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## Main

The progress and development of experimental technologies1,2,3,4 and computational tools5,6,7,8,9 for single-cell genomics have enabled the construction of atlases with millions of cells serving as high-resolution coordinate systems10 for biological and therapeutic discoveries11,12,13,14. However, leveraging existing atlases poses a computational challenge known as reference mapping enabling rapid integration of newly generated datasets, denoted as a query. The transfer of knowledge from the reference to the query allows the rapid annotation of the query data7, imputation of missing modalities in the query8,15 and the discovery of novel populations8,15.

Single-cell reference mapping is growing in popularity8,15,16,17,18 to map query datasets by minimal modification of the reference atlas19. Existing reference mapping methods embed new query data into a reference latent space by removing technical differences, such as batch effects between the reference and the query, without access to reference data. However, the implicitly used latent dimensions for joint data representation are not directly interpretable.

An important trend in machine learning is the development of interpretable models, for example, by adding statistical assumptions to learned latent spaces or including prior information from validated mechanisms or other data20. As the former disentanglement approaches have not yielded sufficiently useful latent spaces in our context21,22,23, we hypothesize that using prior information may help identifiability. In particular, we aim to leverage known or newly learned gene programs (GPs) to contextualize query data by answering various questions, including ‘which GPs are disturbed in a disease query data compared with the healthy reference?’ and ‘which biological programs explain a novel population in the query?’ By thus making reference mapping interpretable, it can move beyond mere data alignment between query and reference and be used for further interpretation of query data for example, in the case of disease perturbation versus a healthy atlas. Currently, the standard approach for identifying biological programs in query cells compared with a reference atlas is to test for differentially expressed genes and downstream gene set enrichment. However, the differential expression on an atlas consisting of cells from an arbitrary number of studies with variable degrees of biological and technical heterogeneity represents a challenge for statistical analysis. The currently accepted best practices24,25 suggest that differential expression should be performed on non-integrated expression data and not on the corrected expression values after integration; hence, statistical models should account for complex experimental designs and adjust for batch effects, which is further hampered by modelling constraints such as parameter identifiability. Instead of using simpler non-parametric statistical tests, both biologically relevant and irrelevant genes may be captured, which may compromise the accuracy of enriched gene set terms.

Collectively, it may be useful to have interpretable embeddings directly associated with validatable GPs in the context of atlas-wide comparisons to capture the relevant biological signals while accounting for nonlinear batch effects. This end-to-end approach is common in deep learning and has been shown to outperform classical approaches that use sequential regularization and analysis20. Interpretable reference mapping requires incorporating domain knowledge20, such as curated GPs, into the representation learning model to guide interpretation and exploration. Including domain knowledge to design ‘domain-informed’ deep learning architectures has been shown to improve the performance on challenging prediction tasks, from tumour type26 to protein structure27. Earlier works proposed incorporating regularized linear decodes to include domain knowledge into autoencoders for single-cell data28, with scalable and expressive embeddings compared with existing factor models, such as f-scLVM29. Recent approaches such as VEGA30, scETM31 and pmVAE32 also feature variational autoencoders with linear decoders or training separate VAEs for each GP yet connected via a global loss in the case of pmVAE. Yet, accounting for the incompleteness of domain knowledge and learning new knowledge de novo from the data, rather than being locked into prior-based feature design, are not fully addressed by existing methods. Finally, going beyond single dataset analysis towards large-scale data integration and reference mapping while injecting domain knowledge remains challenging.

In this Technical Report, to address these challenges, we propose to build a machine learning system that exploits the knowledge of the underlying biological phenomenon for single-cell representation learning (as outlined more generally in the idea of ‘differential programs’20 recently). We construct an ‘explainable programmable mapper’ (expiMap) as an interpretable conditional variational autoencoder7,33,34 that allows the incorporation of domain knowledge by performing ‘architecture programming’, that is, constraining the network architecture to ensure that each latent dimension captures the variability of known GPs. We apply an attention-like mechanism35 to select the relevant GPs for each reference dataset. This helps with the prioritization of essential gene sets but also allows the inclusion of genes not initially included in annotated GPs, thereby addressing the incomplete nature of the knowledge database. To identify new variations unique to the query data, such as disease effects, we identify de novo GPs in addition to the known GPs in knowledge bases by learning disentangled latent representations. The framework can be used to automatically identify and explore biological processes in normal and disease states when mapping new query datasets to the atlas while maintaining comparable integration performance to existing data integration methods.

## Results

### Interpretable single-cell reference mapping using expiMap

Linear methods, such as principal component analysis (PCA)36,37 or matrix factorization38,39, learn a representation of the data where each dimension of the latent space can be explained using a weighted combination of the input, such as gene expression. This interpretability comes at the cost of the model’s limited capacity (for example, only capturing linear relationships) to fit the data. In contrast, nonlinear methods using deep neural networks40,41 come with a larger capacity at the expense of reduced model interpretability.

