work
The parametersto optimize in the design of HUNet design architecture include the depth ofthe HUNet and the pooling ratios used in the hPool layers. Although wedemonstrated the generalizability and scalability of our method, we notethat if the depth parameter is increased too much the generalization abilityweakens, which results in over-fitting. On the other hand, a large decreasein the pooling ratio, which means lowering the number of nodes to keep ateach level of the HUNet, may result in having too few nodes to train on whencoupled with a high depth parameter –particularly for small-scale datasets.In future work, one can integrate a hyper attention mechanism to improvethe quality of our hyper pooling and unpooling layers as in graph attentionnetwork (Veliˇckovi´c et al., 2017).
Broader impact
Dimensionality reduction or sample embedding is afundamental step in many machine learning tasks such as classification, re-gression, and clustering to overcome the curse of dimensionality. Hence,learning how to embed samples into a low-dimensional space will have abroader impact in many real-world artificial intelligence applications. In this23ork, we leveraged the structure of a hypergraph to incorporate the high-order relationships existing among subsets of samples in real-life data. Infact, hypergraph representation learning has been lagging behind comparedto its graph counterpart in geometric deep learning (Bronstein et al., 2017).In this work we extended the field of hypergraph representation learningto encoder-decoder architectures through the generalization of pooling andunpooling operations to hypergraphs within a U-Net architecture.Specifically, we addressed a fundamental scientific question:
How to en-code and decode many-to-many high-order relationships present in real lifedatasets to improve predictive learning tasks?
More importantly, we rootedthe application of our proposed hypergraph encoder-decoder architecture inthe field of network neuroscience with the aim of driving precision medicineforward. Our interdisciplinary work combined three research fields: dataembedding, network neuroscience, and precision medicine, having differentsocietal and economic impacts.
First , it propelled the development of au-tomated neurological disorder diagnosis systems. This can alleviate the so-cietal burden of brain disorders by ensuring early and accurate automateddiagnosis for effective treatment. Our proposed HUNet architecture treatsthe subjects as different nodes in a hypergraph, giving insights into the dis-ordered brain alterations in neurological disorder patients, which can helptease apart variations in disorders. This will impact the future of disordered brain connectivity related treatment and diagnosis methods.
Second , the de-velopment of the field of hypergraph-based data embedding can widen thehorizon of geometric deep learning and its applications to complex sampleswith various interaction patterns. 24 . Conclusion
In this paper we proposed the HUNet architecture, where the key idea isto learn a many-to-many node embedding with a high-order feature aggre-gation rule by leveraging the structure of hypergraphs as they are able tomodel high-order interactions between subsets of nodes compared to exist-ing deep graph-based embedding methods. With this mindset, we proposedthe hypergraph pooling (hPool) and hypergraph unpooling (hUnpool) layersand generalized the U-Net architecture to hypergraphs. Using HUNet, weoutperformed state-of-the-art graph and hypergraph sample embedding ar-chitectures using brain connectome datasets of varying scales, disorders, anddistributions. Since our proposed HUNet architecture captures interactionsbetween subsets of patients as hypernodes to pool in the feature embeddingprocess, one can investigate the interpretability of the learned pooling andunpooling weights in order to group subjects with similar disordered brainalterations together. Designing an interpretable
HUNet as for generative ad-versarial networks (GANs) in (Chen et al., 2016) can help propel the field ofprecision medicine with the aim of gaining further insights into disorderedbrain alterations caused by neurological disorders and most importantly theirvariation across subsets of patients. We refer interested readers to our GitHubHUNet source code available at https://github.com/basiralab/HUNet .
6. Acknowledgements
This work was funded by generous grants from the European H2020 MarieSklodowska-Curie action (grant no. 101003403) to I.R.25 eferences
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