MAGAN: Aligning Biological Manifolds
11 MAGAN: Aligning Biological Manifolds
Matthew [email protected] Smita [email protected]
Abstract —It is increasingly common in many types of naturaland physical systems (especially biological systems) to have dif-ferent types of measurements performed on the same underlyingsystem. In such settings, it is important to align the manifoldsarising from each measurement in order to integrate such dataand gain an improved picture of the system. We tackle thisproblem using generative adversarial networks (GANs). Recently,GANs have been utilized to try to find correspondences betweensets of samples. However, these GANs are not explicitly designedfor proper alignment of manifolds. We present a new GANcalled the Manifold-Aligning GAN (MAGAN) that aligns twomanifolds such that related points in each measurement spaceare aligned together. We demonstrate applications of MAGANin single-cell biology in integrating two different measurementtypes together. In our demonstrated examples, cells from thesame tissue are measured with both genomic (single-cell RNA-sequencing) and proteomic (mass cytometry) technologies. Weshow that the MAGAN successfully aligns them such that knowncorrelations between measured markers are improved comparedto other recently proposed models.
I. I
NTRODUCTION
We commonly face the situation of having samples from apair of related domains and want to ask the natural questionof how samples from one relate to samples from the other.Our motivational system for this is two types of measurementson cells sampled from the same population in a biologicalsystem. It is important for the discovery of new biology tointegrate these datasets, which are often generated at great costand expense. However, a fundamental challenge is that thereare exponentially many possible relationships that could existbetween the two domains of measurement and the system mustlearn a logical way to map between them.The first approaches for teaching neural networks to learnthese relationships required supervised paired examples fromeach domain, an impractical demand for many applications[1]. Recently, there have been attempts at performing the sametask without the supervision of paired data [2, 3, 4]. Like theseprevious models, the MAGAN learns to map between distinctdomains from unsupervised, unpaired data without pretraining.It can take a point in the first domain and generate a point thatis indistinguishable from points in the other domain. However,unlike previous models, the MAGAN learns the most coherent mapping, rather than an arbitrary one. The MAGAN will notjust take a point in the first domain and generate any point fromthe second domain, but it will generate the most closely relatedone. This is achieved by aligning, rather than superimposing,the manifolds of the two domains.The high-dimensional inputs that are typical for neuralnetwork applications can typically be modeled very well witha lower-dimensional manifold [5]. Much work has framed the
Fig. 1. There are exponentially many mappings that superimpose the twomanifolds, fooling a GAN’s discriminator. By aligning the manifolds, wemaintain pointwise correspondences. generation problem of GANs as sampling points from thismanifold [6, 7]. Here, each domain lies on a manifold and wewant to find an alignment between them.We first consider an example of the difference betweensuperimposing and aligning manifolds on image domains.Earlier work [4] has demonstrated that an image of an objectin the first domain can be mapped to an object of anotherobject in the second domain while preserving the orientationwith respect to the picture frame. However, in those cases,the orientation axis can be completely reversed. An image inthe first domain facing ◦ maps to an image in the seconddomain facing ◦ , and vice versa. The mappings successfullyfool the discriminator in each domain at the level of an entirebatch (the manifolds are superimposed), but there are othermappings that also fool the discriminator that preserve theindividual pointwise structure of the original domain. Namely,the optimal alignment would map first domain images at ◦ to second domain images also at ◦ . Without aligning themanifolds, only random initialization will determine whichsuperimposition is learned on any particular attempt.While the preference for the logical mappings of manifoldalignment over the arbitrary ones of manifold superimpositionis of general interest to all domains, we present multipleapplications in single-cell biological analysis where it isessential. We propose the novel concept of using adversarialneural networks for alignment of manifolds arising fromdifferent biological experimental data measurement types.Single-cell biological experiments create many situationswhere manifold alignment problems are of interest. New tech-nologies allow for measurements to be made at the granularityof each cell, rather than older technologies which could onlyacquire aggregate summary statistics for whole populations of a r X i v : . [ c s . C V ] F e b Fig. 2. The MAGAN architecture with two generators, two discriminators, reconstruction