Discovering the neural correlate informed nosological relation among multiple neuropsychiatric disorders through dual utilisation of diagnostic information

1 INTRODUCTION . 1.1 NOSOLOGICAL RELATION AMONG NEUROPSYCHIATRIC DISORDERS . Instead of the traditional discrete , categorical view on nosology of multiple neuropsychological disorders ( Frances , 2009 ) , an alternative dimensional , continuous view , which suggests all mental illnesses lie along a single low dimensional spectrum ( Adam , 2013 ; Casey et al. , 2013 ; Helzer et al. , 2009 ) , uprises to be a more promising perspective to view the relation among diverse disorders . Despite its commonplace in the symptom-based clinical domain ( Caspi & Moffitt , 2018 ) , it was not until recently that the finding of shared genetic components across various neuropsychiatric disorders ( Anttila et al. , 2018 ) offered the first computational study to reveal the nosological relation among various disorders . Aside from the discovered commonalities on the genetic makeup of various disorders , as the restingstate functional connectivity ( FC ) has proved itself as a valuable biomarker in distinguishing disorder patients from healthy controls in a plethora of computational studies ( Woodward & Cascio , 2015 ) , they may also hold great promise for uncovering the complex , continuous relation among different neuropsychiatric disorders . Unfortunately , due to their inherent high dimensionality , it poses a formidable data-analytic challenge in discovering the targeted nosological relation on the highdimensional neuropsychiatric FC feature space . Thence , the crux of unveiling the FC informed nosological relation rests on finding an optimal low-dimensional embedding space to be informative of high-dimensional FC feature space . 1.2 THE OBJECTIVE OF THIS RESEARCH . One straightforward approach is to apply dimensionality reduction techniques to these highdimensional FC features . However , as stated in Huys et al . ( 2016 ) , without the supplied supervision signals , over-reliance on data-driven methods may not be sufficient to derive the optimal low-dimensional embeddings of FC features . Hence , the objective of this research is to seek the utilisation of supervision signals , e.g. , diagnostic information , in learning a neural correlate informed low-dimensional embedding space , which allows us to attain the nosological relation among diverse disorders . To harness diagnostic information in learning a targeted low-dimensional embedding space , in the following sections , we chiefly propose a novel type of conditional variational auto-encoder ( VAE ) to incorporate dual utilisation of this information , which involves regularising the learning of embeddings through the introduced implicit clustering effect , and encoding the diagnostic difference among disorders . Based on the demonstrated empirical superiority of our approach in a simulation study , we implemented it in mining a consistent nosological relation among three common disorders across two curated neuropsychiatric FC datasets . In summarisation , main contributions of our work include : • putting forward a novel type of conditional VAE , which involves two utilisations of diagnostic information in learning an optimal embedding space for high dimensional FC features ; • mining a consistent nosological relation among autism spectrum disorder , major depressive disorder , and schizophrenia across two curated neuropsychiatric FC datasets . 2 PREVIOUS WORKS . Harnessing the extracted brain features , recent efforts in computational psychiatry are largely acknowledged on developing increasingly complicated classification models to mine discriminative features to aid clinical diagnosis ( Guo et al. , 2017 ; Heinsfeld et al. , 2018 ) and discovering the subtypes of a single mental illness via advanced data-driven approaches , e.g. , the multi-view clustering method ( Tokuda et al. , 2018 ) . Unfortunately , little attention was paid to unveiling complex nosological relations among diverse types of neuropsychiatric disorders based on brain correlates . Among few attempts , a worth mentioned study that shares a similar motivation with ours is the work of Xia et al . ( 2018 ) , where dimensions of psychopathology were closely tied to certain common brain features at the macro level , e.g. , resting-state networks . However , this finding – in support of the differentiation of disorders on the brain feature level – reveals little on the nosological relationship among disorders , let alone their symptom-based methodology , which does not fully harness the essence of brain features . Narrowing our focus on prevailing methodological modifications on VAE , a plethora of variants ranging from the improved generalisation , e.g. , beta-VAE ( Burgess et al. , 2018 ) , Vamp-VAE ( Tomczak & Welling , 2018 ) , the diversified generative process , e.g. , EVAE ( Bai et al. , 2019 ) , Multi-entity VAE ( Nash et al. , 2017 ) , the optimised inference ( encoding ) process , e.g. , Info-VAE ( Zhao et al. , 2017 ) , the discretised latent space , e.g. , vq-VAE ( Oord et al. , 2017 ) , to the conditional variant and its recent advents , e.g. , a semi-supervised VAE ( Kingma et al. , 2014 ) , NVAE ( Vahdat & Kautz , 2020 ) , CVAE ( Sohn et al. , 2015 ) , and the Gaussian mixture VAE ( Dilokthanakul et al. , 2016 ) . Falling into the last category , our proposed method can be regarded as a novel type of conditional VAE that embeds a clustering concept . To interested readers , we attach a detailed methodological comparison between our approach and mentioned alternative approaches in A.1 . Additionally , rather than the conventional usage in the work of Zhu et al . ( 2020 ) , our implementation of the contrastive learning is directly applied in the embedding space instead of the original feature space . 