deep clustering with convolutional autoencoders github

Ostmeyer, J. et al. Further information on research design is available in theNature Research Reporting Summary linked to this article. Awesome machine learning for combinatorial optimization papers. Cancer Immunol. https://doi.org/10.1038/s41467-021-21879-w, DOI: https://doi.org/10.1038/s41467-021-21879-w. The trained network is then used to take a given TCR and represent it in a continuous numerical domain for downstream analysis such as clustering. There was a problem preparing your codespace, please try again. In Neural Information Processing Systems Workshop on Bayesian Deep Learning (eds. a Network architecture schema: Previously described TCR featurization block is implemented to featurize a TCR sequence and then either output a label (i.e. Efficient test and visualization of multi-set intersections. Robot Packing with Known Items and Nondeterministic Arrival Order. Given these factors, highly effective deep learning agents are likely only a desired choice in games that have a large competitive scene, where they can function as an alternative practice option to a skilled human player. supervised the project, interpreted the results, and provided editorial and conceptual input into the manuscript. Google Scholar. Each model is a PyTorch module, and can be imported like so: More details about each model are in the "Models" section below. the encoder is a 5-layer CNN, with kernel sizes (10, 8, 4, 4, 4) and strides (5, 4, 2, 2, 2), and covers 30ms of audio. For all benchmarks and case studies with GLUE, we used the default hyperparameters unless explicitly stated. 5d)51. NeurIPS, 2020. paper, code, Combinatorial Learning of Robust Deep Graph Matching: An Embedding Based Approach. Information on machine learning techniques in the field of games is mostly known to public through research projects as most gaming companies choose not to publish specific information about their intellectual property. PubMed Graph connectivity (GC) was also used to evaluate the extend of mixing among omics layers and was defined as in a recent benchmark study73: where LCCj is the number of cells in largest connected component of the cell k-nearest neighbors graph (K=15) for cell type j, Nj is the number of cells in cell type j and M is the total number of cell types. apply a deep convolutional autoencoder network to prestack seismic data to learn a feature representation that can be used in a clustering algorithm for facies mapping. Learning Practically Feasible Policies for Online 3D Bin Packing Arxiv, 2021. paper, Hang Zhao and Chenyang Zhu and Xin Xu and Hui Huang and Kai Xu, Attend2Pack: Bin Packing through Deep Reinforcement Learning with Attention ICML Workshop, 2021. paper, Solving 3D bin packing problem via multimodal deep reinforcement learning AAMAS, 2021. paper, Learning to Solve 3-D Bin Packing Problem via Deep Reinforcement Learning and Constraint Programming IEEE transactions on cybernetics, 2021. paper, Jiang, Yuan and Cao, Zhiguang and Zhang, Jie, Learning to Pack: A Data-Driven Tree Search Algorithm for Large-Scale 3D Bin Packing Problem CIKM, 2021. paper, Zhu, Qianwen and Li, Xihan and Zhang, Zihan and Luo, Zhixing and Tong, Xialiang and Yuan, Mingxuan and Zeng, Jia, Learning Efficient Online 3D Bin Packing on Packing Configuration Trees. 9, 5345 (2018). The cell latent variable u is shared across different omics layers. & Regev, A. Sun, Haoran and Chen, Wenbo and Li, Hui and Song, Le. Following featurization via the described TCR Featurization Block, we needed an architecture that could handle applying a label to a collection of these featurized sequences. 2019. Procedia Manufacturing, 2018. journal, Boosting combinatorial problem modeling with machine learning. Text-to-Speech Synthesis, 2009. A. et al. Science 370, eaba7721 (2020). This area of research is often termed as improving the explainability of neural networks. AAAI, 2019. paper. Single-cell multiomics sequencing reveals the functional regulatory landscape of early embryos. In comparison, we also attempted to perform integration using online iNMF, which was the only other method capable of integrating the data at full scale, but the result was far from optimal (Supplementary Figs. While CD83 was highly expressed in both monocytes and B cells, the inferred TFs showed more constrained expression patterns (Supplementary Fig. However, undergraduate students with demonstrated strong backgrounds in probability, statistics (e.g., linear & logistic regressions), numerical linear algebra and optimization are also welcome to register. MMD-MA25 was executed using the Python script provided at https://bitbucket.org/noblelab/2020_mmdma_pytorch. Google Scholar. For example, we noted that the Flu-MP TCR is more sensitive to perturbation in the -chain whereas the BMLF1 epitope shows similar sensitivity to perturbations in either the - or -chain. Posts ordered by most recently publishing date Stat. 310) to the derived TCR distances on the nine murine and seven human tetramer-sorted antigen-specific T cells and assessed classification performance via fivefold cross-validation strategy, measuring AUC, Recall, Precision, and F1 Score. The GLUE alignment successfully revealed a shared manifold of cell states across the three omics layers (Fig. [2]Vincent P, Larochelle H, Lajoie I, et al. Nature 577, 706710 (2020). Deepfakes (a portmanteau of "deep learning" and "fake") are synthetic media in which a person in an existing image or video is replaced with someone else's likeness. Methods 391, 1421 (2013). Cell 174, 10151030 (2018). And while findings from our proposed methods would ultimately need to be validated through these more rigorous methods, we do believe that these proposed methods are capable of learning the salient signal from the noise present as is evidenced from the predictive power of these models presented in this work. Meanwhile, DNA methylation in the gene promoter is usually assumed to suppress expression, so they can be connected with a negative edge (sij=1). Finally, the authors mention that self-training is likely complimentary to pre-training and their combination may yield even better results. [34] Most attempted methods have involved the use of ANN in some form. Papalexopoulos, Theodore, Christian Tjandraatmadja, Ross Anderson, Juan Pablo Vielma and Daving Belanger. Methods 14, 10831086 (2017). MathSciNet We also observed varying associations with gene characteristics. The two atlases consist of large numbers of cells but with low coverage per cell. Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. Nat. At cell receptor sequencing-based assay identifies cross-reactive recall CD8+ T cell clonotypes against autologous HIV-1 epitope variants. Variational autoencoder for deep learning of images, labels and captions. Here, we propose a computational framework called GLUE (graph-linked unified embedding), which bridges the gap by modeling regulatory interactions across omics layers explicitly. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. First, to benchmark these various methods of featurization in clustering antigen-specific TCRs, we ran an agglomerative clustering algorithm varying the number of clusters from 5 to 100 and then assessed the variance ratio criterion of the clustering solutions and the adjusted mutual information from the clustering solutions to the ground truth antigen labels (scikit-learn)34,35. Regression . As a data-driven science, genomics largely utilizes machine learning to capture dependencies in data and derive novel biological hypotheses. 46, D380D386 (2018). Specifically, we may compile all orthologs into a GLUE guidance graph and perform integration without explicit ortholog conversion. OConnell, K. A. et al. In particular, the incorporation of batch correction could further enable effective curation of new datasets with the integrated atlas as a global reference49. 1 were created using an image set downloaded from Servier Medical Art (https://smart.servier.com/, CC BY 3.0). NeurIPS, 2020. paper. Models are optimized by minimizing a CTC loss. 3e). 7b,d). 17), which was also supported by marker expression and accessibility (Supplementary Figs. [J] arXiv preprint arXiv:1812.11317. Recent studies have generated human cell atlases for gene expression28 and chromatin accessibility29 containing millions of cells. Stuart, T. & Satija, R. Integrative single-cell analysis. Proc. To unify the cell type labels, we performed a nearest neighbor-based label transfer with the snmC-seq dataset as a reference. Parts of Fig. This learns general representations on huge amounts of data, and can supposedly improve the performance on the new task with limited data. A variational autoencoder provides superior antigen-specific clustering. The data autoencoders in GLUE are customizable with appropriate generative models that conform to omics-specific data distributions. Decoupled Neural Interfaces using Synthetic Gradients [ax1608] Understanding