Original: D ISCOVERING THE NEURAL CORRELATE INFORMED NOSOLOGICAL RELATION AMONG MULTIPLE NEUROPSYCHIATRIC DISORDERS THROUGH DUAL UTILISATION OF DIAGNOSTIC INFORMATION
Revised: D ISCOVERING THE NEURAL CORRELATE INFORMED NOSOLOGICAL RELATION AMONG MULTIPLE NEUROPSYCHIATRIC DISORDERS THROUGH DUAL UTILISATION OF DIAGNOSTIC INFORMATION

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: The unravelled nosological relation among diverse types of neuropsychiatric disorders serves as an important precursor in advocating the dimensional approach to psychiatric classiﬁcation.
Revised: The unravelled nosological relation among diverse types of neuropsychiatric disorders serves as an important precursor in advocating the dimensional approach to psychiatric classiﬁcation.

Original: Leveraging high-dimensional abnormal resting-state functional connectivity, the crux of mining corresponded nosological relations is to derive a low-dimensional embedding space that preserves the diagnostic attributes of represented disorders.
Revised: Leveraging high-dimensional abnormal resting-state functional connectivity, the crux of mining corresponded nosological relations is to derive a low-dimensional embedding space that preserves the diagnostic attributes of represented disorders.

Original: To accomplish this goal, we seek to exploit the available diagnostic information in learning the optimal embedding space by proposing a novel type of conditional variational auto-encoder that incorporates dual utilisation of diagnostic information.
Revised: To accomplish this goal, we seek to exploit the available diagnostic information in learning the optimal embedding space by proposing a novel type of conditional variational auto-encoder that incorporates dual utilisation of diagnostic information.

Original: Encouraged by the achieved promising results in challenging the conventional approaches in low dimensional density estimation of synthetic functional connectivity features, we further implement our approach on two empirical neuropsychiatric neuroimaging datasets and discover a reliable nosological relation among autism spectrum disorder, major depressive disorder, and schizophrenia.
Revised: Encouraged by the achieved promising results in challenging the conventional approaches in low dimensional density estimation of synthetic functional connectivity features, we further implement our approach on two empirical neuropsychiatric neuroimaging datasets and discover a reliable nosological relation among autism spectrum disorder, major depressive disorder, and schizophrenia.

Original:  I NTRODUCTION 1. N OSOLOGICAL RELATION AMONG NEUROPSYCHIATRIC DISORDERS
Revised:  I NTRODUCTION 1. N OSOLOGICAL RELATION AMONG NEUROPSYCHIATRIC DISORDERS

Original: 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.
Revised: 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.

Original: 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.
Revised: 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.

Original: 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.
Revised: 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.

Original: 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.
Revised: 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.

Original: 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.
Revised: 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.

Original: 1. T HE OBJECTIVE OF THIS RESEARCH
Revised: 1. T HE OBJECTIVE OF THIS RESEARCH

Original: One straightforward approach is to apply dimensionality reduction techniques to these highdimensional FC features.
Revised: One straightforward approach is to apply dimensionality reduction techniques to these highdimensional FC features.

Original: However, as stated in Huys et al.
Revised: However, as stated in Huys et al.

Original: (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.
Revised: (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.

Original: 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.
Revised: 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.

Original: 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.
Revised: 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.

Original: 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.
Revised: 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.

Original: In summarisation, main contributions of our work include:
Revised: In summarisation, main contributions of our work include:

Original: • 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.
Revised: • 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.

Original:  P REVIOUS WORKS
Revised:  P REVIOUS WORKS

Original: 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).
Revised: 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).

Original: Unfortunately, little attention was paid to unveiling complex nosological relations among diverse types of neuropsychiatric disorders based on brain correlates.
Revised: Unfortunately, little attention was paid to unveiling complex nosological relations among diverse types of neuropsychiatric disorders based on brain correlates.

Original: Among few attempts, a worth mentioned study that shares a similar motivation with ours is the work of Xia et al.
Revised: Among few attempts, a worth mentioned study that shares a similar motivation with ours is the work of Xia et al.

Original: (2018), where dimensions of psychopathology were closely tied to certain common brain features at the macro level, e.g., resting-state networks.
Revised: (2018), where dimensions of psychopathology were closely tied to certain common brain features at the macro level, e.g., resting-state networks.

Original: 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.
Revised: 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.

Original: 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
Revised: 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

Original: 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),
Revised: 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),

Original: CVAE (Sohn et al., 2015), and the Gaussian mixture VAE (Dilokthanakul et al., 2016).
Revised: CVAE (Sohn et al., 2015), and the Gaussian mixture VAE (Dilokthanakul et al., 2016).

Original: Falling into the last category, our proposed method can be regarded as a novel type of conditional VAE that embeds a clustering concept.
Revised: Falling into the last category, our proposed method can be regarded as a novel type of conditional VAE that embeds a clustering concept.

Original: To interested readers, we attach a detailed methodological comparison between our approach and mentioned alternative approaches in A.1.
Revised: To interested readers, we attach a detailed methodological comparison between our approach and mentioned alternative approaches in A.1.

Original: Additionally, rather than the conventional usage in the work of Zhu et al.
Revised: Additionally, rather than the conventional usage in the work of Zhu et al.

Original: (2020), our implementation of the contrastive learning is directly applied in the embedding space instead of the original feature space.
Revised: (2020), our implementation of the contrastive learning is directly applied in the embedding space instead of the original feature space.

Original:  D UAL UTILISATION OF DIAGNOSTIC INFORMATION
Revised:  D UAL UTILISATION OF DIAGNOSTIC INFORMATION

Original: 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.
Revised: 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.

Original: 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.
Revised: 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.

Original: The z associated diagnostic difference is further encoded in the embedding space via contrastive learning .
Revised: The z associated diagnostic difference is further encoded in the embedding space via contrastive learning .

Original: 3. M ANNER -1: REGULARISE THE LEARNING OF z IN A NOVEL CONDITIONAL VAE
Revised: 3. M ANNER -1: REGULARISE THE LEARNING OF z IN A NOVEL CONDITIONAL VAE

Original: Under the coinage of VAE an amortised joint inference distribution q φ ( z x ) ≡ q φ ( z | x ) p ( x ) is commonly employed to approximate the intractable joint generative distribution p θ ( z, x ) ≡ p θ ( x | z ) p ( z ) .
Revised: Under the coinage of VAE an amortised joint inference distribution q φ ( z, x ) ≡ q φ ( z | x ) p ( x ) is commonly employed to approximate the intractable joint generative distribution p θ ( z, x ) ≡ p θ ( x | z ) p ( z ) .

Original: Aside from the empirical distribution p ( x ), we further assume p ( z ) as a Normal distribution and both q φ ( z | x ) and p θ ( x | z ) are conditional Gaussian distributions in which location and scale parameters can be parameterised by complex neural networks.
Revised: Aside from the empirical distribution p ( x ) we further assume p ( z ) as a Normal distribution and both q φ ( z | x ) and p θ ( x | z ) are conditional Gaussian distributions in which location and scale parameters can be parameterised by complex neural networks.

Original: 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: D KL { q φ ( z, x ) ||
Revised: 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: D KL { q φ ( z, x ) ||

Original: p
Revised: p

Original: θ ( z, x ) } .
Revised: θ ( z, x ) } .

Original: 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 D KL { q φ ( z, x, c )
Revised: 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 D KL { q φ ( z, x, c )

Original: || p θ
Revised: || p θ

Original: ( z, x, c ) } .
Revised: ( z, x, c ) } .

Original: Crucially, to further specify the inner components of two joint densities q φ ( z, x, c ) , and p θ ( z, x, c ) , we adopt a hierarchical encoding process that admits the following derivation: q ( φ z , φ c ) ( c, z | x ) ∝ q φ c ( c | z ) q φ z ( z | x ) .
Revised: Crucially, to further specify the inner components of two joint densities q φ ( z, x, c ) , and p θ ( z, x, c ) , we adopt a hierarchical encoding process that admits the following derivation: q ( φ z ,φ c ) ( c, z | x ) ∝ q φ c ( c | z ) q φ z ( z | x ) .