3 DUAL UTILISATION OF DIAGNOSTIC INFORMATION . To exploit diagnostic information , i.e. , the discrete diagnostic labels c ∈ C , in learning low dimensional latent variables in the embedding space z ∈ Z that are aware of both projected high dimensional FC input features x ∈ X , and their diagnostic attributes , as shown in Figure 1 , we delineate dual utilisation of diagnostic information in our proposed conditional VAE . 1 . The provided diagnostic information first serves as observable variables to regularise the learning of z in a novel type of conditional VAE , ensuring the learned ( lowdimensional ) z to be clustered around close-to-empirical cluster centers . 2 . The z associated diagnostic difference is further encoded in the embedding space via contrastive learning . 3.1 MANNER-1 : REGULARISE THE LEARNING OF z IN A NOVEL CONDITIONAL VAE Under the coinage of VAE , an amortised joint inference distribution qφ ( z , x ) ≡ qφ ( z|x ) p ( x ) is commonly employed to approximate the intractable joint generative distribution pθ ( z , x ) ≡ pθ ( x|z ) p ( z ) . Aside from the empirical distribution p ( x ) , we further assume p ( z ) as a Normal distribution , and both qφ ( z|x ) and pθ ( x|z ) are conditional Gaussian distributions in which location and scale parameters can be parameterised by complex neural networks . The optimisation objective is acknowledged as the minimisation of the K-L divergence between two joint densities of latent variable and input , i.e. , p ( z , x ) and q ( z , x ) , as : DKL { qφ ( z , x ) ||pθ ( z , x ) } . To harness provided diagnostic information , we explicitly add discrete variables c ∈ C to represent available diagnostic information ; it leads to a conditional VAE ( Figure 1 ( b ) ) , which the general optimisation objective can be derived as DKL { qφ ( z , x , c ) ||pθ ( z , x , c ) } . Crucially , to further specify the inner components of two joint densities qφ ( z , x , c ) , and pθ ( z , x , c ) , we adopt a hierarchical encoding process that admits the following derivation : q ( φz , φc ) ( c , z|x ) ∝ qφc ( c|z ) qφz ( z|x ) . In a similar vein , a hierarchical decoding process permits : p ( θz , θc ) ( x , z|c ) ∝ pθz ( x|z ) pθc ( z|c ) . As a result , the foregoing joint inference and generative distributions can be derived as : q ( φz , φc ) ( x , z , c ) ∝ qφc ( c|z ) qφz ( z|x ) p ( x ) and p ( θz , θx ) ( x , z , c ) ∝ pθx ( x|z ) pθz ( z|c ) p ( c ) , respectively . From now on , we ignore subscripts for brevity . The general optimisation objective of this novel type of conditional VAE Lcvae with specified hierarchical inference and generative processes can be rewritten into a novel format : DKL { qφ ( z , x , c ) ||pθ ( z , x , c ) } = ∑ c ∫∫ q ( c|z ) q ( z|x ) p ( x ) log q ( c|z ) q ( z|x ) p ( x ) p ( x|z ) p ( z|c ) p ( c ) dzdx = Ex∼p ( x ) [ Ez∼q ( z|x ) [ − log p ( x|z ) ] ] + Ex∼p ( x ) [ DKL ( q ( c|z ) ||p ( c ) ) + ∑ c q ( c|z ) log q ( z|x ) p ( z|c ) ] , ( 1 ) where we ignore the optimisation irrelevant term log p ( x ) ( an empirical distribution ) for brevity . With N number of classes , we define a categorical prior distribution Cat ( c|π ) , π ∈ < N+ for p ( c ) . Three different parameterised Gaussian distributions are further defined as p ( x|z ) = N ( x ; µx , I ) , q ( z|x ) = N ( z ; µx , σ2xI ) , and p ( z|c ) = N ( z ; µc , I ) , respectively , where I is the identity matrix . q ( c|z ) can be modelled by a multi-layer feedforward soft-max network to produce a categorical representation ( logits ) for each encoded latent representation . The first component in equation 1 , i.e. , Ez∼q ( z|x ) [ − log p ( x|z ) ] , can be realised as the conventional reconstruction loss in a vanilla VAE to ensure the low-dimensional latent variables encoded from high-dimensional FC features . The entire second component , i.e. , DKL ( q ( c|z ) ||p ( c ) ) +∑ c q ( c|z ) log q ( z|x ) p ( z|c ) , can be understood as the imposed ( implicit ) clustering effect in the lowdimensional embedding space . The first term in the above-mentioned second component , i.e. , DKL ( q ( c|z ) ||p ( c ) ) , serves as a preventative measure to prevent creating a dominant cluster in the face of a class-imbalanced dataset . Intuitively , it encourages the produced categorical representation ( logits ) c from q ( c|z ) to be close to its categorical prior distribution p ( c ) . However , since the K-L divergence is an intractable divergence measure , we compute an easy-to-measure alternative quantity , i.e. , the cross-entropy of two distributions H ( q ( c|z ) , p ( c ) ) in practice . Letting N to indicate the number of input classes , the second term in the second component , i.e. , ∑ c q ( c|z ) log q ( z|x ) p ( z|c ) , encourages the learned embedding space to be divided into N different subspaces to ensure each encoded latent representation can be ascribed to 1 of N clusters . Since the loss term that is on the clustering effect can be pinned down to 12 ||z − µc|| 2 ( see the derivation in attached A.2 ) , where the cluster centre µc is provided directly from the precomputed empirical cluster mean , the first utilisation of diagnostic information can be regarded as enforcing each encoded embedding z to be closer to its belonged cluster centre µc within each of N clusters .

Summary: This paper presents a new conditional variational autoencoder (VAE) approach to learn a low dimensional embedding of neuropsychiatric disorders from resting-state functional connectivity data. The proposed approach uses diagnostic information in 2 ways: 1) to cluster samples from the same disorder together via the conditional VAE model, and 2) to separate the clusters using contrastive learning.  The method is compared against other dimension reduction approaches on both a synthetic dataset and 2 real datasets which showed a consistent nosological relation among 3 disorders.