deep clustering with convolutional autoencoders github

Sfaira accelerates data and model reuse in single cell genomics. For each cell, average precision (AP) computes the average cell type precision up to each cell type-matched neighbor, and mean average precision is the average average precision across all cells. Finally, a new acoustic model is trained on the pseudo-labeled data as well as the original labeled data. A representative positive cognate epitope is shown where the AUC for the cognate epitope in this classification problem is 1.0, suggesting that the repertoire against this epitope is statistically distinguishable from the controls leading us to believe that this is an antigen-specific response (Fig. Hidden layer dimensionality is the dimensionality of hidden layers in the data encoders and modality discriminator. antigen specificity) or continuous regressed value (i.e. add git ignore, update dynamic dispatch papers. affinity measurement). machinelearning. Preprocessing dimensionality is the reduced dimensionality used for the first transformation layers of the data encoders (see Methods). Molecular identity of human outer radial glia during cortical development. Stuart, T. et al. See Supplementary Table 1 for detailed information on single-cell omics datasets used in this study, including access codes and URLs. Trends Genet. Genet. & LeCun, Y.) In the second step, the data and graph autoencoders are updated according to equation (20). Dota 2 is a multiplayer online battle arena (MOBA) game. In Proc. It implements three different autoencoder architectures in PyTorch, and a predefined training loop. a contrastive loss \(L_m\), where the model needs to identify the true quantized latent speech representation, and distractors. This part briefly introduces the fundamental ML problems-- regression, classification, dimensionality reduction, and clustering-- and the traditional ML models and numerical algorithms for solving the problems. To better use the large data size, the hidden layer dimensionality was doubled to 512 from the default 256. 6a), while the Nephron data profiled four donors, all of which showed substantial batch effect against each other in both scRNA-seq and scATAC-seq (Supplementary Fig. In other words, we sum up the cosine similarities (raised to the power of 4 to increase contrast) between cluster i and all its matching clusters in other layers with cosine similarity >0.5, and then normalize by cluster size, which effectively balances the contribution of matching clusters regardless of their sizes. The network is trained with an Adam Optimizer (learning rate=0.001) to minimize the cross-entropy loss between the soft-maxed logits and the one-hot encoded representation of the discrete categorical outputs of the network. Student researchers from Carnegie Mellon University used computer vision techniques to create an agent that could play the game using only image pixel input from the game. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. 