loss, and correspondence loss. Domain 1 comprises upright imagesof 3’s and 7’s, Domain 2 comprises rotated images of 3’s and 7’s. cells. While these instruments allow us to discover biologicalphenomena that were not apparent before, it is a challenge tointegrate and analyze this information in a unified fashion forbiological discovery. Further, even for the same technology,experiments run on different days or in different batches canshow variations even on the same populations, possibly due tocalibration differences. In such cases even replicate experimentsneed alignment before comparison. Two such technologies thatwe examine are single-cell RNA sequencing which measurescells in thousands of gene (mRNA) dimensions and masscytometry which measures protein abundances in several dozendimensions [8, 9].In all of these examples we have two data manifolds witha latent physical cell being measured analogously in eachmanifold. In some applications it might be adequate to simplysuperimpose these manifolds in any way. In many applicationsthough, including the ones demonstrated here, we would like tobe able to align them such that the two representations of eachlatent cell are aligned. The MAGAN presented here improvesupon neural models for manifold alignment by finding themapping between the manifolds ( correspondence ) that modelsthese latent points by penalizing differences in each point’srepresentation in the two manifolds.We summarize the contributions of this paper as follows:1) The introduction of a novel GAN architecture that alignsrather than superimposes manifolds to find relationshipsbetween points in two distinct domains2) The demonstration of novel applications made possibleby the new architecture in the analysis of single-cellbiological dataThe rest of this paper is organized as follows. First, there is adetailed description of the MAGAN architecture. Next, there isa validation of its performance on artificial data and the standardMNIST dataset. Then, there are demonstrations on three real- world biological applications: mapping between two replicatecytometry domains, mapping between two different cytometrydomains, and mapping between one cytometry domain and asingle-cell RNA sequencing domain.II. M
ODEL
In this section we detail the MAGAN architecture andspecify the notation used thereafter. We then elaborate onthe key novel aspects of the model individually, discussing inturn the unsupervised correspondence loss, semi-supervisedcorrespondence loss, and data augmentation.
A. Architecture
The MAGAN (Figure 2) is composed of two GANs, eachwith a generator network G that takes as input X and outputsa target dataset X (cid:48) . We refer to each generator as a mapping from the input domain to the output domain. Each generatorattempts to make its output G ( X ) indistinguishable by D from X (cid:48) . Denote the two datasets X and X . Let the generatormapping from X to X be G and the generator mappingfrom X to X be G . The discriminator that tries to separatetrue points from mapped ones for the first domain is D andthe discriminator doing so for the second domain is D .The loss for G on minibatches x and x is: x = G ( x ) x = G ( x ) L r = L reconstruction = L ( x , x ) L d = L discriminator = − E x ∼ P X [ log D ( x )] L c = L correspondence = L ( x , x ) L G = L r + L d + L c where L is any loss function, here mean-squared error (MSE). Similarly, the loss for G is: x = G ( x ) x = G ( x ) L r = L ( x , x ) L d = − E x ∼ P X [ log D ( x )] L c = L ( x , x ) L G = L r + L d + L c The losses for D and D are: L D = − E x ∼ P X [ log D ( x ) − log D ( x )] − E x ∼ P X [ log (1 − D ( x ))] L D = − E x ∼ P X [ log D ( x ) − log D ( x )] − E x ∼ P X [ log (1 − D ( x ))] A crucial implementation decision is to tie the weightsof G and G for both directions of the mapping, as thisensures that the mappings will be between similar data points.For example, after mapping a point from X to X , in orderto reconstruct the original point x , the first mapping mustpreserve enough information in x . Then, since the weightsare tied, G must use this information in the same way whenasked to map an original point x . B. Correspondence Loss1) Unsupervised Correspondence:
Previous models includedonly two restrictions: (1) that the two generators be able toreconstruct a point after it moves to the other domain and back,and (2) that the discriminators not be able to distinguish batchesof true and mapped points. To do this, the generators couldlearn arbitrarily complex mappings as long as they superimposethe two manifolds.To instead enforce the manifolds be fully aligned, theMAGAN includes a correspondence loss between a point inits original domain and that point’s representation after beingmapped to the other domain. This correspondence loss canbe chosen appropriately for the manifolds in any particularproblem. In the biological domains considered here, there area subset of shared features measured in both experiments. Weuse the MSE over these subsets as the correspondence loss,as we do not want to reorder sets of points or change theirrepresentations more than what is required to match the otherdomain.
