Surreal-GAN:Semi-Supervised Representation Learning via GAN for uncovering heterogeneous disease-related imaging patterns

1 INTRODUCTION . Neuroimaging , conventional machine learning , and deep learning have offered unprecedented opportunities to understand the underlying mechanism of brain disorders over the past decades ( Davatzikos ( 2018 ) ) , and pave the road towards individualized precision medicine ( Bzdok & MeyerLindenberg ( 2018 ) ) . A large body of case-control studies leverage mass univariate group comparisons to derive structural or functional neuroimaging signatures ( Habeck et al . ( 2008 ) , Hampel et al . ( 2008 ) , Ewers et al . ( 2011 ) ) . However , these studies suffer from under-powered statistical inferences since the heterogeneous nature of neurological diseases usually violate the assumption that each group population are pathologically homogeneous . A developing body of methodologies aim to disentangle this heterogeneity with various machine learning methods . Zhang et al . ( 2016 ) used the Bayesian Latent Dirichlet Allocation ( LDA ) model to identify latent atrophy patterns from voxel-wise gray matter ( GM ) density maps . However , due to the nature of the LDA model , the density maps need to be first discretized , and the model hypothesized to exclusively capture brain atrophy while ignoring the potential regional enlargement . Young et al . ( 2018 ) proposed another method , Sustain , to uncover temporal and phenotypic heterogeneity by inferring both subtypes and stages . However , the model only handles around 10-20 large brain regions without more detailed information . Notably , both methods applied directly in the PT domain , confront main limitations in avoiding potential disease-unrelated brain variations . In con- trast , semi-supervised clustering methods ( Varol et al . ( 2016 ) , Dong et al . ( 2015 ) , Wen et al . ( 2021 ) , Yang et al . ( 2021 ) ) were proposed to cluster patients via the patterns or transformations between the reference group ( CN ) and the target patient group ( PT ) . A recent proposed deep learning-based semi-supervised clustering method , termed Smile-GAN ( Yang et al . ( 2021 ) ) , achieved better clustering performance by learning multiple mappings from CN to PT with an inverse clustering function for both regularization and clustering . Albeit these methods demonstrated potential in clinical applications , they have a major limitation . In particular , these methods model disease heterogeneity as a dichotomous process and derive a discrete output , i.e. , subtype , ignoring the fact that brain disorders progress along a continuum . Moreover , variations among subtypes were contributed by both spatial and temporal differences in disease patterns , thus confounding further clinical analysis. ( Fig . 1a ) To address the aforementioned limitations , we propose a novel method , Surreal-GAN ( SemiSupervised Representation Learning via GAN ) , for deriving heterogeneity-refined imaging signatures . Surreal-GAN is a significant extension of Smile-GAN under the same principle of semisupervised learning by transforming data from CN domain X to PT domain Y. Herein , the key extension is to represent complex disease-related heterogeneity with low dimensional representations with each dimension indicating the severity of one relatively homogeneous imaging patterns ( Fig . 1b ) . We refer to these low dimensional representations as R-indices ( ri , where i represent the ith dimension ) of disease pattern . The first key point of the method is modelling disease as a continuous process and learning infinite transformation directions from CN to PT , with each direction capturing a specific combination of patterns and severity . The idea is realized by learning one transformation function which takes both normal data and a continuous latent variable as inputs and output synthesized-PT data whose distribution is indistinguishable from that of real PT data . The second key point is controlling the monotonicity ( from light to severe with increasing index value ) of disease patterns through one monotonicity loss and double sampling procedure . The third key point is boosting the separation of captured disease patterns through one orthogonality regularization . The fourth key point is introducing a inverse mapping function consisting of a decomposition part and a reconstruction part which further ensure that patterns synthesized through transformation are exactly captured for inferring representations of disease patterns , the R-indices . Besides the above mentioned regularization , function continuity , transformation sparsity and inverse consistency further guide the model to capture meaningful imaging patterns . To support our claims , we first validated Surreal-GAN on semi-synthetic data with known ground truth of disease patterns and severity . We compared performance of Surreal-GAN to NMF ( Lee & Seung ( 1999 ) ) , Factor Analysis and LDA ( Blei et al . ( 2001 ) ) models , and subsequently studied the robustness of model under various conditions . Finally , we applied the model to one real dataset for Alzheimer ’ s disease ( AD ) , and demonstrated significant correlations between the proposed imaging signatures and clincial variables . 2 METHODS . The schematic diagram of Surreal-GAN is shown in Fig . 