20, 241 (2019). By modeling regulatory interactions across omics layers explicitly, GLUE uniquely supports integrative regulatory inference for unpaired multi-omics datasets. Machine Learning for Combinatorial Optimization: a Methodological Tour d'horizon. b. Classification of SAT problem instances by machine learning methods. (Curran Associates, Inc., 2016). Text-to-Speech Synthesis, 2009. Posts ordered by most recently publishing date Across these various datasets, the level of non-specific signal varies given the technical difficulties associated with extracting true non-specific signatures of TCR responses, and we seek to demonstrate the value of applying deep learning in these scenarios to leverage knowledge about sequence homology to extract the true antigen-specific signals. In this tutorial, you will discover how to use Keras to develop and evaluate neural network models for multi-class classification problems. As a data-driven science, genomics largely utilizes machine learning to capture dependencies in data and derive novel biological hypotheses. AAAI Workshop, 2020. paper. 7b,d). By creating an adaptive ISRU function with a trainable and parameter (3), we found this improved the training of our network and allowed us to make a sequence-level assignment between 0 and 1 for each sequence to each learned concept in the model. Models are optimized by minimizing a CTC loss. Online iNMF16, LIGER17, Harmony18, bindSC33, and Seurat v3 (ref. For example, ATAC peaks located near the promoter of a gene would be encouraged to have similar embeddings to that of the gene, while DNA methylation in the gene promoter would be encouraged to have a dissimilar embedding to that of the gene. 46, 175185 (1992). To unify the cell type labels, we performed a nearest neighbor-based label transfer with the snmC-seq dataset as a reference. Tareen, A. [7] t-SNEt-SNEAn illustrated introduction to the t-SNE algorithmSNEt-SNELargeVist-SNE-CSDN, [8] DECIDECPython-GithubDEC-Keras-Githubpiiswrong/dec-GithubDCEC-Github. Zeng, H., Edwards, M. D., Liu, G. & Gifford, D. K. Convolutional neural network architectures for predicting DNAprotein binding. 14, 390403 (2013). Cicero predicts cis-regulatory DNA interactions from single-cell chromatin accessibility data. This part briefly introduces the fundamental ML problems-- regression, classification, dimensionality reduction, and clustering-- and the traditional ML models and numerical algorithms for solving the problems. clear whether they learn similar patterns or if they can be effectively combined. 13, 599604 (2018). 15) on the positive predicted sequences (prob>0.99) from the initial screen against non-cognate epitopes to learn TCR sequence-specific features that could distinguish responses between variants of the GAG TW10 epitope family. Deep Learning can do image recognition with much complex structures. & Reddy, S. T. Advanced methodologies in high-throughput sequencing of immune repertoires. Trends Biotechnol. Visualization of cell embeddings confirmed that the GLUE alignment was correct and accurate (Supplementary Fig. The number of studies in this area using DL is growing as new efficient models are proposed. 36, 428431 (2018). 