2) Semi-supervised Correspondence:
The MAGAN’s corre-spondence loss additionally provides a natural opportunity forsemi-supervised learning. If each point in X already had aknown correspondence with a point in X , no framework ofdual GANs would be necessary to discover relationships. Insome domains, though, it is difficult to define a meaningfuldistance measure but easy to acquire a very small numberof labeled pairs. We would like a model that learns fromunsupervised data but can improve with any small numberof labels that can be acquired. In those situations, we wantto leverage both (1) the information that the unsupervisedmodel has learned on all of the data and (2) incorporate theinformation the labels provide where they exist. The MAGAN can be used in this setting without any furthermodification using the following choice of correspondence lossfunction. We choose the loss function to be nonzero only atthe paired points in each domain. Its value is then the sum ofthe losses on each labeled pair, where the loss for a particularlabeled pair ( x i , x j ) , x i ∈ X , x j ∈ X is: L c = M SE ( G ( x i ) , x j ) + M SE ( G ( x j ) , x i ) C. Manifold Data Augmentation
The MAGAN utilizes a novel technique for data augmenta-tion, leveraging the imperfect reconstructions each generatorproduces within its domain. It has been well established thatautoencoders model and reconstruct from the data manifold[10, 11]. We note that the dual GANs within each domainfunction as an autoencoder, meaning their reconstruction x (cid:48) i of a sample x i is another point near the underlying manifold,but importantly x (cid:48) i (cid:54) = x i . By letting each discriminator seethe reconstructions as true samples from the real domain, weboth (1) augment the original data with new samples from themanifold and (2) prevent the discriminators from learning toseparate real from generated examples by modeling the noisearound the manifold, which differs between X and G ( X ) and between X and G ( X ) . This is especially importantin biological settings, where the number of measurements percell dwarfs the number of cells measured and dropout in themeasuring process produces sparsity.III. E XPERIMENTS
All experiments were performed with the MAGAN frame-work with discriminators of five layers each and generators ofthree layers each. Layer sizes depended on the dataset, whileLeaky ReLU activations were used on all layers except theoutput layers of the discriminators (which were sigmoid) andthe generators (which were linear). Dropout of 0.9 was appliedduring training and for images convolutional layers were used.Optimization was performed on 100,000 iterations of batchesof size 256 by the ADAM optimizer with learning rate 0.001.As with other GANs, the generators and discriminators aretrained alternatively, so they each must get progressively betteras their adversaries make their tasks harder and harder. Oneknown difficulty in the adversarial training process is preventinga collapse of the generator into mapping all inputs to one point,chasing the minimum probability region of the discriminator asit moves. To combat this, the MAGAN includes the approachoutlined in [12]. This involves giving the discriminator accessto minibatch information by having a subset of the networkprocess a rotation of the original data matrix.
A. Artificial Data
We first test the MAGAN on a generated example ofpoints sampled from Gaussian distributions with varying means.Figure 3a shows the three subpopulations in the first domain X in blue and the three in the second domain X in red withan example mapping where, without the correspondence loss,each subpopulation in X is mapped to a subpopulation in X , but not to the closest one. Even though the distribution Fig. 3. Both models superimpose the manifolds, meaning the first domain ( X ) is mapped to the second domain ( X ) such that the dataset of the first domainafter mapping ( G ( X )) matches the second domain. Without the correspondence loss, though, this mapping is arbitrary and thus the relationships foundvary. With the correspondence loss, the relationships found are coherent. This is confirmed with (a) a GAN without correspondence loss on artificial data (b)MAGAN on artificial data (c) a GAN without correspondence loss on MNIST and (d) MAGAN on MNIST. of G ( X ) matches the distribution of X , for an individualpoint x i ∈ X , G ( x i ) is not the member of X that ismost closely analogous to it. The MAGAN finds a mappingthat fools the discriminator, too: the one that least alters theoriginal input (Figure 3b).Without the correspondence loss, not only is a less-preferredmanifold superimposition chosen, but the one chosen variesfrom run to run of the model. We compare the variability of thelearned mappings across multiple runs of each model with 100independent trials. In each trial we evaluate the relationshipsby calculating G ( x i ) for each x i ∈ X and calculating itsnearest neighbor x j in the real X . Then, this is repeated forthe other domain. Figure 4a confirms that for the GAN withoutthe correspondence loss, the learned manifold superimposition(and thus the correspondences) varies with repeated trainingthe model. Figure 4b confirms the MAGAN instead aligns themanifolds and finds the same correspondence every time. B. MNIST
Next we test a subset of the MNIST handwritten digit data bytaking only 3’s and 7’s as the first domain X , and a 120 degreerotation of each image as the second domain X . Without thecorrespondence loss (Figure 3c), each subpopulation in X maps to one of the subpopulations in X , but the original3’s go to the rotated 7’s and vice versa. There is no term inthe objective function to create a preference for the mappingthat sends original 3’s to rotated 3’s. It would be difficult todefine a distance measure that captures the notion of alignmentwith these manifolds, but it is a natural place where a smallnumber of labeled pairs could be easily acquired. The semi-supervised correspondence loss with just a single labeled pairof points finds the desired manifold alignment and gets thecorrect correspondences for all of the other points that areunlabeled (Figure 3d). Fig. 4. In simulations of 100 complete training runs of each model, withoutcorrespondence loss the resulting relationships learned varied randomly inboth the (a) toy and (c) MNIST datasets. With correspondence loss, the mostcoherent relationship was found repeatedly for both (b) toy and (d) MNISTdatasets.