1d . The model is applied on regional volume data ( volumes of different graymatter ( GM ) and whitematter ( WM ) regions ) derived from structural MRI . The essential idea is to learn one transformation function f : X ∗ Z → Y which transforms CN data x to different synthesized PT data y′ = f ( x , z ) , with latent variable z specifying distinct combinations of patterns and severity . Here , the Latent ( LAT ) domain , Z = { z : 0 ≤ zi ≤ 1 ∀1 ≤ i ≤ M } , is a class of vectors with dimension M ( predefined number of patterns ) . We denote four different data distributions as x ∼ pcn ( x ) , y ∼ ppt ( y ) , y′ ∼ psyn ( y′ ) and z ∼ plat ( z ) , respectively , where z ∼ plat ( z ) is sampled from a multivariate uniform distribution U [ 0 , 1 ] M , rather than a categorical distribution as in Smile-GAN . A continuous z variable lay the foundation for learning continuous representations . In addition to the transformation function , an adversarial discriminator D is introduced to distinguish between real PT data y and synthesized PT data y′ , thereby ensuring that the synthesized PT data are indistinguishable from real PT data ( i.e . minimizing the distance between real PT data distribution and synthesized PT data distribution ) . A continuous latent variable does not necessarily lead to desired properties of CN to PT transformation ( e.g . separation/monotonicity of captured patterns ) . That is , there are a number of functions potentially achieving equality in distributions , thus making it hard to guarantee that the transformation function learned by the model is closely related to the underlying pathology progression process . Moreover , during the training procedure , the transformation function backboned by the neural network tends to ignore the latent variable z . Even when the Latent variable z is emphasized through regularization , different components of z variable do not naturally lead to separate disease patterns and are not positively correlated with pattern severity . Therefore , with the assumption that there is one true underlying function for real PT data y = h ( x , z ) , Surreal-GAN aims to boost the transformation function f to approximate the true underlying function h , while having the latent variable , z , contributing to recognizable imaging patterns rather than being random ” activators ” of the GAN . For this purpose , we constrain the function class of f via different types of regularization as follows : 1 ) encourage sparse transformations ; 2 ) enforce Lipschitz continuity of functions ; 3 ) introduce one inverse function g : Y → Z serving to decompose and reconstruct ; 4 ) boost orthogonality among synthesized/captured patterns ; 5 ) enforce monotonicity of synthesized patterns and their positive correlations with components of z . The training procedure of Surreal-GAN is thus a adversarial learning process constrained by these regularization . 2.1 ADVERSARIAL LOSS . The adversarial loss ( Goodfellow et al . ( 2014 ) ) is used for iterative training of discriminator D and the transformation function f , which can be written as : LGAN ( D , f ) = Ey∼ppt ( y ) [ log ( D ( y ) ) ] + Ex∼pcn ( x ) , z∼Plat ( z ) [ log ( 1−D ( f ( x , z ) ) ) ] ( 1 ) = Ey∼ppt ( y ) [ log ( D ( y ) ) ] + Ey′∼psyn ( y′ ) [ log ( 1−D ( y ′ ) ) ] ( 2 ) The transformation function f attempts to transform CN data to synthesized PT data so that they follow similar distributions as real PT data . The discriminator , providing a probability that y comes from the real data rather than transformation function , tries to identify the synthesized PT data and distinguish it from real PT data . Therefore , the discriminator D attempts to maximize the adversarial loss function while the transformation function f attempts to minimize against it . The training process can be denoted as : min f max D LGAN ( D , f ) = Ey∼ppt ( y ) [ log ( D ( y ) ) ] + Ey′∼psyn ( y′ ) [ log ( 1−D ( y ′ ) ) ] ( 3 ) 2.2 REGULARIZATION . 2.2.1 SPARSE TRANSFORMATION . We assumed that disease process will not change brain anatomy dramatically and primarily only affect certain regions throughout most of disease stages , which means that the true underlying transformation only moderately changes some regions while keeping the rest mildly affected with most of z variable values . Therefore , to encourage sparsity and control the distance of transformations , we defined one change loss to be the l1 distance between the synthesized PT data and the original CN data : Lchange ( f ) = Ex∼pcn ( x ) , z∼plat ( z ) [ ||f ( x , z ) − x||1 ] ( 4 ) 2.2.2 LIPSCHITIZ CONTINUITY AND INVERSE MAPPING . First , by enforcing the transformation function f to be K1-Lipschitz continuous , we can derive that , for fixed latent variable z = a and ∀x1 , x2 ∈ X , ||f ( x1 , a ) − f ( x2 , a ) ||2 ≤ K1||x1−x2||2 . Thus , by controlling the constant K1 , we constrained the transformation function to preserve the original distances among CN data instead of scattering them into the PT domain if a same combination of disease patterns is imposed to them . Second , to prevent latent variable z from being ignored , we introduced an inverse mapping function from PT domain to Latent domain , g : Y → Z . By Lemma1 ( section A.1 ) , with function g being K2-Lipschitz continuous and d ( · , · ) being any distance metric satisfying triangular inequality , we can derive that , ∀z1 , z2 ∼ plat ( z ) , z1 6= z2 , and x̄ ∼ pcn ( x ) , d ( f ( x̄ , z1 ) , f ( x̄ , z2 ) ) is lower bounded by ( d ( z1 , z2 ) K2 − 1 K2 ( d ( g ( f ( x̄ , z1 ) ) , z1 ) + d ( g ( f ( x̄ , z2 ) ) , z2 ) ) ) . Therefore , by minimizing the distance between sampled latent variable z and reconstructed latent variable g ( f ( x , z ) ) , we can control differences between synthesized patterns to be positively related to the distances between latent variables , and thus to be non-trivial ( i.e. , same CN data is transformed to significantly different PT data with different z variables ) . Therefore , we define a reconstruction loss as the l2 distance between g ( f ( x , z ) ) and z : Lrecons ( f , g ) = Ex∼pcn ( x ) , z∼plat ( z ) [ ||g ( f ( x , z ) ) − z||2 ] ( 5 )

Summary: In this paper, the authors pointed out the lack of a good method to predict disease progression in a continuum based on representations explicitly modeling specific disease effects. To tackle this problem, they proposed a novel method called Surreal-GAN, a tailored version of generative adversarial networks, to learn separate representations of neurological and neuropsychiatric diseases in an unsupervised manner. The method frames the diseased brains as multiple disease-related features at various severity imposed on normal brains, and by learning these features and severities, it can generate a representation with improved reliability and explanability.