Here we aim to design a system that can provide biologically interpretable answers to queries of an integrated representation of multiple (denoted by N) reference single-cell datasets and custom GPs. These can be gene lists from existing curated databases42,43, lists extracted from literature44 or individually curated gene sets (Fig. 1a). This knowledge is transformed into a binary GP matrix, in which each row is a GP. Each column denotes the membership of a gene in that program (Methods and Fig. 1b).

We wire the network weights using the GP matrix such that each latent variable contributes to the reconstruction of a set of genes defined by the GP similar to28,30. The model receives a gene expression matrix from N different single-cell studies (X) and an additional vector for corresponding one-hot encoded study labels (S1:N) for each cell, for example, the experimental laboratories or sequencing technologies (Fig. 1c). The adopted variational autoencoder architecture7,40 leverages a nonlinear encoder for flexibility and a linear decoder45 for interpretability. The latent space dimension chosen is equal to the number of GPs. The weights from each latent dimension (that is, latent GP) to output are programmed according to the GP matrix so that a latent GP can only contribute to the reconstruction of genes in a particular GP (denoted as ‘fixed membership’ in Fig. 1c). As annotated GPs are often incomplete, we allow the inclusion of other genes in each GP by applying L1 sparsity regularization to genes not initially labelled to belong to that GP (denoted as ‘soft membership’ in Fig. 1c). This enables the model to leverage the sparse selection of other genes, which helps in the reconstruction and therefore accounts for incomplete domain knowledge, to refine ontologies and pave the way towards a data-driven alternative means to learn GPs (see later results).

However, the number of GPs may be very large, and potentially redundant, and not all are relevant for every atlas. To select only informative GPs, an attention-like mechanism is implemented with a group lasso regularization layer in latent space (Methods), which de-activates GPs that are redundant or do not contribute to the reconstruction error of the model. The model is trained end to end and can thus be used to construct reference atlases with interpretable embedding dimensions, which we can leverage to analyse integrated datasets.

On the basis of this pre-trained, interpretable reference model, we propose employing transfer learning, as outlined in architectural surgery15 (Methods), to map new datasets into the reference. We modify the strategy of fine-tuning conditional weights in scArches allowing the model to learn new GPs that are not included in the reference model. This is achieved by adding new latent space dimensions, that is, nodes with trainable weights in the bottleneck layer of the model (Fig. 1d and Methods), while keeping the rest frozen. We implement two ways of learning these new GPs: either by learning GPs confined to pre-defined genes (denoted as ‘new constrained’ in Fig. 1d) that were not present or those that have been de-activated in the reference model. In addition, the model may also learn de novo GPs as realized by an L1-regularized gene before capture of new variations in the query data without pre-defined gene sets (denoted as ‘new unconstrained’ in Fig. 1d). The limited learning capacity of the model at the reference mapping stage, due to frozen weighting, enforces an information bottleneck (that is, a reduced capacity to learn and store information), encouraging the new nodes to learn important and potentially disentangled46 sources of variations in the query data. We further employ the Hilbert–Schmidt independence criterion (HSIC)22,47, a kernel-based measure of latent variable independence47, to further enforce independence between old and new unconstrained GPs learned during query optimization (Fig. 1d).

The probabilistic representation learned by expiMap as a Bayesian model allows the performance of hypothesis testing on the integrated latent space of the query and the reference accounting for technical factors (Methods). The hypothesis testing is performed at the GP level, identifying differential GPs between two groups of cells by sampling from the group’s posterior distribution of the latent variables. The ratio between two hypothesis probabilities is reported by the Bayes factor. Later, we demonstrate how this ability helps to identify GPs associated with a perturbation in the query data compared with the healthy reference. When talking about the results of the expiMap Bayes test, we call the GPs ‘enriched’ if their absolute logarithmic Bayes score is greater than or equal to 2.3.

Collectively, through expiMap, we propose an approach to learning interpretable, domain-aware representations of single-cell datasets for the integrative analysis of reference and query data. Further, we propose a modified version of architecture surgery that goes beyond pre-defined domain knowledge while retaining interpretability. This allows contextualizing the query data with the reference data within a specific GP to answer the user’s biological questions.

### expiMap parses transcriptional response to IFN-β

One of the ultimate goals in building large, single-cell atlases is studying the effect of perturbations (for example, disease) and contextualizing it within a given healthy reference. To demonstrate the applicability of our model in this scenario, we constructed a human immune cell atlas from four studies of bone marrow48 and peripheral blood mononuclear cells (PBMCs)49,50,51. We then mapped a query PBMC dataset of samples from eight patients diagnosed with systemic lupus erythematosus whose cells were either untreated (control) or treated with interferon (IFN)-β, a potent cytokine inducing a strong transcriptional response in immune cells52. Successful mapping should align untreated cells to matching cell types in the healthy reference while preserving the strong effect of IFN-β. The expiMap model trained with GPs extracted from the Reactome42,43 pathway knowledgebase successfully mapped the query untreated cells to the healthy reference while forming clusters indicative of the IFN-β-treated cells (Fig. 2a).