Original: In a similar vein, a hierarchical decoding process permits: p ( θ z , θ c ) ( x, z | c ) ∝ p θ z ( x | z ) p θ c ( z | c ) .
Revised: In a similar vein, a hierarchical decoding process permits: p ( θ z ,θ c ) ( x, z | c ) ∝ p θ z ( x | z ) p θ c ( z | c ) .

Original: As a result, the foregoing joint inference and generative distributions can be derived as: q ( φ z , φ c ) ( x, z, c ) ∝ q φ c ( c | z ) q φ z ( z | x ) p ( x ) and p ( θ z , θ x ) ( x, z, c ) ∝ p θ x ( x | z ) p θ z ( z | c ) p ( c ) , respectively.
Revised: As a result, the foregoing joint inference and generative distributions can be derived as: q ( φ z ,φ c ) ( x, z, c ) ∝ q φ c ( c | z ) q φ z ( z | x ) p ( x ) and p ( θ z ,θ x ) ( x, z, c ) ∝ p θ x ( x | z ) p θ z ( z | c ) p ( c ) , respectively.

Original: From now on, we ignore subscripts for brevity.
Revised: From now on, we ignore subscripts for brevity.

Original: The general optimisation objective of this novel type of conditional
Revised: The general optimisation objective of this novel type of conditional

Original: VAE L cvae with speciﬁed hierarchical inference and generative processes can be rewritten into a novel format:
Revised: VAE L cvae with speciﬁed hierarchical inference and generative processes can be rewritten into a novel format:

Original: [Equation]
Revised: [Equation]

Original: where we ignore the optimisation irrelevant term log p ( x ) (an empirical distribution) for brevity.
Revised: where we ignore the optimisation irrelevant term log p ( x ) (an empirical distribution) for brevity.

Original: With N number of classes, we deﬁne a categorical prior distribution Cat ( c | π ) , π ∈ ℜ N + for p ( c ) .
Revised: With N number of classes, we deﬁne a categorical prior distribution Cat ( c | π ) , π ∈ (cid:60) N + for p ( c ) .

Original: Three different parameterised Gaussian distributions are further deﬁned as p ( x | z ) =
Revised: Three different parameterised Gaussian distributions are further deﬁned as p ( x | z ) =

Original: N ( x ; µ x , I ) , q ( z | x ) =
Revised: N ( x ; µ x , I ) , q ( z | x ) =

Original: N ( z ; µ x , σ 2 x I ) , and p ( z | c ) =
Revised: N ( z ; µ x , σ 2 x I ) , and p ( z | c ) =

Original: N ( z ; µ c , I ) , respectively, where I is the identity matrix.
Revised: N ( z ; µ c , I ) , respectively, where I is the identity matrix.

Original: 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.
Revised: 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.

Original: The ﬁrst component in equation 1, i.e., E z ∼ q ( z | x ) [ − log p ( x | z )] , can be realised as the conven- tional reconstruction loss in a vanilla VAE to ensure the low-dimensional latent variables encoded from high-dimensional FC features.
Revised: The ﬁrst component in equation 1, i.e., E z ∼ 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.

Original: The entire second component, i.e., D KL ( q ( c | z ) || p ( c ))
Revised: The entire second component, i.e., D KL ( q ( c | z ) || p ( c ))

Original: + % 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.
Revised: + (cid:80) 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.

Original: The ﬁrst term in the above-mentioned second component, i.e., D KL ( q ( c | z ) || p ( c )) , serves as a preventative measure to prevent creating a dominant cluster in the face of a class-imbalanced dataset.
Revised: The ﬁrst term in the above-mentioned second component, i.e., D KL ( q ( c | z ) || p ( c )) , serves as a preventative measure to prevent creating a dominant cluster in the face of a class-imbalanced dataset.

Original: Intuitively, it encourages the produced categorical representation (logits) c from q ( c | z ) to be close to its categorical prior distribution p ( c ) .
Revised: Intuitively, it encourages the produced categorical representation (logits) c from q ( c | z ) to be close to its categorical prior distribution p ( c ) .

Original: 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.
Revised: 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.

Original: 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.
Revised: Letting N to indicate the number of input classes, the second term in the second component, i.e., (cid:80) 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.

Original: Since the loss term that is on the clustering effect can be pinned down to 12 || z − µ c
Revised: Since the loss term that is on the clustering effect can be pinned down to 12 || z − µ c

Original: || 2 (see the derivation in attached A.2), where the cluster centre µ c 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
Revised: || 2 (see the derivation in attached A.2), where the cluster centre µ c 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

Original: µ c within each of N clusters.
Revised: µ c within each of N clusters.

Original: 3. M ANNER -2: ENCODING DIAGNOSTIC DIF FERENCE IN THE EMBEDDING SPACE
Revised: 3. M ANNER -2: ENCODING DIAGNOSTIC DIFFERENCE IN THE EMBEDDING SPACE

Original: Corresponding to the second manner of utilising diagnostic information, i.e., embedding the diagnostic difference between disorder phenotypes, we resort to contrastive learning (Hadsell et al., 2006) to explicitly model this diagnostic difference.
Revised: Corresponding to the second manner of utilising diagnostic information, i.e., embedding the diagnostic difference between disorder phenotypes, we resort to contrastive learning (Hadsell et al., 2006) to explicitly model this diagnostic difference.

Original: As shown in Figure 1(c), we deﬁne a pair of latent vectors −→ z 1 , −→ z 2 ∈ Z to represent two encoded high dimensional brain features, a distance function D to measure the dissimilarity of two latent vectors, and a binary class indicator C = 0 , 1 to verify whether latent vectors come from the same disorder phenotype.
Revised: As shown in Figure 1(c), we deﬁne a pair of latent vectors −→ z 1 , −→ z 2 ∈ Z to represent two encoded high dimensional brain features, a distance function D to measure the dissimilarity of two latent vectors, and a binary class indicator C = 0 , 1 to verify whether latent vectors come from the same disorder phenotype.

Original: The contrastive loss function
Revised: The contrastive loss function

Original: L clr (Hadsell et al., 2006) in our case can be deﬁned as:
Revised: L clr (Hadsell et al., 2006) in our case can be deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: where m is a pre-deﬁnable margin to weigh the contribution of dissimilar pairs, and we adopt the simple euclidean distance to capture D ( −→ z 
Revised: where m is a pre-deﬁnable margin to weigh the contribution of dissimilar pairs, and we adopt the simple euclidean distance to capture D ( −→ z 1 , −→

Original: [Equation]
Revised: [Equation]

Original: i
Revised: 2 ) , i.e., D (

Original: [Equation]
Revised: [Equation]

Original: e. [Equation]
Revised: 1 , −→ z 2 ) =

Original: [Equation] || −→ [Equation]{+||−→ z 1 −−→ z 
+}
Revised: {+||−→ z 1 −−→ z 
+}

Original: z 2 || 2 .
Revised: || 2 .

Original: To preserve the authentic diagnostic differences among encoded latent variables, the m is set to 0 , i.e., we weigh the equal importance of dissimilar and similar pairs.
Revised: To preserve the authentic diagnostic differences among encoded latent variables, the m is set to 0 , i.e., we weigh the equal importance of dissimilar and similar pairs.

Original: Importantly, bridging the two loss terms, i.e., L clr and L cvae together, an overall loss function L total through a convex combination of two terms can be derived as:
Revised: Importantly, bridging the two loss terms, i.e., L clr and L cvae together, an overall loss function L total through a convex combination of two terms can be derived as:

Original: [Equation]
Revised: [Equation]

Original: where we enforce the equal contribution of two loss terms, i.e., λ = 0 .
Revised: where we enforce the equal contribution of two loss terms, i.e., λ = 0 .

Original: 5 , to stress the equal importance of two manners of utilising the diagnostic information in deriving our targeted embedding space.
Revised: 5 , to stress the equal importance of two manners in utilising the diagnostic information to derive our targeted embedding space.

Revised: In attached A.3, we show the effects of differentiated λ in shaping the embedding space for interested readers.