Using the same simulation design as in the previous section,we can test the robustness of the models in finding theseparticular mappings. The GAN without the correspondenceloss discovers either relationship with roughly even probability(Figure 4c). Remarkably, the MAGAN is able to use the singlelabeled example to learn that (except for a few sloppily written3’s that in fact look more like 7’s) the original 3’s correspondto the rotated 3’s and that the original 7’s correspond to therotated 7’s every time (Figure 4d).
C. Correspondence Between CyTOF Replicates
We now test the MAGAN on real biological data from single-cell time-of-flight mass cytometry (CyTOF) measurements of
Fig. 5. Selected markers illustrating large batch effects that separate the twodata manifolds. protein abundance. Each protein, also referred to as a marker , ismeasured individually for each cell, allowing for more granularanalysis than processes that only measure population totalsfor the cells in a given sample. Here the same sample wasrun twice in different batches ( replicates ), but due to machinecalibration and other experimental details that are impossibleto reproduce precisely each time, there are distortions betweenthe batches. Thus, even though the same physical blood sampleis being measured, the data manifold of each batch is different.The type of noise introduced by these distortions is not known a priori , need not fit any parametric assumption, and is likelyto be highly nonlinear.To analyze these two batches together, we need to knowwhich cells in the first batch correspond to which cells inthe second batch. To do this, we learn a mapping with theMAGAN between the batches, each of which contains 75,000cells with 34 individual markers measured. Figure 5 showsthat the two batches indeed contain distinct differences in boththe values of each marker and their distribution. For example,the mean value of HLA-DR in the second batch is higher thanthe maximum value in the first batch.We demonstrate that the MAGAN with its correspondenceloss preserves crucial information that is lost with the mappingfrom the GAN without the correspondence loss. Often, analysisstarts by identifying subpopulations of interest. For example,naive T-cells and central memory T-cells serve distinct functionsand can be identified by looking at two isoforms of the CD45marker, CD45RA and CD45RO [13]. In naive T-cells, CD45RAis present while CD45RO is not (CD45RA+CD45RO-), and incentral memory T-cells CD45RA is not present while CD45ROis (CD45RA-CD45RO+). Figure 6a shows that very few cellshad any CD45RA readings in the first batch, a typical case ofinstrument-induced dropout. Figure 6b shows proper readingsfor CD45RA in the second batch, where the two distinctsubpopulations are clearly seen.Both models learn a mapping for the first batch of cells x such that G ( x ) fools their discriminators by looking likethe second batch of cells x . However, in the GAN without thecorrespondence loss (Figure 7a), naive T-cells in the first batchare mapped to central memory T-cells in the second batchand vice versa. If we went through the manual process ofgating (selecting cells by manually looking at relative markerexpression) central memory T-cells in the first batch and wantedto know whether their expression was similar in the secondbatch, we would be led to believe incorrectly that either thereare none of these cells in the second batch or their expressionprofile is radically different. Fig. 6. Two distinct populations of T-cells (CD45RA+CD45RO- andCD45RA-CD45RO+) with severe dropout in the CD45RA marker thatcauses a difference between that between the (a) first batch and (b)second batch.Fig. 7. (a) Without correspondence loss, the GAN corrects the batcheffect but subpopulations are reversed. (b) The MAGAN still correctsthe batch effect and subpopulations are preserved.