Synthetic Gradients and Decoupled Neural Interfaces [ax1703] Efficient. Commun. Illustrations for Panel a provided by Tim Phelps Copyright 2020 JHU AAM, Department of Art as Applied to Medicine, The Johns Hopkins University School of Medicine. Google Scholar. SPI1 (highlighted with a green box) is a known regulator of NCF2. Since 2016, substantial research has been done to detect epilepsy using DL models such as convolutional neural networks (CNNs), recurrent neural networks (RNNs), deep belief networks (DBNs), Autoencoders (AEs), CNN-RNNs, and CNN-AEs [30,31,32,33]. [14], Supreme Commander 2 is a real-time strategy (RTS) video game. Finally, we correlated information from these RSLs to the crystal structure (Fig. Carousel with three slides shown at a time. Annals of Operations Research, 2004. journal. In contrast to the GAG TW10 epitope family, the GAG IW9 family had only two variants (the consensus epitope - ISPRTLNAW and the I147M escape variant - MSPRTLNAW) that both generated immune responses. Do Deep Nets Really Need to be Deep [nips14] Synthetic Gradients. The data likelihoods \(p\left( {{{{\mathbf{x}}}}_k|{{{\mathbf{u}}}},{{{\mathbf{V}}}};\theta _k} \right)\) (that is, data decoders) in equation (3) are built on the inner product between the cell embedding u and feature embeddings Vk. PubMed Nonetheless, graphs, as intuitive and flexible representations of regulatory knowledge, can embody more complex regulatory patterns, including within-modality interactions, nonfeature vertices and multi-relations. Biotechnol. 20 and 21). Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. 16, e9438 (2020). Nat. The images or other third party material in this article are included in the articles Creative Commons license, unless indicated otherwise in a credit line to the material. If V/D/J gene information is provided as an input to the network, this data are represented first as categorical variable with a one-hot encoding to the network. Can be trained on sequences of varying length. Rush, A.) Since T cell receptor function is ultimately tied to its 3D structure (a derivative from the linear sequence) and its interaction with its cognate epitope, it is plausible that our models are capable of learning information about local 3D structure of the T cell receptor. Commun. Finally, while most perturbations lowered the predictive binding affinity of the given TCR to its cognate antigen, we noted that for the BMLF1 TCR, the G at -6 demonstrated that many perturbations at that site would actually increase the binding affinity of this TCR, suggesting that this approach could also be utilized for TCR engineering to design high-affinity TCRs. In order to demonstrate the utility of these algorithms, we collected a variety of TCR-Seq datasets including samples sorted by antigen specificity20,21,22, samples collected from single-cell RNA-seq experiments (10x_Genomics), and samples collected from a novel experimental assay used in detecting functional expansion of T cells31 (full dataset details in Supplementary Fig. As a demonstration, we used the official peripheral blood mononuclear cell Multiome dataset from 10X34 and fed it to GLUE as unpaired scRNA-seq and scATAC-seq data. PubMed Prediction of specific tcr-peptide binding from large dictionaries of tcr-peptide pairs. Despite the emergence of experimental methods for simultaneous measurement of multiple omics modalities in single cells, most single-cell datasets include only one modality. Supposing that the cell type of the ith cell is y(i) and that the cell types of its K ordered nearest neighbors are \(y_1^{\left( i \right)},y_2^{\left( i \right)}, \ldots, y_K^{\left( i \right)}\), the mean average precision is then defined as follows: where \(1_{y^{\left( i \right)} = y_k^{\left( i \right)}}\) is an indicator function that equals 1 if \(y^{\left( i \right)} = y_k^{\left( i \right)}\) and 0 otherwise. Deep Learning-based Hybrid Graph-Coloring Algorithm for Register Allocation. \end{array}$$, $$\begin{array}{*{20}{c}} {{{{\mathbf{\mu }}}}_i = \mathbf{\alpha} \odot {{{\mathbf{V}}}}_k^ \top \mathbf{u} + \mathbf{\beta} } \end{array}$$, \({{{\mathbf{\mu }}}} \in {\Bbb R}^{\left| {{{{\mathcal{V}}}}_k} \right|},{{{\mathbf{\sigma }}}} \in {\Bbb R}_ + ^{\left| {{{{\mathcal{V}}}}_k} \right|},{{{\mathbf{\delta }}}} \in \left( {0,1} \right)^{\left| {{{{\mathcal{V}}}}_k} \right|}\), https://doi.org/10.1038/s41587-022-01284-4. Trends Biotechnol. For the scRNA-seq data, 4,000 highly variable genes were selected using the organ-balanced subsample. 