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 method to reduce the dimensions of resting-state functional connectivity for psychiatry disorders such as autism spectrum disorder, major depressive disorder, and schizophrenia. Their method is based on a conditional variational auto-encoder that utilized the diagnostic label. They evaluated their method on two neuroimaging datasets and obtained clustered 2-dimensional representations of multi psychiatry disorder.