Original:  A SSESSING THE IMPORTANCE OF DIAGNOSTIC INFORMATION
Revised:  A SSESSING THE IMPORTANCE OF DIAGNOSTIC INFORMATION

Original: To assess the importance of diagnostic information in deriving the optimal low-dimensional embedding of neuropsychiatric FC features, we implemented our proposed method in a simulation study, along with the ground truth embeddings.
Revised: To assess the importance of diagnostic information in deriving the optimal low-dimensional embedding of neuropsychiatric FC features, we implemented our proposed method in a simulation study, along with the ground truth embeddings.

Original: The purpose of this simulation study is twofold.
Revised: The purpose of this simulation study is twofold.

Original: (1) We seek to assess the importance of diagnostic information in recovering ground truth embeddings.
Revised: (1) We seek to assess the importance of diagnostic information in recovering ground truth embeddings.

Original: We also aim to demonstrate the empirical superiority of our proposed approach over several existing supervised representation learning approaches.
Revised: We also aim to demonstrate the empirical superiority of our proposed approach over several existing supervised representation learning approaches.

Original: In preparing for the synthetic high-dimensional FC features, we strictly follow the following threestep generation procedure.
Revised: In preparing for the synthetic high-dimensional FC features, we strictly follow the following threestep generation procedure.

Original: We ﬁrstly pre-train a U-map projection function F on a real FC dataset, e g., the SRPBS multi-disorder brain connectivity database (Yamashita et al., 2019).
Revised: We ﬁrstly pre-train a U-map projection function F on a real FC dataset, e.g., the SRPBS multi-disorder brain connectivity database (Tanaka et al., 2021).

Original: 300 low dimensional embeddings are then sampled from three 2D multivariate Gaussians N 2 (0 . 8 , 1); N 2 (0 . 4 , 1); N 2 (0 . 2 , 1) .
Revised: 300 low dimensional embeddings are then sampled from three 2D multivariate Gaussians N 2 (0 . 8 , 1); N 2 (0 . 4 , 1); N 2 (0 . 2 , 1) .

Original: The pair-wise distance relations among three 2D Gaussians are served as the ground truth relation (see Figure 2).
Revised: The pair-wise distance relations among three 2D Gaussians are served as the ground-truth relation (see Figure 2).

Original: The inverted projection function F − 1 is further utilised to map the foregoing 300 embeddings from the low-dimensional (2D) ground truth embedding space into the highdimensional FC space.
Revised: The inverted projection function F − 1 is further utilised to map the foregoing 300 embeddings from the low-dimensional (2D) ground truth embedding space into the highdimensional FC space.

Original: In the case of the SRPBS dataset, this produces a [300 , 9730] synthetic FC feature matrix.
Revised: In the case of the SRPBS dataset, this produces a [300 , 9730] synthetic FC feature matrix.

Original: 4. MODELS AND THEIR IMPLEMENTATIONS 4.1. W ITHIN OUR APPROACH
Revised: M 4.1. W ITHIN OUR APPROACH

Original: The crux of implementing our proposed approach is to compute the ﬁnal loss function equationwith its involved loss terms L cvae in equation 1, and L clr in equation 2.
Revised: The crux of implementing our proposed approach is to compute the ﬁnal loss function equationwith its involved loss terms L cvae in equation 1, and L clr in equation 2.

Original: Aside from the easy computed reconstruction loss, in the remaining components of equation 1, i.e., 
Revised: Aside from the easy computed reconstruction loss, in the remaining components of equation 1, i.e., (cid:80) c

Original: [Equation]
Revised: q ( c | z ) log q ( z | x ) p ( z | c ) +

Original: D KL ( q ( c | z ) || p ( c )) , the parameters that are pertaining to the optimisation of 
Revised: D KL ( q ( c | z ) || p ( c )) , the parameters that are pertaining to the optimisation of (cid:80) c

Original: [Equation] can be reduced to σ 2 x , µ x , z , and µ c .
Revised: q ( c | z ) log q ( z | x ) p ( z | c ) can be reduced to σ 2 x , µ x , z , and µ c .

Original: As σ x and µ x can be parameterised by a feed-forward neural network, it allows the production of z via the re-parameteristion trick Kingma & Welling (2013), i.e., z = $ & σ x + µ x , $ ∼ N (0 , 1) .
Revised: As σ x and µ x can be parameterised by a feed-forward neural network, it allows the production of z via the re-parameteristion trick Kingma & Welling (2013), i.e., z = (cid:15) (cid:78) σ x + µ x , (cid:15) ∼ N (0 , 1) .

Original: The cluster-wise mean µ c can be produced by a simple one layer
Revised: The cluster-wise mean µ c can be produced by a simple one layer

Original: Gaussian network with supplied supervision signals, e.g., cluster labels in this simulation study.
Revised: Gaussian network with supplied supervision signals, e.g., cluster labels in this simulation study.

Original: A simple softmax network is sufﬁcient to measure the KL-divergence between q ( c | z ) and p ( c ) .
Revised: A simple softmax network is sufﬁcient to measure the KL-divergence between q ( c | z ) and p ( c ) .

Original: The second loss term L clr in equation 2 can be easily captured via adding a margin (distance) based feed-forward neural network on top of encoded latent representations.
Revised: The second loss term L clr in equation 2 can be easily captured via adding a margin (distance) based feed-forward neural network on top of encoded latent representations.

Original: Essentially, to demonstrate the importance of dual utilisation of diagnostic information in our proposed approach, corresponding to above-mentioned two utilisations, three instantiations of our approach are established ranging from the full implementation of our model ( Full model ) to the mere implementation of L cvae ( Manner-1 model ), and L clr ( Manner-2 model ).
Revised: Essentially, to demonstrate the importance of dual utilisation of diagnostic information in our proposed approach, corresponding to above-mentioned two utilisations, three instantiations of our approach are established ranging from the full implementation of our model ( Full model ) to the mere implementation of L cvae ( Manner-1 model ), and L clr ( Manner-2 model ).

Original: Their layerwise conﬁgurations and respective parameter estimation processes are concisely summarised in attached B.1.
Revised: Their layerwise conﬁgurations and respective parameter estimation processes are concisely summarised in attached B.1.

Original: 4.1. A LTERNATIVE APPROACHES
Revised: 4.1. A LTERNATIVE APPROACHES

Original: Three categories of alternative methods are also included in this simulation study, ranging from the conventional semi-supervised VAE ( S-VAE ) (Kingma et al., 2014), the mainstream manifold learning based dimensionality reduction approach, e.g., supervised Umap ( S-Umap ) (McInnes et al.,
Revised: Three categories of alternative methods are also included in this simulation study, ranging from the conventional semi-supervised VAE ( S-VAE ) (Kingma et al., 2014), the mainstream manifold learning based dimensionality reduction approach, e.g., supervised Umap ( S-Umap ) (McInnes et al.,

Original: , to the metric learning based neighbourhood component analysis ( NCA ) (Goldberger et al., 2004).
Revised: , to the metric learning based neighbourhood component analysis ( NCA ) (Goldberger et al., 2004).

Original: The implementation details of these alternative approaches are also demonstrated in attached
Revised: The implementation details of these alternative approaches are also demonstrated in attached

Original: B.1.
Revised: B.1.

Original: 4. P ERFORMANCE EVALUATION
Revised: 4. P ERFORMANCE EVALUATION

Original: As shown in Figure 2, provided with ground truth embeddings and their pairwise distance relations, we target on evaluating the performance of various models regarding the capability of recovering the ground truth embeddings from synthetic high-dimensional FC features.
Revised: As shown in Figure 2, provided with ground truth embeddings and their pairwise distance relations, we target on evaluating the performance of various models regarding the capability of recovering the ground truth embeddings from synthetic high-dimensional FC features.

Original: The evaluation metrics include the qualitative visual inspection of yielded embedding spaces, and a quantitative assessment to verdict whether the distance relations among estimated clusters are preserved.
Revised: The evaluation metrics include the qualitative visual inspection of yielded embedding spaces, and a quantitative assessment to verdict whether the distance relations among estimated clusters are preserved.