14 and Supplementary Figs. An Exact Symbolic Reduction of Linear Smart Predict+Optimize to Mixed Integer Linear Programming. ICML (2022). These proportions of concepts in the repertoire are then sent into a final traditional classification layer. Computationally, one major obstacle faced when integrating unpaired multi-omics data (also known as diagonal integration) is the distinct feature spaces of different modalities (for example, accessible chromatin regions in scATAC-seq versus genes in scRNA-seq)14. Transformers use self-attention to encode the input sequence as well as an optional source sequence. NeurIPS, 2020. paper, code. The developers use a form of neuroevolution called cgNEAT to generate new content based on each player's personal preferences.[30]. Zhang, R., Zhou, T. & Ma, J. Multiscale and integrative single-cell Hi-C analysis with Higashi. Deep Voice: Real-time Neural Text-to-Speech, ICML 2017. \(\phi _1,\phi _2,\phi _3,\phi _{{{\mathcal{G}}}}\) represent learnable parameters in data and graph encoders. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. For analyses where V/D/J gene usage, these genes were represented as categorical variables and one-hot encoded as inputs for the neural network. An effective integration method should match the corresponding cell states from different omics layers, producing cell embeddings where the biological variation is faithfully conserved and the omics layers are well mixed. Pre-training reduces WER by 36 % on nov92 when only about eight hours of transcribed data is However, in the area of the biological sciences, there is not only the desire to create predictive tools but use these tools to inform our own understanding of the mechanisms at play. Tickotsky, N., Sagiv, T., Prilusky, J., Shifrut, E. & Friedman, N. McPAS-TCR: a manually curated catalogue of pathology-associated T cell receptor sequences.Bioinformatics 33, 29242929 (2017). fine-tune it on the available labeled data in an end-to-end fashion using CTC loss and a letter-based output vocabulary, then use the model to label the unlabeled data using self-training. If nothing happens, download GitHub Desktop and try again. The game uses Multilayer Perceptrons (MLPs) to control a platoons reaction to encountered enemy units. Sci Rep. 5, 16923 (2015). While feature conversion may seem to be a straightforward solution, the inevitable information loss19 can be detrimental. Training involves learning a vector encoding of each input sequence, reconstructing the original sequence from the encoding, and calculating the loss (mean-squared error) between the reconstructed input and the original input. Therefore, we trained a repertoire classifier to predict if the well had been treated by the cognate epitope, or non-cognate conditions (CEF, AY9, No Peptide) given its T cell repertoire (Fig. During model training, 10% of the cells were used as the validation set. 2a, each quantified by three separate metrics as shown in Extended Data Fig. Since different genes can have different numbers of connected ATAC peaks, and the ATAC peaks vary in length (longer peaks can contain more ChIP peaks by chance), we devised a sampling-based approach to evaluate TF enrichment. einops - Deep learning operations reinvented (for pytorch, tensorflow, jax and others). 11, 591 (2020). Useful if you don't want to create your own training loop. Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion, Unsupervised deep embedding for clustering analysis, Improved deep embedded clustering with local structure preservation, Deep clustering with convolutional autoencoders, Deep Fuzzy K-Means with Adaptive Loss and Entropy Regularization, An illustrated introduction to the t-SNE algorithm, Deep Clustering: methods and implements-Github. Google Scholar. Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. Preprint at https://arxiv.org/abs/1606.05908 (2016). An autoencoder is a type of artificial neural network used to learn efficient codings of unlabeled data (unsupervised learning). B. Logomaker: beautiful sequence logos in python. Due to this complex layered approach, deep learning models often require powerful machines to train and For example, for count-based scRNA-seq and scATAC-seq data, we used the negative binomial (NB) distribution: where \({{{\mathbf{\mu }}}},{{{\mathbf{\theta }}}} \in {\Bbb R}_ + ^{\left| {{{{\mathcal{V}}}}_k} \right|}\) are the mean and dispersion of the negative binomial distribution, respectively, \({{{\mathbf{\alpha }}}} \in {\Bbb R}_ + ^{\left| {{{{\mathcal{V}}}}_k} \right|},{{{\mathbf{\beta }}}} \in {\Bbb R}^{\left| {{{{\mathcal{V}}}}_k} \right|}\) are scaling and bias factors, is the Hadamard product, Softmaxi represents the ith dimension of the softmax output and \(\mathop {\sum}\nolimits_{j \in {{{\mathcal{V}}}}_k} {{\mathbf{x}_{k}}_{j}}\) gives the total count in the cell. & Zhang, B. Classification of SAT problem instances by machine learning methods. In this work, we present DeepTCR, a collection of unsupervised and supervised deep learning approaches to characterize TCR-Seq data for both descriptive and predictive purposes. To obtain the balancing weights in an unsupervised manner, we devised the following two-stage training procedure. 3 Integration performance of GLUE under different hyperparameter settings. Online iNMF was the only other method that could scale to millions of cells, so we applied it to the full dataset. Predicting the sequence specificities of DNA-and RNA-binding proteins by deep learning. Cao, Z. J., Wei, L., Lu, S., Yang, D. C. & Gao, G. Searching large-scale scRNA-seq databases via unbiased cell embedding with Cell BLAST. Self-training and Pre-training are Complementary for Speech Recognition, Comparison of Deep Learning Methods for Spoken Language Identification, Wav2Spk: A Simple DNN Architecture for Learning Speaker Embeddings from Waveforms, How to install (py)Spark on MacOS (late 2020), Wav2Spk, learning speaker emebddings for Speaker Verification using raw waveforms. Nat. Hamilton, W., et al. I hope this wav2vec series summary was useful. Word sequences are decoded using beam-search. Learning to solve circuit-SAT: An unsupervised differentiable approach ICLR, 2019. paper, code. Lin, Bo and Ghaddar, Bissan and Nathwani, Jatin. This allows us to identify the most predictive sequences against a given epitope. To verify whether the alignment was correct, we tested for significant overlap in cell type marker genes. Zhao, Hang and She, Qijin and Zhu, Chenyang and Yang, Yin and Xu, Kai. Commun. AlphaZero is a modified version of AlphaGo Zero which is able to play Shogi, chess, and Go. 13, 599604 (2018). We mark work contributed by Thinklab with . For the purpose of the algorithm, the maximum length can be altered but we chose 40 as we did not expect any real sequences to be longer than this length. Graph neural reasoning may fail in certifying boolean unsatisfiability Arxiv, 2019. paper, Guiding high-performance SAT solvers with unsat-core predictions SAT, 2019. paper, G2SAT: Learning to Generate SAT Formulas NeurIPS, 2019. paper, code. This vector of proportion features is then fed directly into the classification layer. Every autoencoder inherits from torch.nn.Module and has an encoder attribute and a decoder attribute, both of which also inherit from torch.nn.Module. The RMSprop optimizer with no momentum term is used to ensure the stability of adversarial training. Finally, the weights of the neural network are trained via gradient descent to jointly minimize both the reconstruction and variational loss. Sequencing files were parsed to take the amino acid sequence of the CDR3 after removing unproductive sequences. Simultaneous Planning for Item Picking and Placing by Deep Reinforcement Learning IROS, 2020. paper, Tanaka, Tatsuya and Kaneko, Toshimitsu and Sekine, Masahiro and Tangkaratt, Voot and Sugiyama, Masashi, Monte Carlo Tree Search on Perfect Rectangle Packing Problem Instances GECCO, 2020. paper, PackIt: A Virtual Environment for Geometric Planning ICML, 2020. paper, code, Online 3D Bin Packing with Constrained Deep Reinforcement Learning. [19] The team expanded their work to create a learning algorithm called MuZero that was able to "learn" the rules and develop winning strategies for over 50 different Atari games based on screen data. First, any of the available - or -chain CDR3 variable length sequences are provided to the network and are embedded via the use of a trainable embedding layer, as described by Sidhom et al.7, to learn properties/features of the amino acids and transform the sequences from a discrete to continuous numerical space. Cao, K., Bai, X., Hong, Y. [11] Various deep learning methods have been tested on both games, though most agents usually have trouble outperforming the default AI with cheats enabled or skilled players of the game.