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. https://doi.org/10.5281/zenodo.4498967 (2021). Science 357, 600604 (2017). dtaidistance - High performance library for time series distances (DTW) and time series clustering. Process. & A.S.B. A version in 2014 used n-grams to generate levels similar to the ones it trained on, which was later improved by making use of MCTS to guide generation. Residue Sensitivity Logos are shown for select antigen-specific TCRs. PubMed Central This demonstrates that ultra-low resource speech recognition is possible with self-supervised learning on unlabeled data. [4] Alphastar was initially trained with supervised learning, it watched replays of many human games in order to learn basic strategies. The GLUE training process was repeated four times with different random seeds. [6][4], Reinforcement learning is the process of training an agent using rewards and/or punishments. We first benchmarked GLUE against multiple popular unpaired multi-omics integration methods15,16,17,18,23,24,25,33 using three gold-standard datasets generated by recent simultaneous scRNA-seq and scATAC-seq technologies (SNARE-seq8, SHARE-seq9 and 10X Multiome34), along with two unpaired datasets (Nephron35 and MOp36). Springer, I., Besser, H., Tickotsky-Moskovitz, N., Dvorkin, S. & Louzoun, Y. Product quantization amounts to choosing quantized representations from multiple codebooks and concatenating them, which is then used as an input to a Gumbel softmax to select the quantized representations. e Following training of the model, sequence-level predictions can be obtained by running each TCR sequence in the cognate wells through the repertoire classifier allowing extraction of the antigen-specific sequences from the background noise of the T cell culture. Furthermore, using both sequence and V/D/J gene usage resulted in the highest AUC performance for both the murine and human antigens, suggesting both types of inputs provide distinct and contributary information to antigen specificity assignment in addition to encouraging a featurization of the TCR that is length invariant (Supplementary Fig. Mikolov, T., Sutskever, I., Chen, K., Corrado, G. & Dean, J. in Advances in Neural Information Processing Systems (eds. A Survey of Reinforcement Learning and Agent-Based Approaches to Combinatorial Optimization. 46, D794D801 (2018). UnionCom23, Pamona24 and GLUE were executed using the Python packages unioncom (v.0.3.0), Pamona (v.0.1.0) and scglue (v.0.2.0), respectively. Nat. Multi-omics single-cell data integration and regulatory inference with graph-linked embedding, \({{{\mathbf{X}}}}_1 \in {\Bbb R}^{N_1 \times \left| {{{{\mathcal{V}}}}_1} \right|},{{{\mathbf{X}}}}_2 \in {\Bbb R}^{N_2 \times \left| {{{{\mathcal{V}}}}_2} \right|},{{{\mathbf{X}}}}_3 \in {\Bbb R}^{N_3 \times \left| {{{{\mathcal{V}}}}_3} \right|}\), \({{{\mathcal{V}}}}_1,{{{\mathcal{V}}}}_2,{{{\mathcal{V}}}}_3\), \({{{\mathcal{G}}}} = \left( {{{{\mathcal{V}}}},{{{\mathcal{E}}}}} \right)\), \({{{\mathcal{V}}}} = {{{\mathcal{V}}}}_1 \cup {{{\mathcal{V}}}}_2 \cup {{{\mathcal{V}}}}_3\), \({{{\mathbf{V}}}} = \left( {{{{\mathbf{V}}}}_1^ \top ,{{{\mathbf{V}}}}_2^ \top ,{{{\mathbf{V}}}}_3^ \top } \right)^ \top\), \(\phi _1,\phi _2,\phi _3,\phi _{{{\mathcal{G}}}}\), \(\theta _1,\theta _2,\theta _3,\theta _{{{\mathcal{G}}}}\), \({{{\mathcal{V}}}}_k,k = 1,2, \ldots ,K\), \({{{\mathcal{X}}}}_k \subseteq {\Bbb R}^{\left| {{{{\mathcal{V}}}}_k} \right|}\), \({{{\mathbf{x}}}}_k^{(n)} \in {{{\mathcal{X}}}}_k,n = 1,2, \ldots ,N_K\), \({{{\mathbf{x}}_{k}}_{i}}^{(n)},i \in {{{\mathcal{V}}}}_k\), $$\begin{array}{*{20}{c}} {p\left( {{{{\mathbf{x}}}}_k;\theta _k} \right) = {\int} p \left( {{{{\mathbf{x}}}}_k|{{{\mathbf{u}}}};\theta _k} \right)p\left( {{{\mathbf{u}}}} \right){\mathrm{d}}{{{\mathbf{u}}}}} \end{array}$$, \(p\left( {{{{\mathbf{x}}}}_k|{{{\mathbf{u}}}};\theta _k} \right)\), \(q\left( {{{{\mathbf{u}}}}|{{{\mathbf{x}}}}_k;\phi _k} \right)\), $$\begin{array}{rcl}{{{\mathcal{L}}}}_{{{{\mathcal{X}}}}_k}\left( {\phi _k,\theta _k} \right) & = & {\Bbb E}_{{{{\mathbf{x}}}}_k \sim p_{{{{\mathrm{data}}}}}\left( {{{{\mathbf{x}}}}_k} \right)}\left[ {{\Bbb E}_{{{{\mathbf{u}}}} \sim q\left( {{{{\mathbf{u}}}}|{{{\mathbf{x}}}}_k;\phi _k} \right)}\log p\left( {{{{\mathbf{x}}}}_k|{{{\mathbf{u}}}};\theta _k} \right)}\right. The binding of a TCR to a peptide-major histocompatibility complex (pMHC) is not usually considered a binary phenomenon but rather one that is characterized by a binding affinity. 3 and Supplementary Fig. 9, 5345 (2018). Callaway, E. it will change everything: Deepminds AI makes gigantic leap in solving protein structures. State Key Laboratory of Protein and Plant Gene Research, School of Life Sciences, Biomedical Pioneering Innovative Center (BIOPIC) and Beijing Advanced Innovation Center for Genomics (ICG), Center for Bioinformatics (CBI), Peking University, Beijing, China, You can also search for this author in 0. For convenience, we also introduce the notation \({{{\mathbf{V}}}}_k \in {\Bbb R}^{m \times \left| {{{{\mathcal{V}}}}_k} \right|}\), which contains only feature embeddings in the kth omics layer, and uk, which emphasizes that the cell embedding is from a cell in the kth omics layer. The research by G.G. 81, 68136822 (2009). Commun. Scavuzzo, Lara, F. Chen, Didier Ch'etelat, Maxime Gasse, Andrea Lodi, N. Yorke-Smith and Karen Aardal. Notably, in a Bayesian interpretation, the GLUE regulatory inference can be seen as a posterior estimate, which can be continuously refined on the arrival of new data. & A.S.B. GPT-3 appears to be so good at this that you can use it for Question Answering on generic topics without fine-tuning, and get proper replies. Davis, C. A. et al. It gained a lot of attention lately, especially on Twitter with this headline that just 10 minutes of labeled speech can reach the same WER than a recent system trained on 960 hours of data, from just a year ago. Zhou, Z.-H. A brief introduction to weakly supervised learning. Commun. Cancer Immunol. This is solved by taking the discrete speech representation as an input to a Transformer architecture. Due to this complex layered approach, deep learning models often require powerful machines to train and Extended Data Fig. Generalize a Small Pre-trained Model to Arbitrarily Large TSP Instances. Google Scholar. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. 12c). Then, we cluster the coarsely aligned cell embeddings per omics layer using Leiden clustering. Arxiv, 2020. paper. b For the shown epitopes, experimentally derived antigen-specific CDR3 TCR sequences were collected from the McPAS-TCR database and models trained on the 10x Genomics dataset were applied to this independent dataset of TCRs to assess the classification performance via examining the ROC curves and their corresponding AUCs. 49, 659665 (2017). You signed in with another tab or window. 10b). 215223 (PMLR, 2018). Despite the emergence of experimental methods for simultaneous measurement of multiple omics modalities in single cells, most single-cell datasets include only one modality. Supervised pre-training is clear. Please send your up-to-date resume via yanjunchi AT sjtu.edu.cn. This last work combines both self-supervised training and pre-training for speech recognition. Awesome machine learning for combinatorial optimization papers. The scRNA-seq and scATAC-seq atlases have highly unbalanced cell type compositions, which are primarily caused by differences in organ sampling sizes (Supplementary Fig. 