Original: The ground truth distance relation among three clusters can be expressed as the cosine distance-based inequality:
Revised: To quantify the between-cluster distance on the derived low dimensional embedding space, we rely on the pair-wise Wasserstein distance-based Frechet inception distance (FID) based metric (Heusel et al., 2017), i.e., FID ( µ 1 , Σ 1 ,µ 1 , Σ 2 ) , over the conventional L − 1 and L − 2 distance metrics for its consideration on both ﬁrst and second degree statistics in measuring the discrepancy between two multivariate Gaussians. Its computation can be deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: ) , where D is the simple cosine distance between two cluster centres.
Revised: where µ ; Σ are the ﬁrst and second degree statistics of inferred low-dimensional multivariate Gaussians, i.e., z ∼ N ( µ, Σ) . On the basis of suggested FID metric to assess the between-cluster relation, the provided ground-truth relation among three clusters can be expressed into the three-way inequality: [Equation]

Original: Juxtaposed in demonstrated Figure 2, among three instantiations of our approach, the maximum utilisation of diagnostic information, i.e., the Full model , achieved the most promising result on recovering the correct distance relation among three instantiations of our approach.
Revised: Juxtaposed in demonstrated Figure 2, among three instantiations of our approach, the maximum utilisation of diagnostic information, i.e., the Full model , achieved the most promising result on recovering the correct distance relation among three instantiations of our approach, along with the assessed signiﬁcant pair -wist

Original:  This observation substantiates the importance of diagnostic information in deriving an optimal low-dimensional space for high-dimensional FC like features.
Revised: t -tests among three between-cluster distances in our Full model (see attached C). This observation, in conjunction with the consistent result in the second simulation study (see attached D), greatly substantiates the importance of diagnostic information in deriving an optimal low-dimensional space for high-dimensional FC like features.

Original: The superiority of our approach over included alternative ones are presented less-distorted low-dimensional embedding space and recovered accurate distance matrix.
Revised: The superiority of our approach over included alternative ones are presented less-distorted low-dimensional embedding space and recovered accurate distance matrix.

Original: This observed empirical superiority of our approach assures the further implementation on real neuropsychiatric FC datasets.
Revised: This observed empirical superiority of our approach assures the further implementation on real neuropsychiatric FC datasets.

Original:  M INING THE NOSOLOGICAL RELATION AMONG MULTIPLE DISORDERS
Revised:  M INING THE NOSOLOGICAL RELATION AMONG MULTIPLE DISORDERS

Original: After demonstrating the importance of diagnostic information in learning low-dimensional embeddings of FC features, we seek to implement our approach, i.e., the Full model , on curated neuropsychiatric FC datasets to explore nosological relations among disorder phenotypes.
Revised: After demonstrating the importance of diagnostic information in learning low-dimensional embeddings of FC features, we seek to implement our approach, i.e., the Full model , on curated neuropsychiatric FC datasets to explore nosological relations among disorder phenotypes.

Original: In short, high-dimensional FC features of included neuropsychiatric disorders are ﬁrstly projected into a common low-dimensional (2D) embedding space, where their pairwise distances are served as the nosological relation in this embedding space.
Revised: In short, high-dimensional FC features of included neuropsychiatric disorders are ﬁrstly projected into a common low-dimensional (2D) embedding space, where their pairwise distances are served as the nosological relation in this embedding space.

Original: 5. D ATASETS & M ETRICS
Revised: 5. D ATASETS & M ETRICS

Original: To acquire a high-quality neuropsychiatric FC dataset that contains multiple types of disorders is a time-consuming task, which demands expertise in diagnosis, and consistency in neuroimaging scanning protocols.
Revised: To acquire a high-quality neuropsychiatric FC dataset that contains multiple types of disorders is a time-consuming task, which demands expertise in diagnosis, and consistency in neuroimaging scanning protocols.

Original: The SRPBS multi-disorder brain connectivity database (Yamashita et al., 2019)that pools the fMRI scans of diverse neuropsychiatric patients from 8 different scanning sites with consistent diagnosis guidelines, perfectly serves as our research database.
Revised: The SRPBS multi-disorder brain connectivity database (Tanaka et al., 2021)that pools the fMRI scans of diverse neuropsychiatric patients from 8 different scanning sites with consistent diagnosis guidelines, perfectly serves as our research database.

Original: UTO dataset From the opted SRPBS database, two datasets were formed to suit our need in studying the nosological relation among multiple neuropsychiatric disorders.
Revised: UTO dataset From the opted SRPBS database, two datasets were formed to suit our need in studying the nosological relation among multiple neuropsychiatric disorders.

Original: The primary dataset comes from the data that were acquired at the University of Tokyo Hospital (AKA.
Revised: The primary dataset comes from the data that were acquired at the University of Tokyo Hospital (AKA.

Original: , UTO dataset), which is comprised of fMRI scans from 3 chosen types of clearly diagnosed disorders: autism spectrum disorder (ASD; 10 subjects), major depressive disorder (MDD; 62 subjects), and schizophrenia (SCZ; 35 subjects).
Revised: , UTO dataset), which is comprised of fMRI scans from 3 chosen types of clearly diagnosed disorders: autism spectrum disorder (ASD; 10 subjects), major depressive disorder (MDD; 62 subjects), and schizophrenia (SCZ; 35 subjects).

Original: These fMRI scans were acquired in the same facility under the identical scanning protocol, which is free from the common threat of site difference in most fMRI studies.
Revised: These fMRI scans were acquired in the same facility under the identical scanning protocol, which is free from the common threat of site difference in most fMRI studies.

Revised: The brief introduction of the SRPBS database can be found in attached E.

Original: The subject-wise brain features, i.e., functional connectivities, were obtained from 10-min restingstate fMRI BOLD signals that underwent the identical preprocessing steps.
Revised: The subject-wise brain features, i.e., functional connectivities, were obtained from 10-min restingstate fMRI BOLD signals that underwent the identical preprocessing steps.

Original: To compute the subjectwise functional connectivities from these preprocessed BOLD signals, the BAS parcellation scheme (Riviere et al., 2002) and the standard Pearson R correlation method were adopted for time series extraction, and the computation of connectivity matrix 2 , respectively.
Revised: To compute the subjectwise functional connectivities from these preprocessed BOLD signals, the Brainvisa Sulci Atlas parcellation scheme, which each individual image was divided into 140 regions (Perrot et al., 2011) and the standard Pearson R correlation method were adopted for time series extraction, and the computation of connectivity matrix 2 , respectively.

Original: Hence, for each subject,connectivity features were crafted to serve as high dimensional brain features, forming a [107 , 9730] brain feature matrix.
Revised: Hence, for each subject, 9730 connectivity features were crafted to serve as high dimensional brain features, forming a [107 , 9730] brain feature matrix.

Original: HuShoWa dataset Aside from the main UTO dataset, an independent validation dataset from the SRPBS database were further curated from the Hiroshima University and ShoWa University (HuShoWa dataset).
Revised: HuShoWa dataset Aside from the main UTO dataset, a replication dataset from the SRPBS database were further curated from the Hiroshima University and ShoWa University (HuShoWa dataset).

Original: A total of 307 subjects were included in our HuShoWa dataset, occupying categories of MDD ( 173 subjects), ASD ( 115 subjects), and SCZ ( 19 subjects).
Revised: A total of 307 subjects were included in our HuShoWa dataset, occupying categories of MDD ( 173 subjects), ASD ( 115 subjects), and SCZ ( 19 subjects).

Original: Underwent the identical preprocessing steps, and the feature extraction process, we harvested a [307 , 9730] FC feature matrix.
Revised: Underwent the identical preprocessing steps, and the feature extraction process, we harvested a [307 , 9730] FC feature matrix.

Original: To minimise the effect of site-difference on acquired fMRI scans in HuShoWa dataset, we resorted to the harmonisation technique, e.g., the combat approach (Yu et al., 2018), to correct site-difference in our input features.
Revised: To minimise the effect of site-difference on acquired fMRI scans in HuShoWa dataset, we resorted to the harmonisation technique, e.g., the combat approach (Yu et al., 2018) to correct site-difference in our input features.