[1]. a contrastive loss \(L_m\), where the model needs to identify the true quantized latent speech representation, and distractors. Exploring the three-dimensional organization of genomes: Interpreting chromatin interaction data. The GLUE alignment helped improve the effects of cell typing in all omics layers, including the further partitioning of the scRNA-seq MGE cluster into Pvalb+ (mPv) and Sst+ (mSst) subtypes (highlighted with green circles/flows in Fig. D 58, 899907 (2002). 2016LOSS1. An Efficient Graph Convolutional Network Technique for the Travelling Salesman Problem Arxiv, 2019. paper, code, Chaitanya K. Joshi, Thomas Laurent, Xavier Bresson, POMO: Policy Optimization with Multiple Optima for Reinforcement Learning. Nat Commun 12, 1605 (2021). In order to initially assess the value of using deep learning as method of TCR featurization, we collected data for tetramer-sorted antigen-specific cells for nine murine (Db-F2, Db-M45, Db-NP, Db-PA, Db-PB1, Kb-M38, Kb-SIY, Kb-TRP2, Kb-m139) and seven human (A1-CTELKLSDY, A1-VTEHDTLLY, A2-GILGFVFTL, A2-GLCTLVAML, A2-NLVPMVATV, B7-LPRRSGAAGA, B7-TPRVTGGGAM) antigens where the ground truth label corresponds to a particular antigen specificity for an individual sequence20,21,22. Despite the emergence of experimental methods for simultaneous measurement of multiple omics modalities in single cells, most single-cell datasets include only one modality. In the final stage of training, the learning rate would be reduced by factors of 10 if the validation loss did not improve for consecutive epochs. [J] arXiv preprint arXiv:1812.11317. Z.J.C. First, we jointly cluster cells from all omics layers in the aligned cell embedding space using k-means. SCENIC: single-cell regulatory network inference and clustering. Widrich, M. et al. PubMedGoogle Scholar. Das, Dibyendu and Ahmad, Shahid Asghar and Venkataramanan, Kumar. Benchmarking atlas-level data integration in single-cell genomics. Higher mean average precision indicates higher cell type resolution, and higher Seurat alignment score indicates better omics mixing. At Skillsoft, our mission is to help U.S. Federal Government agencies create a future-fit workforce skilled in competencies ranging from compliance to cloud migration, data strategy, leadership development, and DEI.As your strategic needs evolve, we commit to providing the content and support that will keep your workforce skilled and ready for the roles of tomorrow. Such explicit feature conversion is straightforward, but has been reported to result in information loss19. Music Gesture for Visual Sound Separation, CVPR 2020 CVPR, 2021. paper, code. (A.6) Deep Learning in Image Classification. Therefore, we established a method by which we could identify the most predictive (i.e. Pre-training of neural networks has proven to be a great way to overcome limited amount of data on a new task. Particularly, recent advances in hypergraph modeling62,63 could facilitate the use of prior knowledge on regulatory interactions involving multiple regulators simultaneously, as well as enable regulatory inference for such interactions. 0. 34, 653665 (2018). Extended Data Fig. EfficientNet:Rethinking Model Scaling for Convolutional Neural Networks [icml2019] Beausang, J. F. et al. 4 Integration performance of GLUE with different numbers of highly variable genes. d, Increases in FOSCTTM at different prior knowledge corruption rates for integration methods that rely on prior feature relations (n=8 repeats with different corruption random seeds). In biological sequence analytics such as DeepTCR, investigators want to be able to extract the features/motifs the neural network learned to accomplish its task. 12d). While the aligned atlas was largely consistent with the original annotations29 (Supplementary Fig. Nat. clear whether they learn similar patterns or if they can be effectively combined. Google Scholar. Our results highlight the flexibility and capacity for deep neural networks to extract meaningful information from complex immunogenomic data for both descriptive and predictive purposes. [10]. As shown in previous work31, canonical adversarial alignment amounts to minimizing a generalized form of JensenShannon divergence among the cell embedding distributions of different omics layers: where \(q_k\left( {{{\mathbf{u}}}} \right) = {\Bbb E}_{{{{\mathbf{x}}}}_k \sim p_{{{{\mathrm{data}}}}}\left( {{{{\mathbf{x}}}}_k} \right)}q\left( {{{{\mathbf{u}}}}|{{{\mathbf{x}}}}_k;\phi _k} \right)\) represents the marginal cell embedding distribution of the kth layer.

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