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. Cell 183, 11031116 (2020). 2016LOSS1. Hybrid Models for Learning to Branch NeurlPS, 2020. paper, code, Gupta, Prateek and Gasse, Maxime and Khalil, Elias B and Kumar, M Pawan and Lodi, Andrea and Bengio, Yoshua, Accelerating Primal Solution Findings for Mixed Integer Programs Based on Solution Prediction. Article Science 370, eaba7612 (2020). Preprint at https://www.biorxiv.org/content/10.1101/2019.12.18.880146v2.full (2020). & Kim, D. Deep convolutional neural networks for pan-specific peptide-MHC class I binding prediction. Estimating the ratio of CD4+ to CD8+ T cells using high-throughput sequence data. 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. Here the graph likelihood has no trainable parameters, so \(\theta _{{{\mathcal{G}}}} = \emptyset\). experimental exposures, therapies, clinical outcomes) apply to an entire repertoire of TCR sequences and not to any individual sequence. Aguet, F. et al. Regression . 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. Regarding the language model decoding, the authors considered a 4-gram language model, a word-based convolutional language model, and a character-based convolutional language model. This finding was consistent with the initial IFN- based approaches that found both the consensus and the escape variant both generated immune responses and furthermore, the I127M escape variant was recognized by other subjects who did not have that acquired mutation, confirming our model learned the true cross-reactive nature of this repertoire46. Test-Time Training with Masked Autoencoders Test-time training with MAE MAE ICML-14 DeCAFDeCAF: A Deep Convolutional Activation Feature for Generic Visual Recognition. Of note, this assignment of a sequence to a concept is done through an adaptive activation function that outputs a value between 0 and 1, allowing the network to put attention on the sequences that are relevant to the learning task. For the / CDR3 sequences, we take variable length right-padded sequence data which has been encoded in one-hot representation and first apply an embedding layer which transforms this one-hot representation to a trainable continuous representation of dimensionality 64. Bioinformatics 36, 22722274 (2020). Graduated Assignment for Joint Multi-Graph Matching and Clustering with Application to Unsupervised Graph Matching Network Learning. PubMed Central CVPR, 2018. paper, Zanfir, Andrei and Sminchisescu, Cristian, Learning Combinatorial Embedding Networks for Deep Graph Matching. A rapid and robust method for single cell chromatin accessibility profiling. 