Original: Validation scheme & Evaluation metric Since empirical neuropsychiatric FC features are seldom accompanied with ground truth low-dimensional embeddings, it is essential to obtain a strict validation scheme to ensure the reliability of our attained nosological relations.
Revised: Validation scheme & performance assessment Since empirical neuropsychiatric FC features are seldom accompanied with ground truth low-dimensional embeddings, it is essential to obtain a strict validation scheme to ensure the reliability of our attained nosological relations.

Original: For this, we opt for 10-fold cross validation scheme that demands 10 times running of our model on two empirical datasets.
Revised: For this, we opt for 10-fold cross validation scheme that demands 10 times running of our model on two empirical datasets.

Original: To quantify the targeted nosological relation among diverse neuropsychiatric phenotypes on the derived low dimensional embedding space, we rely on the pair-wise Wasserstein distance-based Frechet inception distance (FID) based metric (Heusel et al., 2017), i.e., FID ( µ 1 , Σ 1 ,µ 1 , Σ 2 ) , over the conventional L − 1 and L − 2 distance metrics for its consideration on both ﬁrst and second degree statistics in measuring the discrepancy between two multivariate Gaussians. Its computation can be deﬁned as:
Revised: In consist with the previous harnessed FID metric in assessing the between-cluster relation, in both UTO and HuShoWa datasets, we applied the FID metric to compute the cluster-wise

Original: [Equation] where µ ; Σ are the ﬁrst and second degree statistics of inferred low-dimensional multivariate Gaussians, i.e., z ∼ N ( µ, Σ ) .
Revised: FID distance on the basis of yielded low-dimensional embeddings of three disorders, e.g.,

Original: On the basis of computed pair-wise FID between embeddings of three disorders, e.g., FID ( ASD − MDD ) ; FID ( ASD − SCZ ) ; FID ( MDD − SCZ ) , their inequality may inform us the nosological relation among ASD, MDD, and SCZ.
Revised: FID ( ASD − MDD ) ; FID ( ASD − SCZ ) ; FID ( MDD − SCZ ) , their inequality may inform us the noso- logical relation among ASD, MDD, and SCZ.

Original: Over the implemented 10-fold cross validation, multiple paired t-tests in testing the statistical signiﬁcance of these pair-wise FID distances were also conducted to assess the reliability of attained nosological relations.
Revised: Over the implemented 10-fold cross validation, multiple paired t-tests in testing the statistical signiﬁcance of these pair-wise FID distances were also conducted to assess the reliability of attained nosological relations.

Original: 5. D ISCOVERED NOSOLOGICAL RELATIONSHIPS AMONG ASD, MDD AND SCZ
Revised: 5. D ISCOVERED NOSOLOGICAL RELATIONSHIPS AMONG ASD, MDD AND SCZ

Original: As shown in Figure 3 (left panel), on low-dimensional embedding spaces that are derived from the
Revised: As shown in Figure 3 (left panel), on low-dimensional embedding spaces that are derived from the

Original: UTO and HuShoWa datasets, with maximum utilisation of diagnostic information, the learned embeddings are compact, clear-cut
Revised: UTO and HuShoWa datasets, with maximum utilisation of diagnostic information, the learned embeddings are compact, clear-cut

Original: clustered in accordance with their represented disorder phenotypes.
Revised: clustered in accordance with their represented disorder phenotypes.

Original: In respect of the attained highly consistent embedding spaces (Figure 3 left panel), two similar FID matrices (Figure 3 middle panel), and high statistical power in conducted paired t-tests (Figure 3 right panel) across two curated neuropsychiatric datasets, it unveils us a consistent three-way FID based inequality among  ASD, MDD, and SCZ: FID ( ASD − MDD ) 
Revised: In respect of the attained highly consistent embedding spaces (Figure 3 left panel), two similar FID matrices (Figure 3 middle panel), and high statistical power in conducted paired t-tests (Figure 3 right panel) across two curated neuropsychiatric datasets, it unveils us an interestingly between-phenotype relation among the considered ASD, MDD and SCZ, i.e., FID ( ASD − MDD ) >

Original: > FID ( ASD − SCZ )  > FID ( MDD − SCZ ) .
Revised: FID ( ASD − SCZ ) and FID ( ASD − MDD ) > FID ( MDD − SCZ ) .

Original: This suggests that ASD and MDD may belong to two independent nosology entities, which are loosely connected regarding their large differences in corresponded neural correlates (functional connectivities), whereas the closely linked ASD-SCZ, MDD-SCZ nexuses indicate the potential ’nosological closeness’ between SCZ and the other two disorder phenotypes, which is in line with the previous ﬁndings on high rates of comorbidity on SCZ-ASD (Wood, 2017) and SCZ-MDD (Tsai & Rosenheck, 2013).
Revised: This suggests that ASD and MDD may belong to two independent nosology entities, which are loosely connected regarding their large differences in corresponding neural correlates (functional connectivities), whereas the closely linked ASD-SCZ, MDD-SCZ nexuses indicate the potential ’nosological closeness’ between SCZ and the other two disorder phenotypes, which is in line with the previous ﬁndings on high rates of comorbidity on SCZ-ASD (Wood, 2017) and SCZ-MDD (Tsai & Rosenheck, 2013).

Original: Based on this ﬁnding, we are able to align these disorder phenotypes on an FC informed dimensional coordinate, allowing ASD and MDD to situate at two ends of this coordinate, in correspondence with two domains of psychopathology, i.e., the neurodevelopmental and affective pathology, respectively (see attached D)
Revised: Based on this ﬁnding, we are able to align these disorder phenotypes on an FC informed dimensional coordinate, allowing ASD and MDD to situate at two ends of this coordinate, in correspondence with two domains of psychopathology, i.e., the neurodevelopmental and affective pathology, respectively (see attached G)

Original: (Craddock & Owen, 2010).
Revised: (Craddock & Owen, 2010).

Original: 5. N EUROSCIENTIF IC INSIGHT
Revised: 5. N EUROSCIENTIFIC INSIGHT

Original: Furthermore, since the ﬁnal projected FCs were represented as two-dimensional features, it is of increasing research interest to see whether these two-dimensional features are in agreement with existing disorder related connectivities in the brain space.
Revised: Furthermore, since the ﬁnal projected FCs were represented as two-dimensional features, it is of increasing research interest to see whether these two-dimensional features are in agreement with existing disorder related connectivities in the brain space.

Original: For this,  we have projected each ’cluster centre’ (the most representative feature in each of three clusters) back into the high dimensional FC space, and tentatively identiﬁed a signiﬁcant connection between Caudate and Cuneus that obtains the highest impact in discriminating three disorders (MDD, SCZ, ASD) among the overall 9730 FCs.
Revised: For this, harnessing the layer-wise relevance propagation method (Montavon et al., 2019), we have projected each ’cluster centre’ (the most representative feature in each of three clusters) back into the high dimensional FC space, and tentatively identiﬁed a signiﬁcant connection between Caudate and Cuneus that obtains the highest relevance in discriminating three disorders (MDD, SCZ, ASD) among the overall 9730 FCs.

Original: However, the validity of this preliminary neuroscientiﬁc insight on involved functional connectivity is under the threat of used nonlinear projections (multiple ReLU connections in both inference and generative networks), where the isomorphism between the low-dimensional embedding and high-dimensional FC space is not guaranteed.
Revised: However, the validity of this preliminary neuroscientiﬁc insight on involved functional connectivity is under the threat of used nonlinear projections (multiple ReLU connections in both inference and generative networks), where the isomorphism between the low-dimensional embedding and high-dimensional FC space is not guaranteed.

Original: Lastly, to assess the interpretability of learned low-dimensional embedding space in our context, we focus on probing the potential association between learned embeddings and measured clinical or psychological assessments.
Revised: Lastly, to assess the interpretability of learned low-dimensional embedding space in our context, we focus on probing the potential association between learned embeddings and measured clinical or psychological assessments.