The MAGAN learns a different mapping (Figure 6b), the onein which subpopulation correspondences are preserved. Notably,the resulting mapped dataset G ( x ) is not negatively affectedby the correspondence loss. Instead, out of the two mappingsthat have similar results at the aggregate level, the one thatmaintains pointwise correspondences is learned. With the cellcorrespondences from other manifold superimpositions, thewrong biological conclusions could be made. This applicationnecessitates the MAGAN’s manifold alignment. D. Correspondence Between Different CyTOF Panels
Next we demonstrate the MAGAN’s ability to align twomanifolds in domains whose dimensionality only partly overlap.Despite the other advantages of CyTOF instruments, onedisadvantage is that CyTOF experiments can only measurethe expression of 30-40 markers per cell. Each experimentchooses which 30-40 markers to measure and refers to this setas the panel . Even though each panel has a limited capacity,different panels can be run on different samples from the samephysical blood or tissue. The MAGAN provides the opportunityto combine the results from these multiple panels and effectivelyincrease the number of expression measurements acquired foreach cell.
To test this, we use the datasets from two experimentspublished in [14] where each experiment had a different panelthat was run on samples from the same population of cells. Thefirst panel measured 35 markers, the second panel measured31 markers, and 16 of those were measured in both. Withoutany advanced methods, all we would be able to do acrossexperiments is compare population summary statistics — andlose all of the information at a single-cell resolution thatmotivated these experiments being done in the first place.If we can identify points in each panel that measure thesame cell, we can combine the measurements and have anaugmented 50-dimensional dataset. To accomplish this, wetake the first experiment’s panel as one domain and the secondexperiment’s panel as the other domain and use the MAGANto learn a mapping between the two. We then combine theoriginal 35 dimensions of a cell in the first experiment x i with the 15 dimensions unique to the second experiment fromthat cell after mapping G ( x i ) .For combining the measurements from each experiment tobe meaningful, the mapped point G ( x i ) must correspondaccurately to the true point x i . This notion can be captured bytaking the correspondence loss function to be the MSE acrossthe 16 dimensions that are shared between the experiments. Inother words, the MAGAN should use the shared measurementsto match cells between experiments, and then learn the requiredmapping for all of the measurements that are not shared.Without incorporating this correspondence measure into themodel, x i need not be analogous to G ( x i ) in any way, andtheir information could not be combined.We evaluate the accuracy of each model’s learned corre-spondence by removing one of the markers measured in bothexperiments, CD3, from the first experiment. Then, we mappoints from the first experiment to the second experimentand evaluate how well the discovered CD3 values correspondwith the true, held-out CD3 values for each cell from the firstexperiment.Figure 8a shows that the GAN without the correspondenceloss finds a manifold superimposition that does not preservethe values of CD3 for each cell accurately. Quantitatively, wecan evaluate this with the correlation coefficient between thereal, held-out CD3 values and the CD3 values predicted aftermapping each point to the other domain. For the GAN withoutthe correspondence loss (Figure 8a), the correlation is -.275,while for the MAGAN (Figure 8b) it is .801. The negativecorrelation means that without the correspondence loss, theGAN will systematically map cells in one panel to differentcells in the other panel.We perform cross-validation by repeating this test witheach of the 16 shared markers in turn for the GAN withoutcorrespondence loss (Figure 8c) and the MAGAN (Figure 8d).While some of the markers have more shared informationthan others and are recovered more accurately, in all cases thecorrelation is better with the MAGAN.If we had not measured one of these in the first experiment,we would have been able to use the learned value from themapping in its place with remarkable accuracy. The MAGANcan powerfully increase the impact of CyTOF experimentsby expanding their limited capacity of markers that can be Fig. 8. Using the MAGAN’s correspondence loss, measurements from eachexperiment can be combined. Their true values are known because they aremeasured in both experiments. Performing cross-validation by holding eachout from the first experiment, we can measure the correlation between thepredicted value and the real, correct value. measured at any one time.