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. { - {{{\mathrm{KL}}}}\left( {q\left( {{{{\mathbf{u}}}}|{{{\mathbf{x}}}}_k;\phi _k} \right)\parallel p\left( {{{\mathbf{u}}}} \right)} \right)} \right]\end{array}$$, \({{{\mathcal{V}}}} = \mathop {\bigcup}\nolimits_{k = 1}^K {{{{\mathcal{V}}}}_k}\), \({{{\mathcal{E}}}} = \left\{ {\left( {i,j} \right)|i,j \in {{{\mathcal{V}}}}} \right\}\), \(s_{ii} = 1,w_{ii} = 1,\forall i \in {{{\mathcal{V}}}}\), \({{{\mathbf{v}}}}_i \in {\Bbb R}^m,i \in {{{\mathcal{V}}}}\), \({{{\mathbf{v}}}}_i \in {\Bbb R}^m,i \in {{{\mathcal{V}}}}_k\), \({{{\mathbf{V}}}} \in {\Bbb R}^{m \times \left| {{{\mathcal{V}}}} \right|}\), $$\begin{array}{*{20}{c}} {p\left( {{{{\mathbf{x}}}}_k,{{{\mathcal{G}}}};\theta _k,\theta _{{{\mathcal{G}}}}} \right) = {\int} p \left( {{{{\mathbf{x}}}}_k|{{{\mathbf{u}}}},{{{\mathbf{V}}}};\theta _k} \right)p\left( {{{{\mathcal{G}}}}|{{{\mathbf{V}}}};\theta _{{{\mathcal{G}}}}} \right)p\left( {{{\mathbf{u}}}} \right)p\left( {{{\mathbf{V}}}} \right){\mathrm{d}}{{{\mathbf{u}}}}{\mathrm{d}}{{{\mathbf{V}}}}} \end{array}$$, \(p\left( {{{{\mathbf{x}}}}_k|{{{\mathbf{u}}}},{{{\mathbf{V}}}};\theta _k} \right)\), \(p\left( {{{{\mathcal{G}}}}|{{{\mathbf{V}}}};\theta _{{{\mathcal{G}}}}} \right)\), $$\begin{array}{*{20}{c}} {p\left( {{{\mathbf{u}}}} \right) = N\left( {{{{\mathbf{u}}}};\mathbf{0},{{{\mathbf{I}}}}_m} \right)} \end{array}$$, $$\begin{array}{*{20}{c}} {p\left( {{{{\mathbf{v}}}}_i} \right) = N\left( {{{{\mathbf{v}}}}_i;\mathbf{0},{{{\mathbf{I}}}}_m} \right),p\left( {{{\mathbf{V}}}} \right) = \mathop {\prod }\limits_{i \in {{{\mathcal{V}}}}} p\left( {{{{\mathbf{v}}}}_i} \right)} \end{array}$$, \({{{\mathbf{V}}}}_k \in {\Bbb R}^{m \times \left| {{{{\mathcal{V}}}}_k} \right|}\), $$\begin{array}{rcl} {\log p\left( {{{{\mathcal{G}}}}|{{{\mathbf{V}}}};\theta _{{{\mathcal{G}}}}} \right)} = {{\Bbb E}_{i,j \sim p\left( {i,j;w_{ij}} \right)}} \\ {\left[ {\log \sigma \left( {s_{ij} {{{\mathbf{v}}}}_i^ \top {{{\mathbf{v}}}}_j} \right) + {\Bbb E}_{j\prime \sim p_{{{{\mathrm{ns}}}}}\left( {j\prime |i} \right)}\log \left( {1 - \sigma \left( {s_{ij} {{{\mathbf{v}}}}_i^ \top {{{\mathbf{v}}}}_{j\prime }} \right)} \right)} \right]} \end{array}$$, \(\theta _{{{\mathcal{G}}}} = \emptyset\), $$\begin{array}{*{20}{c}} {p\left( {{{{\mathbf{x}}}}_k|{{{\mathbf{u}}}},{{{\mathbf{V}}}};\theta _k} \right) = \mathop {\prod }\limits_{i \in {{{\mathcal{V}}}}_k} {{{\mathrm{NB}}}}\left( {{{\mathbf{x}}_{k}}_{i};\mathbf{\mu} _i,\mathbf{\theta} _i} \right)} \end{array}$$, $$\begin{array}{*{20}{c}} {{\mathrm{NB}}\left( {{{\mathbf{x}}_{k}}_{i};{{{\mathbf{\mu }}}}_i,{{{\mathbf{\theta }}}}_i} \right) = \frac{{{{{\mathrm{{\Gamma}}}}}\left( {{{\mathbf{x}}_{k}}_{i} + {{{\mathbf{\theta }}}}_i} \right)}}{{{{{\mathrm{{\Gamma}}}}}\left( {{{{\mathbf{\theta }}}}_i} \right){{{\mathrm{{\Gamma}}}}}\left( {{{\mathbf{x}}_{k}}_{i} + 1} \right)}}\left( {\frac{{{{{\mathbf{\mu }}}}_i}}{{{{{\mathbf{\theta }}}}_i + {{{\mathbf{\mu }}}}_i}}} \right)^{{\mathbf{x}_{k}}_{i}}\left( {\frac{{{{{\mathbf{\theta }}}}_i}}{{{{{\mathbf{\theta }}}}_i + {{{\mathbf{\mu }}}}_i}}} \right)^{{{{\mathbf{\theta }}}}_i}} \end{array}$$, $$\begin{array}{*{20}{c}} {{{{\mathbf{\mu }}}}_i = {{{\mathrm{Softmax}}}}_i\left( {{{{\mathbf{\alpha }}}} \odot {{{\mathbf{V}}}}_k^ \top {{{\mathbf{u}}}} + {{{\mathbf{\beta }}}}} \right) \cdot \mathop {\sum }\limits_{j \in {{{\mathcal{V}}}}_k} {\mathbf{x}_{k}}_{j}} \end{array}$$, \({{{\mathbf{\mu }}}},{{{\mathbf{\theta }}}} \in {\Bbb R}_ + ^{\left| {{{{\mathcal{V}}}}_k} \right|}\), \({{{\mathbf{\alpha }}}} \in {\Bbb R}_ + ^{\left| {{{{\mathcal{V}}}}_k} \right|},{{{\mathbf{\beta }}}} \in {\Bbb R}^{\left| {{{{\mathcal{V}}}}_k} \right|}\), \(\mathop {\sum}\nolimits_{j \in {{{\mathcal{V}}}}_k} {{\mathbf{x}_{k}}_{j}}\), \(\theta _k = \left\{ {{{{\mathbf{\theta }}}},{{{\mathbf{\alpha }}}},{{{\mathbf{\beta }}}}} \right\}\), $$\begin{array}{*{20}{c}} {q\left( {{{{\mathbf{u}}}},{{{\mathbf{V}}}}|{{{\mathbf{x}}}}_k,{{{\mathcal{G}}}};\phi _k,\phi _{{{\mathcal{G}}}}} \right) = q\left( {{{{\mathbf{u}}}}|{{{\mathbf{x}}}}_k;\phi _k} \right) \cdot q\left( {{{{\mathbf{V}}}}|{{{\mathcal{G}}}};\phi _{{{\mathcal{G}}}}} \right)} \end{array}$$, \(q\left( {{{{\mathbf{V}}}}|{{{\mathcal{G}}}};\phi _{{{\mathcal{G}}}}} \right)\), $$\begin{array}{*{20}{c}} {q\left( {{{{\mathbf{V}}}}|{{{\mathcal{G}}}};\phi _{{{\mathcal{G}}}}} \right) = \mathop {\prod }\limits_{i \in {{{\mathcal{V}}}}} q\left( {{{{\mathbf{v}}}}_i|{{{\mathcal{G}}}};\phi _{{{\mathcal{G}}}}} \right)} \end{array}$$, $$\begin{array}{*{20}{c}} {q\left( {{{{\mathbf{v}}}}_i|{{{\mathcal{G}}}};\phi _{{{\mathcal{G}}}}} \right) = N\left( {{{{\mathbf{v}}}}_i;{{{\mathrm{GCN}}}}_{{{{\mathbf{\mu }}}}_i}\left( {{{{\mathcal{G}}}};\phi _{{{\mathcal{G}}}}} \right),{{{\mathrm{GCN}}}}_{{{{\mathbf{\sigma }}}}_i^2}\left( {{{{\mathcal{G}}}};\phi _{{{\mathcal{G}}}}} \right)} \right)} \end{array}$$, $$\begin{array}{*{20}{c}} {q\left( {{{{\mathbf{u}}}}|{{{\mathbf{x}}}}_k,{{{\mathbf{V}}}}_k;\phi _k} \right) = N\left( {{{{\mathbf{u}}}};{{{\mathrm{MLP}}}}_{k,{{{\mathbf{\mu }}}}}\left( {{{{\mathbf{x}}}}_k;\phi _k} \right),{{{\mathrm{MLP}}}}_{k,{{{\mathbf{\upsigma }}}}^2}\left( {{{{\mathbf{x}}}}_k;\phi _k} \right)} \right)} \end{array}$$, $$\begin{array}{*{20}{c}} {\mathop {\sum}\limits_{k = 1}^K {{\Bbb E}_{{{{\mathbf{x}}}}_k \sim p_{{{{\mathrm{data}}}}}\left( {{{{\mathbf{x}}}}_k} \right)}} \left[ {\begin{array}{*{20}{c}} {{\Bbb E}_{{{{\mathbf{u}}}} \sim q\left( {{{{\mathbf{u}}}}|{{{\mathbf{x}}}}_k;\phi _k} \right),{{{\mathbf{V}}}} \sim q\left( {{{{\mathbf{V}}}}|{{{\mathcal{G}}}};\phi _{{{\mathcal{G}}}}} \right)}\log p\left( {{{{\mathbf{x}}}}_k|{{{\mathbf{u}}}},{{{\mathbf{V}}}};\theta _k} \right)p\left( {{{{\mathcal{G}}}}|{{{\mathbf{V}}}};\theta _{{{\mathcal{G}}}}} \right)} \\ { - \mathrm{KL}\left( {q\left( {{{{\mathbf{u}}}}|{{{\mathbf{x}}}}_k;\phi _k} \right)q\left( {{{{\mathbf{V}}}}|{{{\mathcal{G}}}};\phi _{{{\mathcal{G}}}}} \right)\parallel p\left( {{{\mathbf{u}}}} \right)p\left( {{{\mathbf{V}}}} \right)} \right)} \end{array}} \right]} \end{array}$$, $$\begin{array}{*{20}{c}} {K \cdot {{{\mathcal{L}}}}_{{{\mathcal{G}}}}\left( {\theta _{{{\mathcal{G}}}},\phi _{{{\mathcal{G}}}}} \right) + \mathop {\sum}\limits_{k = 1}^K {{{{\mathcal{L}}}}_{{{{\mathcal{X}}}}_k}} \left( {\theta _k,\phi _k,\phi _{{{\mathcal{G}}}}} \right)} \end{array}$$, $$\begin{array}{rcl}{{{\mathcal{L}}}}_{{{{\mathcal{X}}}}_k}\left( {\theta _k,\phi _k,\phi _{{{\mathcal{G}}}}} \right) = {\Bbb E}_{{{{\mathbf{x}}}}_k \sim p_{{{{\mathrm{data}}}}}\left( {{{{\mathbf{x}}}}_k} \right)} \\ \left[ {{\Bbb E}_{{{{\mathbf{u}}}} \sim q\left( {{{{\mathbf{u}}}}|{{{\mathbf{x}}}}_k;\phi _k} \right),{{{\mathbf{V}}}} \sim q\left( {{{{\mathbf{V}}}}|{{{\mathcal{G}}}};\phi _{{{\mathcal{G}}}}} \right)}\log p\left( {{{{\mathbf{x}}}}_k|{{{\mathbf{u}}}},{{{\mathbf{V}}}};\theta _k} \right) - {{{\mathrm{KL}}}}\left( {q\left( {{{{\mathbf{u}}}}|{{{\mathbf{x}}}}_k;\phi _k} \right)\parallel p\left( {{{\mathbf{u}}}} \right)} \right)} \right]\end{array}$$, $$\begin{array}{*{20}{c}} {{{{\mathcal{L}}}}_{{{\mathcal{G}}}}\left( {\theta _{{{\mathcal{G}}}},\phi _{{{\mathcal{G}}}}} \right) = {\Bbb E}_{{{{\mathbf{V}}}} \sim