Original: On the basis of provided ASD related autism spectrum quotient (AQ score) (Baron-Cohen et al., 2001), and MDD related Beck Depression Inventory (BDI score) (Beck et al., 1996) in the HuShoWa dataset, the associations between our derived embeddings and the clinical factors are captured, i.e., AQ score: ρ = 0 . 533 , p < 0 .
Revised: On the basis of provided ASD related autism spectrum quotient (AQ score) (Baron-Cohen et al., 2001), and MDD related Beck Depression Inventory (BDI score) (Beck et al., 1996) in the HuShoWa dataset, the associations between our derived embeddings and the clinical factors are captured, i.e., AQ score: ρ = 0 . 533 , p < 0 .

Original: 001 and BDI score: ρ = 0 .
Revised: 001 and BDI score: ρ = 0 .

Original: 540 , p < 0 .
Revised: 540 , p < 0 .

Original: 001 , suggesting the potential clinical awareness of derived low-dimensional embeddings.
Revised: 001 , suggesting the potential clinical awareness of derived low-dimensional embeddings.

Original:  L IMITATIONS
Revised:  L IMITATIONS

Original: Two limitations of this study are the conﬁned capability of our approach in discerning a novel subtype of a disorder or the nosological overlap between two disorders, and the scarcity of high-quality independent FC datasets that are curated other than the harnessed SRPBS database.
Revised: Two limitations of this study are the conﬁned capability of our approach in discerning a novel subtype of a disorder or the nosological overlap between two disorders, and the scarcity of high-quality independent FC datasets that are curated other than the harnessed SRPBS database.

Original: The former limitation may be ameliorated through investigating the ﬁner (possibly hierarchical) structure within low-dimensional embeddings of known phenotypes, whereas the latter one may be tackled by the future release of the high-quality imaging datasets from the Brain/Minds beyond MRI project (Koike et al., 2021).
Revised: The former limitation may be ameliorated through investigating the ﬁner (possibly hierarchical) structure within low-dimensional embeddings of known phenotypes, whereas the latter one may be tackled by the future release of the high-quality imaging datasets from the Brain/Minds beyond MRI project (Koike et al., 2021).

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: Unveiling the complex nosological relation among diverse neuropsychiatric disorders with the help of diagnostic information, we advocate a neural correlate based approach, which targets on deriving a neural correlate informed low-dimensional embedding space.
Revised: Unveiling the complex nosological relation among diverse neuropsychiatric disorders with the help of diagnostic information, we advocate a neural correlate based approach, which targets on deriving a neural correlate informed low-dimensional embedding space.

Original: To learn low-dimensional embeddings of neural correlates that preserves the diagnostic attributes, dual utilisation of diagnostic information that is incorporated in a novel type of conditional VAE, are proposed in this research.
Revised: To learn low-dimensional embeddings of neural correlates that preserves the diagnostic attributes, dual utilisation of diagnostic information that is incorporated in a novel type of conditional VAE, are proposed in this research.

Original: Relying on this method, we discovered a reliable and consistent nosological relation among ASD,
Revised: Relying on this method, we discovered a reliable and consistent nosological relation among ASD,

Original: MDD, and SCZ across two curated neuropsychiatric FC datasets.
Revised: MDD, and SCZ across two curated neuropsychiatric FC datasets.

Original: Undoubtedly, the current approach – at its embryonic stage -- faces several mentioned limitations, a future approach with enhanced interpretability on the obtained low dimensional embedding space, and the inclusion of an independent validation dataset should enable us to attain more reliable nosological relations on various neuropsychiatric disorders to encourage the age-old, discrete categorical view on mental illness to make an exit at a fast pace.
Revised: Undoubtedly, the current approach – at its embryonic stage -- faces several mentioned limitations, a future approach with enhanced interpretability on the obtained low dimensional embedding space, and the inclusion of an independent validation dataset should enable us to attain more reliable nosological relations on various neuropsychiatric disorders to encourage the age-old, discrete categorical view on mental illness to make an exit at a fast pace.

Original: E THICS S TATEMENT
Revised: E THICS S TATEMENT

Original: Concerning the potential ethnic issues of our propose approach in serving as a clinical application, we raise four concerns on future use of our approach.
Revised: Concerning the potential ethnic issues of our propose approach in serving as a clinical application, we raise four concerns on future use of our approach.

Original: Undoubtably, the sole reliance on the discovered nosological relation on basis of our approach is neither optimal nor responsible in the diagnosis procedure.
Revised: Undoubtably, the sole reliance on the discovered nosological relation on basis of our approach is neither optimal nor responsible in the diagnosis procedure.

Original: Given the early stage of our approach, harnessing the discovered nosological relation among disorders is still immature to replace established symptom-based diagnosis in clinical psychiatry.
Revised: Given the early stage of our approach, harnessing the discovered nosological relation among disorders is still immature to replace established symptom-based diagnosis in clinical psychiatry.

Original: Regarding discovered nosological relations among disorders, given the heterogeneous, complex, and developmental nature of most neuropsychiatric disorders, further validation of such relations is demanded.
Revised: Regarding discovered nosological relations among disorders, given the heterogeneous, complex, and developmental nature of most neuropsychiatric disorders, further validation of such relations is demanded.

Original: Meanwhile, since our probed nosological relation among disorders on 2D embedding space is induced from the computed distance relation, it does not imply any potential signs of comorbidity.
Revised: Meanwhile, since our probed nosological relation among disorders on 2D embedding space is induced from the computed distance relation, it does not imply any potential signs of comorbidity.

Original: To yield a wide range of social acceptance on using neuroimaging modality in clinical diagnosis, it demands more future validation studies that come from other aspects, e
Revised: To yield a wide range of social acceptance on using neuroimaging modality in clinical diagnosis, it demands more future validation studies that come from other aspects, e

Original: g., the genetic aspect, and symptom relevance.
Revised: g., the genetic aspect, and symptom relevance.

Original: Solely relying on the neuroimaging modality in clinical diagnosis can be misleading and unethical.
Revised: Solely relying on the neuroimaging modality in clinical diagnosis can be misleading and unethical.

Original: Unless these raised ethical concerns are addressed, along with additional validation studies and evidence, clinical use of our approach should not be considered and recommended at this moment.
Revised: Unless these raised ethical concerns are addressed, along with additional validation studies and evidence, clinical use of our approach should not be considered and recommended at this moment.

Original: R EPRODUCIBILITY S TATEMENT
Revised: R EPRODUCIBILITY S TATEMENT

Original: To boost the reproducibility of our study, the executable Python code snippet that allows the implementation of our model is contained within the uploaded Zip ﬁle.
Revised: To boost the reproducibility of our study, the executable Python code snippet that allows the implementation of our model is contained within the uploaded Zip ﬁle.

Original: Within the uploaded
Revised: Within the uploaded

Original: Zip ﬁle, we also include a code snippet that contains the helper function we used in the computation of FID scores between embeddings.
Revised: Zip ﬁle, we also include a code snippet that contains the helper function we used in the computation of FID scores between embeddings.

Original: The used SRPBS database can be referred to https://bicr-resource.atr.jp/srpbsfc/ .
Revised: The used SRPBS database can be referred to https://bicr-resource.atr.jp/srpbsfc/ .

Original: The public available version of this referred database can be requested upon the site.
Revised: The public available version of this referred database can be requested upon the site.

Original: A M ETHODOLOGICAL DISTINCTIVENESS & DERIVATIONS A. M ETHODOLOGICAL COMPARISON BETWEEN OUR APPROACH AND ALTERNATIVES
Revised: A M ETHODOLOGICAL DISTINCTIVENESS & DERIVATIONS A. M ETHODOLOGICAL COMPARISON BETWEEN OUR APPROACH AND ALTERNATIVES

Original: As summarised in following Table 1, the main difference between our approach and NVAE (Vahdat & Kautz, 2020) lies on the derived hierarchical process in deﬁning the decoding and generative process.
Revised: As summarised in following Table 1, the main difference between our approach and NVAE (Vahdat & Kautz, 2020) lies on the derived hierarchical process in deﬁning the decoding and generative process.