E. Correspondence Between CyTOF and scRNA-seq
To demonstrate the MAGAN aligning manifolds of domainswith radically different dimensionality and underlying structure,we use it to find correspondences between CyTOF and scRNA-seq measurements made on the same set of cells. These twotypes of measurements have advantages and disadvantages,including the throughput, quality, and amount of informationacquired from each. Being able to combine their informationoffers the possibility of getting the best from each and findinginsights that might not otherwise be obtainable. In order todo this, though, it is crucial for pointwise correspondences tobe accurate, or else features of a data point in the scRNA-seqdomain will be ascribed to the incorrect point in the CyTOFdomain and the relationships will be meaningless.To test the MAGAN in this setting, we use a datasetconsisting of 2830 measurements, where the dimensionalityof each domain is 12 and 12496 for CyTOF and scRNA-seq,respectively [15]. The scRNA-seq data was normalized withthe inverse hyperbolic sine transform and preprocessed withMAGIC [16]. Here we know the true correspondences of whichpoints in the two domains are the same cell. In this settingwe use the semi-supervised correspondence loss and show theimpact of providing the pairing of just 10 cells, which caneasily be acquired with a few minutes of inspection. This effortis dwarfed by the time and expense involved in acquiring thedata from the actual biological experiments themselves, andnow substantially improves what we can do with that data.We evaluate the quality of the correspondences learned withtwo metrics. First, we calculate the correspondence error , or
TABLE IW
ITH THE
MAGAN’
S CORRESPONDENCE LOSS , THE ACCURACY OF THELEARNED MAPPING IS DRAMATICALLY IMPROVED , AS MEASURED BY THE
MSE
BETWEEN THE KNOWN REAL POINT x AND THE PREDICTED POINT G ( x ) AFTER MAPPING .Paired CyTOF& scRNA-seq WithoutCorrespondenceLoss WithCorrespondenceLossMSE( x , G ( x )) x , G ( x )) MSE between the true known value x i ∈ X and the predictedcorrespondence G ( x i ) . Table I shows the correspondenceerror for the correspondences mapping to and from each domain.With the correspondence loss, the MAGAN cuts the MSEdramatically. IV. D ISCUSSION
A considerable amount of work has been devoted to GANarchitectures in recent years. After the original paper introducedthe GAN [17], the difficulty in training them prompted theneed for improved training techniques [12].Beyond the unsupervised models discussed earlier, othermodels have tried improving the accuracy of found correspon-dences with supervision or semi-supervision. These modelshave included forcing the networks to model specific additionalcodes or representations and conditioning on external variablessuch as text [18, 19, 20].Other approaches decompose the primary task of the GANinto separate, domain-specific tasks performed sequentially[21, 22, 23]. All of these have focused on image domains.Compared to image domains, relatively little has been donewith GANs in biological domains. [24] does not use a GAN,but performs batch correction with neural networks.We show here how GANs can be used for tasks where thegeneration of new samples from a given distribution is notthe primary goal. In these cases, other terms in the objectivefunction can be used to better match the loss landscape with thetask at hand. The MAGAN illustrates one such re-purposing ofthe GAN architecture that outperforms the existing architecturesat finding point-wise correspondences between domains.V. C
ONCLUSION
The MAGAN discovers relationships between domains byaligning their manifolds rather than just superimposing them.Crucially, this can be used when one system is measured intwo different ways and thus forms two different manifolds.In this case, the point in each manifold for one object in theunderlying system are linked. This preserves information at apointwise (rather than just population aggregate) level.The MAGAN facilitates integration of datasets from multiplebiological modalities. As each type of experiment capturesdifferent information with different strengths and weaknesses,combining them makes possible discoveries that could not befound otherwise. R