q\left( {{{{\mathbf{V}}}}|{{{\mathcal{G}}}};\phi _{{{\mathcal{G}}}}} \right)}\log p\left( {{{{\mathcal{G}}}}|{{{\mathbf{V}}}};\theta _{{{\mathcal{G}}}}} \right) - \mathrm{KL}\left( {q\left( {{{{\mathbf{V}}}}|{{{\mathcal{G}}}};\phi _{{{\mathcal{G}}}}} \right)\parallel p\left( {{{\mathbf{V}}}} \right)} \right)} \end{array}$$, \(\phi = \left( {\mathop {\bigcup}\nolimits_{k = 1}^K {\phi _k} } \right) \cup \phi _{{{\mathcal{G}}}}\), \(\theta = \left( {\mathop {\bigcup}\nolimits_{k = 1}^K {\theta _k} } \right) \cup \theta _{{{\mathcal{G}}}}\), $$\begin{array}{*{20}{c}} {{{{\mathcal{L}}}}_{{{\mathrm{D}}}}\left( {\phi ,\psi } \right) = - \frac{1}{K}\mathop {\sum }\limits_{k = 1}^K {\Bbb E}_{{{{\mathbf{x}}}}_k \sim p_{{{{\mathrm{data}}}}}\left( {{{{\mathbf{x}}}}_k} \right)}{\Bbb E}_{{{{\mathbf{u}}}} \sim q\left( {{{{\mathbf{u}}}}|{{{\mathbf{x}}}}_k;\phi _k} \right)}\log {{{\mathrm{D}}}}_k\left( {{{{\mathbf{u}}}};\psi } \right)} \end{array}$$, $$\begin{array}{*{20}{c}} {\mathop {{\min }}\limits_\psi \lambda _{{{\mathrm{D}}}} \cdot {{{\mathcal{L}}}}_{{{\mathrm{D}}}}\left( {\phi ,\psi } \right)} \end{array}$$, $$\begin{array}{*{20}{c}} {\mathop {{\max }}\limits_{\theta ,\phi } \lambda _{{{\mathrm{D}}}} \cdot {{{\mathcal{L}}}}_{{{\mathrm{D}}}}\left( {\phi ,\psi } \right) + \lambda _{{{\mathcal{G}}}}K \cdot {{{\mathcal{L}}}}_{{{\mathcal{G}}}}\left( {\theta _{{{\mathcal{G}}}},\phi _{{{\mathcal{G}}}}} \right) + \mathop {\sum }\limits_{k = 1}^K {{{\mathcal{L}}}}_{{{{\mathcal{X}}}}_k}\left( {\theta _k,\phi _k,\phi _{{{\mathcal{G}}}}} \right)} \end{array}$$, $$\frac{1}{K}\mathop {\sum}\limits_{k = 1}^K {{{{\mathrm{KL}}}}} \left( {q_k({{{\mathbf{u}}}})||\frac{1}{K}\mathop {\sum}\limits_{k = 1}^K {q_k} ({{{\mathbf{u}}}})} \right)$$, \(q_k\left( {{{\mathbf{u}}}} \right) = {\Bbb E}_{{{{\mathbf{x}}}}_k \sim p_{{{{\mathrm{data}}}}}\left( {{{{\mathbf{x}}}}_k} \right)}q\left( {{{{\mathbf{u}}}}|{{{\mathbf{x}}}}_k;\phi _k} \right)\), \(q_i\left( {{{\mathbf{u}}}} \right) = q_j\left( {{{\mathbf{u}}}} \right),\forall i \ne j\), $$\begin{array}{*{20}{c}} {{{{\mathcal{L}}}}_{{{\mathrm{D}}}}\left( {\phi ,\psi } \right) = - \frac{1}{K}\mathop {\sum }\limits_{k = 1}^K \frac{1}{{W_k}}\mathop {\sum }\limits_{n = 1}^{N_k} w^{\left( n \right)} \cdot {\Bbb E}_{{{{\mathbf{u}}}} \sim q\left( {{{{\mathbf{u}}}}|{{{\mathbf{x}}}}_k^{\left( n \right)};\phi _k} \right)}\log {{{\mathrm{D}}}}_k\left( {{{{\mathbf{u}}}};\psi } \right)} \end{array}$$, \(W_k = \mathop {\sum}\nolimits_{n = 1}^{N_k} {w^{\left( n \right)}}\), \(q_k\left( {{{\mathbf{u}}}} \right) = \frac{1}{{W_k}}\mathop {\sum}\limits_{n = 1}^{N_k} {w^{\left( n \right)}} q\left( {{{{\mathbf{u}}}}|{{{\mathbf{x}}}}_k^{\left( n \right)};\phi _k} \right)\), \({\boldsymbol{\epsilon}} \sim {{{\mathcal{N}}}}\left( {{\boldsymbol{\epsilon}} ;\mathbf{0},{\mathbf{\Sigma}}} \right)\), $$\begin{array}{*{20}{c}} {w_i = \frac{{\mathop {\sum }\nolimits_{k_i \ne k_j} f\left( {{{{\mathbf{u}}}}_i,{{{\mathbf{u}}}}_j} \right)}}{{n_i}}} \end{array}$$, $$\begin{array}{*{20}{c}} {f\left( {{{{\mathbf{u}}}}_i,{{{\mathbf{u}}}}_j} \right) = \left\{ {\begin{array}{*{20}{l}} {\cos \left( {{{{\mathbf{u}}}}_i,{{{\mathbf{u}}}}_j} \right)^4,} \hfill & {{\mathrm{cos}}({{{\mathbf{u}}}}_i,{{{\mathbf{u}}}}_j) > 0.5} \hfill \\ {0,} \hfill & {{\mathrm{otherwise}}} \hfill \end{array}} \right.}

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