Original: In NVAE, the hierarchical process can be seen in the sequential production of z , i.e., z = z 1 , ..., z l ; q ( z 2 | x ) =
Revised: In NVAE, the hierarchical process can be seen in the sequential production of z , i.e., z = z 1 , ..., z l ; q ( z 2 | x ) =

Original: q ( z 2 | z 1 , x ) .
Revised: q ( z 2 | z 1 , x ) .

Original: This sequential encoding and decoding procedure allows NVAE to implement feature combinations from z 1 to z l .
Revised: This sequential encoding and decoding procedure allows NVAE to implement feature combinations from z 1 to z l .

Original: Differ to NVAE, our hierarchical process focuses on the conditional generation of the latent variable z from both discrete variable c , and the input x , i.e., q ( c, z | x ) ∝ q ( z | x ) q
Revised: Differ to NVAE, our hierarchical process focuses on the conditional generation of the latent variable z from both discrete variable c , and the input x , i.e., q ( c, z | x ) ∝ q ( z | x ) q

Original: ( c | z ) .
Revised: ( c | z ) .

Original: More importantly, with a deﬁned categorical prior distribution for the discrete variable Cat ( c | π ) , π ∈ ℜ N + for p ( c ) , the N groups of z are encouraged to be mutually independent, i.e., the implicit clustering effect.
Revised: More importantly, with a deﬁned categorical prior distribution for the discrete variable Cat ( c | π ) , π ∈ (cid:60) N + for p ( c ) , the N groups of z are encouraged to be mutually independent, i.e., the implicit clustering effect.

Original: Such independence is neither the objective nor the outcome of NVAE approach.
Revised: Such independence is neither the objective nor the outcome of NVAE approach.

Original: The hierarchical process in our inference model entails the conditional chain q ( c | z ) q ( z | x ) p ( x ) , whereas in CVAE (Sohn et al., 2015), such conditional chain is replaced by the conditional density
Revised: The hierarchical process in our inference model entails the conditional chain q ( c | z ) q ( z | x ) p ( x ) , whereas in CVAE (Sohn et al., 2015), such conditional chain is replaced by the conditional density

Original: q ( z | x, y ) .
Revised: q ( z | x, y ) .

Original: In a similar vein, as illustrated in Table 1, the major distinction of our approach to a conventional semi-supervised VAE (Kingma et al., 2014) also lies on their differentiated derivations on inference networks.
Revised: In a similar vein, as illustrated in Table 1, the major distinction of our approach to a conventional semi-supervised VAE (Kingma et al., 2014) also lies on their differentiated derivations on inference networks.

Original: In regards to different instantiations on graphical models, our approach also differs from the Gaussian mixture VAE on both inference and generative networks.
Revised: In regards to different instantiations on graphical models, our approach also differs from the Gaussian mixture VAE on both inference and generative networks.

Original: In Gaussian mixture VAE (Dilokthanakul et al., 2016), the inference network is deﬁned as q ( x, w, z | c ) , and its corresponded generation network is p ( c, x, w, z ) ∝ p ( x | w, z ) p ( c | x ) .
Revised: In Gaussian mixture VAE (Dilokthanakul et al., 2016), the inference network is deﬁned as q ( x, w, z | c ) , and its corresponded generation network is p ( c, x, w, z ) ∝ p ( x | w, z ) p ( c | x ) .

Original: Ignoring the Gaussian mixture variable w , the foregoing networks can be seen as q ( x, z | c ) ∝ q ( z | x ) q ( x | c ) and p ( x | z ) p ( c | x ) .
Revised: Ignoring the Gaussian mixture variable w , the foregoing networks can be seen as q ( x, z | c ) ∝ q ( z | x ) q ( x | c ) and p ( x | z ) p ( c | x ) .

Original: In our networks, these two processes are deﬁned as q ( c | z ) q
Revised: In our networks, these two processes are deﬁned as q ( c | z ) q

Original: ( z | x ) and p ( x | z ) p ( z | c ) , respectively.
Revised: ( z | x ) and p ( x | z ) p ( z | c ) , respectively.

Original: Aside from these differences, we acknowledge the explicit modelling of different Gaussian mixtures w , which cannot be achieved by the current version of our approach.
Revised: Aside from these differences, we acknowledge the explicit modelling of different Gaussian mixtures w , which cannot be achieved by the current version of our approach.

Original: [Table]
Revised: [Table]

Revised: NVAE Π l p ( z l | z <l ) p ( x | z l ) Π l q ( z l | z <l , x ) p ( x ) CVAE p ( z | x ) p ( c | x, z ) q ( z | x, c ) p ( x, c )

Revised: S EMI SUPERVISED -VAE p ( x | z, c ) p ( z | c ) p ( c ) q ( z | x, c ) p ( x ) p ( c )

Revised: G AUSSIAN MIXTURE VAE p ( x | z ) p ( c | x ) p ( z ) q ( z | x ) q ( c | z ) p ( x )

Revised: O UR APPROACH

Revised: p ( x | z ) p ( z | c ) p ( c ) q ( c | z ) q ( z | x ) p ( x )

Original: A. D ERIVATIONS
Revised: A. D ERIVATIONS

Original: In Section 3.1, we put forward a novel type of conditional VAE that allows implicit clustered embeddings.
Revised: In Section 3.1, we put forward a novel type of conditional VAE that allows implicit clustered embeddings.

Original: The core component of our loss function is pertaining to log q ( z | x ) p ( z | c ) , where we embed the implicit clustering effect.
Revised: The core component of our loss function is pertaining to log q ( z | x ) p ( z | c ) , where we embed the implicit clustering effect.

Original: Given two conditional Gaussians to parameterise q ( z | x ) and p ( z | c ) , respectively, i.e., q ( z | x ) ∼ N ( µ x , σ 2 x ) , and p ( z | c ) ∼ N ( µ c , I ) , the general optimisation loss term for this component can be derived into: 
Revised: Given two conditional Gaussians to parameterise q ( z | x ) and p ( z | c ) , respectively, i.e., q ( z | x ) ∼ N ( µ x , σ 2 x ) , and p ( z | c ) ∼ N ( µ c , I ) , the general optimisation loss term for this component can be derived into: [Equation] N

Original: [Equation]
Revised: [Equation]

Original:  where N ( µ x , σ 2 x ) 
Revised: µ [Equation] [Equation] where N ( µ x , σ 2 x ) = (cid:112) 2 πσ 2 x exp (cid:40)

Original: =
Revised: [Equation] (cid:13)(cid:13)(cid:13)(cid:13) z − µ x σ [Equation]

Revised: (cid:13)(cid:13)(cid:13)(cid:13) 2 (cid:41) , and N ( µ c ,

Original: [Equation]
Revised: [Equation]

Original: −
Revised: = [Equation] exp (cid:26) [Equation]

Revised: (cid:107)

Original: [Equation]
Revised: [Equation]

Revised: (cid:107)

Revised: [Equation]

Original: −
Revised: (cid:27) , [Equation] (cid:107) [Equation]

Revised: (cid:107)

Original: [Equation]
Revised: [Equation]

Revised: (cid:13)(cid:13)(cid:13)(cid:13)

Revised: [Equation]

Revised: [Equation]

Original:  , and N ( µ c , 
Revised: [Equation] ) (cid:13)(cid:13)(cid:13)(cid:13) , as z = µ x + (cid:15) ⊗ σ x , (cid:15) ∼ N (0 , 1) = [Equation] (cid:15)

Revised: |

Original: [Equation]
Revised: [Equation]

Original: Ignoring the last constant term 1 2 || $ || 2 , we arrive at
Revised: Ignoring the last constant term 1 2 || (cid:15) || 2 , we arrive at

Original: [Equation]
Revised: [Equation]

Original:  From the following general derivation, it clearly indicates the contribution of µ c in producing our targeted low dimensional embeddings z .
Revised: [Equation] From the following general derivation, it clearly indicates the contribution of µ c in producing our targeted low dimensional embeddings z .