EFERENCES [1] P. Isola, J.-Y. Zhu, T. Zhou, and A. A. Efros, “Image-to-image translation with conditional adversarial networks,” arXiv preprint arXiv:1611.07004 , 2016.[2] J.-Y. Zhu, T. Park, P. Isola, and A. A. Efros, “Unpairedimage-to-image translation using cycle-consistent adver-sarial networks,” arXiv preprint arXiv:1703.10593 , 2017.[3] Z. Yi, H. Zhang, P. Tan, and M. Gong, “Dualgan: Un-supervised dual learning for image-to-image translation,” arXiv preprint , 2017.[4] T. Kim, M. Cha, H. Kim, J. Lee, and J. Kim, “Learn-ing to discover cross-domain relations with generativeadversarial networks,” arXiv preprint arXiv:1703.05192 ,2017.[5] P. Domingos, “A few useful things to know about machinelearning,”
Communications of the ACM , vol. 55, no. 10,pp. 78–87, 2012.[6] N. Park, A. Anand, J. R. A. Moniz, K. Lee, T. Chakraborty,J. Choo, H. Park, and Y. Kim, “Mmgan: Manifoldmatching generative adversarial network for generatingimages,” arXiv preprint arXiv:1707.08273 , 2017.[7] J.-Y. Zhu, P. Kr¨ahenb¨uhl, E. Shechtman, and A. A. Efros,“Generative visual manipulation on the natural imagemanifold,” in
European Conference on Computer Vision .Springer, 2016, pp. 597–613.[8] S. C. Bendall, G. P. Nolan, M. Roederer, and P. K.Chattopadhyay, “A deep profiler’s guide to cytometry,”
Trends in immunology , vol. 33, no. 7, pp. 323–332, 2012.[9] A. M. Klein, L. Mazutis, I. Akartuna, N. Tallapragada,A. Veres, V. Li, L. Peshkin, D. A. Weitz, and M. W.Kirschner, “Droplet barcoding for single-cell transcrip-tomics applied to embryonic stem cells,”
Cell , vol. 161,no. 5, pp. 1187–1201, 2015.[10] G. E. Hinton, P. Dayan, and M. Revow, “Modelingthe manifolds of images of handwritten digits,”
IEEEtransactions on Neural Networks , vol. 8, no. 1, pp. 65–74, 1997.[11] P. Vincent, H. Larochelle, Y. Bengio, and P.-A. Manzagol,“Extracting and composing robust features with denoisingautoencoders,” in
Proceedings of the 25th internationalconference on Machine learning . ACM, 2008, pp. 1096–1103.[12] T. Salimans, I. Goodfellow, W. Zaremba, V. Cheung,A. Radford, and X. Chen, “Improved techniques for train-ing gans,” in
Advances in Neural Information ProcessingSystems , 2016, pp. 2234–2242.[13] J. D. Capra, C. A. Janeway, P. Travers, and M. Walport,
Inmunobiology: the inmune system in health and disease .Garland Publishing,, 1999.[14] M. Setty, M. D. Tadmor, S. Reich-Zeliger, O. Angel, T. M.Salame, P. Kathail, K. Choi, S. Bendall, N. Friedman, andD. Pe’er, “Wishbone identifies bifurcating developmentaltrajectories from single-cell data,”
Nature biotechnology ,vol. 34, no. 6, p. 637, 2016.[15] L. Velten, S. F. Haas, S. Raffel, S. Blaszkiewicz, S. Islam,B. P. Hennig, C. Hirche, C. Lutz, E. C. Buss, D. Nowak et al. , “Human haematopoietic stem cell lineage com- mitment is a continuous process,”
Nature cell biology ,vol. 19, no. 4, p. 271, 2017.[16] D. van Dijk, J. Nainys, R. Sharma, P. Kathail, A. J. Carr,K. R. Moon, L. Mazutis, G. Wolf, S. Krishnaswamy,and D. Pe’er, “Magic: A diffusion-based imputationmethod reveals gene-gene interactions in single-cell rna-sequencing data,”
BioRxiv , p. 111591, 2017.[17] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu,D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio,“Generative adversarial nets,” in
Advances in neuralinformation processing systems , 2014, pp. 2672–2680.[18] L. Tran, X. Yin, and X. Liu, “Disentangled representationlearning gan for pose-invariant face recognition,” in
CVPR ,vol. 3, 2017, p. 7.[19] G. Perarnau, J. van de Weijer, B. Raducanu, and J. M.´Alvarez, “Invertible conditional gans for image editing,” arXiv preprint arXiv:1611.06355 , 2016.[20] S. Reed, Z. Akata, X. Yan, L. Logeswaran, B. Schiele, andH. Lee, “Generative adversarial text to image synthesis,” arXiv preprint arXiv:1605.05396 , 2016.[21] M.-Y. Liu and O. Tuzel, “Coupled generative adversarialnetworks,” in
Advances in neural information processingsystems , 2016, pp. 469–477.[22] H. Zhang, T. Xu, H. Li, S. Zhang, X. Huang, X. Wang,and D. Metaxas, “Stackgan: Text to photo-realistic imagesynthesis with stacked generative adversarial networks,”in
IEEE Int. Conf. Comput. Vision (ICCV) , 2017, pp.5907–5915.[23] X. Wang and A. Gupta, “Generative image modeling usingstyle and structure adversarial networks,” in
EuropeanConference on Computer Vision . Springer, 2016, pp.318–335.[24] U. Shaham, K. P. Stanton, J. Zhao, H. Li, K. Raddassi,R. Montgomery, and Y. Kluger, “Removal of batcheffects using distribution-matching residual networks,”