Original: I.e., from the regularisation viewpoint, µ c can be viewed as one type of regulariser to regularise the learning of z . 
Revised: I.e., from the regularisation viewpoint, µ c can be viewed as one type of regulariser to regularise the learning of z . A. T HE IMPORTANCE OF λ IN DETERMINING THE SHAPE OF z

Revised: distances are enlarged, along with the increasing amount of void space among clusters.

Revised: In the main text, to balance the contribution of two manners, we pre-deﬁne the λ = 0 .

Revised: 5 to enforce the equal weighting between two manners.

Original:  B T RAINING CONF IGURATIONS B. C ONF IGURATIONS IN THE SIMULATION STUDY (S ECTION 4.)
Revised: However, this λ value be tuned ﬂexibly in determining the shape of each embedded cluster. In the attached Figure 4, through the demonstration on MNIST dataset, we show the effect of differentiated λ in controlling the overlapping degree of encoded clusters. I.e., λ → 0 , clusters of embeddings tend to be overlapped with each other, whereas λ → 1 , encoded clusters are pulling away from each other, in the contribution of the independent, nonoverlapping clustering effect. B T RAINING CONFIGURATIONS B. C ONFIGURATIONS IN THE SIMULATION STUDY (S ECTION 4.)

Original: The following two subsections are presented here to demonstrate the default conﬁgurations for implementing three instantiations of our proposed approach, i.e., Full model,
Revised: The following two subsections are presented here to demonstrate the default conﬁgurations for implementing three instantiations of our proposed approach, i.e., Full model,

Original: Manner-1 model, and Manner-2 model , and three alternative models, i.e., S-VAE model, S-Umap model, and NCA model , which were utilised in Section 3 of the main text.
Revised: Manner-1 model, and Manner-2 model , and three alternative models, i.e., S-VAE model, S-Umap model, and NCA model , which were utilised in Section 3 of the main text.

Original: [Table]
Revised: [Table]

Original: 1: FC 9730;200;2;200,9730, ReLU activation Network-2 ( c ): FC 4000; 2000;100, ReLU and Soft-max activations Network-3 ( L clr ):FC 40, ReLU activation, D : Euclidean distance λ = 0.5, Batch Size: 20, num of batch:optimiser:Adam, (Kingma & Ba, 2014) Manner-1 model Network-1 ( z ):FC 9730;200;2;200,9730, ReLU activation Network-2 ( c ): FC 4000;2000;100, ReLU activation Batch Size: 20, num of batch:optimiser:Adam (Kingma & Ba, 2014) Manner-2 model FC 40, ReLU activation D : Euclidean distance, Batch Size: 20, num of batch:optimiser:Adam (Kingma & Ba, 2014)
Revised: [Table]

Original: B. C ONF IGURATIONS IN THE EMPIRICAL STUDY (S ECTION 5.)
Revised: B. C ONFIGURATIONS IN THE EMPIRICAL STUDY (S ECTION 5.)

Original: The following table summarises the training conﬁguration of our approach in exploring the nosological relations among various disorder phenotypes in two empirical datasets (Section 4 of the main text).
Revised: The following table summarises the training conﬁguration of our approach in exploring the nosological relations among various disorder phenotypes in two empirical datasets (Section 4 of the main text).

Original: To improve the reliability of the attained nosological relation, we adopt the nearly identical model conﬁgurations to implement our model in UTO and HuShoWa datasets, respectively.
Revised: To improve the reliability of the attained nosological relation, we adopt the nearly identical model conﬁgurations to implement our model in UTO and HuShoWa datasets, respectively.

Original: Regarding the scarce sample sizes of two datasets, the complex cross validation procedure may not be appropriate here.
Revised: Regarding the scarce sample sizes of two datasets, the complex cross validation procedure may not be appropriate here.

Original: Therefore, in Figure 4 of the main text, the results were yielded from two full datasets.
Revised: Therefore, in Figure 4 of the main text, the results were yielded from two full datasets

Original: [Table] [Table]
Revised: [Table] [Table]

Original: C E MPIRICAL COMPARISON OF DERIVED EMBEDDINGS FROM OUR APPROACH AND ALTERNATIVE MODELS
Revised: C S TATISTICAL TESTS

Revised: To assess the statistical signiﬁcance of attained embedding relation in the above-mentioned simulation study (see Main text Section 4.), we conduct a series of paired t-tests to assess the signiﬁcance of pair-wise between-cluster FID across six different approaches. As shown in the following Figure 5, our proposed Full model achieved the most superior t-test results among the harnessedapproaches, albeit the comparison between C2-C3 and C1-C2 does not achieve the statistical significance. Note here, the observed strong statistical signiﬁcance between C2-C3 and C1-C2 in NCA, i.e., C 2 − C 3 > C 1 − C 2 , contradicts with the ground-truth, i.e., C 2 − C 3 < C 1 − C 2 .

Revised: D T HE SECOND SIMULATION STUDY

Revised: We attach a supplementary (2nd) simulation study (see the following Figure 6) to further solidify the observed empirical superiority of our approach, i.e., the full model, in recovering low-dimensional embedding of FC-like high-dimensional features.

Revised: In this attached 2nd simulation study, all set-ups are kept the same with the 1st simulation study (Section 4.

Revised: main text) except for the non-linear scattering of simulated ground-truth embeddings.

Revised: Inconsistent with the rendered 1st simulation study, the only model that recovers the ground-truth between-cluster relation is our proposed Full model .

Original: Implemented on the same HuShoWa dataset, we empirically evaluate the derivedlow dimensional embeddings from our approach, and several alternative ones, e
Revised: The employed UTO and HuShoWa datasets are derived from the SRPBS database, which is consisted of resting-state fMRI scans from 8 different sites. The detailed curation and pre-precessing steps of SRPBS database can be consulted in the work of Tanaka et al. Here, we highlight some of the key aspects of scanning and data pre-processing.

Original: [Table]
Revised: At each site, the subject-wise restingstate fMRI data were obtained from 10-min resting-state fMRI scanning session with the eye-open condition. For data curated in HuShoWa dataset, the imaging data were acquired in a Siemens Spec- tra and a Siemens Verio scanner, whereas for the data in UTO dataset, data were acquired from a GE MR750W scanner. All resting-state fMRI data underwent the identical pre-processing step include slice-time correlation, realignment, co-registeration, segmentation of T1-weighted structural images, normalisation to Montreal Neurological Institute (MNI) space, and spatial smoothing with an isotropic Gaussian kernel of 6 mm full-width at half-maximum.

Revised: F E MPIRICAL COMPARISON OF DERIVED EMBEDDINGS FROM OUR APPROACH AND ALTERNATIVE MODELS

Original: g., neighbourhood component analysis (NCA) (Goldberger et al., 2004), T-SNE (Van der Maaten & Hinton, 2008), Supervised Umap (S-Umap) (McInnes et al., 2018), and a conventional semi-supervised VAE (S-VAE) (Kingma et al., 2014).Empirical comparison of embeddings derived from our approach and several alternative approaches on HuShoWa dataset.
Revised: Implemented on the same HuShoWa dataset, we empirically evaluate the derived low dimensional embeddings from our approach, and several alternative ones, e.g., neighbourhood component analysis (NCA) (Goldberger et al., 2004), T-SNE (Van der Maaten & Hinton, 2008), Supervised Umap (S-Umap) (McInnes et al., 2018), and a conventional semi-supervised VAE (S-VAE) (Kingma et al., 2014).

Original: D D IMENSIONAL VIEW ON ASD, MDD, & SCZ
Revised: G D IMENSIONAL VIEW ON ASD, MDD, & SCZ

Original: Demonstrated in the following Figure 5, in accordance with the envisioned dimensional view on diverse neuropsychiatric disorders (Craddock & Owen, 2010), we recast the attained the nosological relation among ASD, MDD, SCZ on the following continua, where ASD and MDD reside at two poles of this continua.
Revised: Demonstrated in the following Figure 8, in accordance with the envisioned dimensional view on diverse neuropsychiatric disorders (Craddock & Owen, 2010), we recast the attained the nosological relation among ASD, MDD, SCZ on the following continua, where ASD and MDD reside at two poles of this continua.