By testing between IFN-β and control conditions, we identified the top differential GPs, matching to previously reported GPs53,54 including IFN-related pathways (Fig. 2b), which also separates the control reference and query cells from stimulated query cells (Fig. 2c). Following up with a cell-type-specific analysis, we identified differential GPs across cell types (that is, one versus all) or cell-type-specific IFN-β effects (that is, IFN-β versus control within a cell type). In particular, we detected a group of population-specific GPs that separated one cell type from the rest (Fig. 2d, first row). The population-specific GPs can be used together with perturbation-associated GPs (that is, obtained from IFN-β query cells versus control cells in both query and reference for that cell type) to resolve the heterogeneity of cell state for that cell type (Fig. 2d, second row; for all cell types, see Extended Data Fig. 1 and Supplementary Figs. 13). We found that the general IFN GPs (for example, IFN signalling) are always induced in all cell types (Fig. 2e and Supplementary Figs. 2 and 3). In contrast, some GPs (for example, GPCR-related programs; their genes are provided in Supplementary Tables 1 and 2), including genes from the CXC chemokine family (for example, CXCL10), are present only in the myeloid lineage (for highlighted GPs, see Fig. 2e; for all extended figures, see Supplementary Figs. 2 and 3). Additionally, we detected carbohydrate metabolism activity in CD14+ and CD16+ monocytes and dendritic cells (DCs), and active amino acid metabolism in CD14+ monocytes after IFN-β stimulation (Supplementary Figs. 2 and 3). This is in agreement with previous observations in cancer and viral infection showing that amino acid, lipid and carbohydrate metabolic pathways contribute to the immune response55,56. Specifically, it is known that IFN-β engages with the amino acid metabolic pathway to produce polyamines and clear viral infections57. Still, a direct link to myeloid cells, as revealed by expiMap, has not been reported elsewhere.

Differential expression analysis on atlases is challenging due to the complex experimental designs and the probable presence of nonlinear batch effects that cannot be modelled by linear approaches. Gene set enrichment analysis (GSEA) is a classical approach for inferring the activity of GPs and involves the sequential pipeline of differential expression analysis and gene set enrichment test. To evaluate the robustness of expiMap’s integrated GP test, we hence compared it with the classical GSEA via limma-fry58,59 (Supplementary Note 1 and Extended Data Fig. 2). In our comparisons, we observed that, unlike conventional gene set testing, which tends to detect general, non-specific terms, expiMap was able to identify specialized GPs. For example, in the B-cell population of both IFN-β-treated and control cells, expiMap detected B-cell receptor signalling and antigen presentation activity, which are more descriptive of B-cell biology than the general terms such as ‘adaptive immune response’ or ‘immune response’ that were found to be enriched in these cells by limma-fry (Extended Data Fig. 2c). We postulate that the increased variability in gene expression measurements hinders the detection of specialized biological signals by standard gene set testing on cell atlases. This indicates that expiMap can extract biologically relevant GPs from a single-cell atlas consisting of many datasets while accounting for technical variations such as batch effects, which may not always be feasible with existing pipelines, owing to the presence of nonlinear batch effects.

To further analyse the contribution of individual genes in each GP, we introduce the gene importance score: the absolute value of decoder weights for genes in GPs (see also Methods), which can measure the comparative importance of genes within each GPs. Using the importance score, we analysed the dependence between the expression levels of genes and their importance scores in various GPs (Supplementary Note 2 and Extended Data Fig. 3). We also confirmed the robustness of the model under different data query dataset sizes (Supplementary Note 3 and Extended Data Fig. 4). Finally, we compared reference mapping and individual analysis of query data by applying expiMap on IFN-β dataset alone and repeated analogous analysis as shown in Fig. 2. We found the results similar (Supplementary Note 4 and Extended Data Fig. 5).

### Biologically informed modelling improves the performance

As a means to benchmark the performance of expiMap’s reference mapping component, we compared it with scArches + scVI7, Seurat v48 and Symphony18. Although expiMap and scVI both leverage scArches for reference mapping, scVI did not mix the untreated monocytes from the query data with healthy monocytes in the reference (dotted circle in Fig. 3a; for mixing of studies, see Supplementary Fig. 4), whereas expiMap successfully integrated them into the healthy reference (0.68 versus 0.47 average batch correction scores; see further for a description of the metrics) while preserving the effect of IFN-β treatment in cells that should not be integrated with the rest. We attribute this to the explicit incorporation of the IFN-β-related GPs in the expiMap model, which helps differentiate the perturbed and control states while resolving the transcriptional similarities between control cells, leading to better mixing of control states. We investigated this by removing the top five GPs obtained from the IFN-β versus control comparison (Fig. 2b) and retraining the model. We observed that this led to the incorrect mixing of control and stimulated cells with the reference (Supplementary Fig. 5). In this example, both scArches + scVI and expiMap had better performance than Seurat v4 and Symphony for integrating control query cells into control cells from the reference (Fig. 3b). We also quantitatively evaluated the integration of query control cells into the healthy reference using nine different metrics of biological preservation and mixing60.

We further benchmarked expiMap in de novo integration against scVI and non-amortized scVI (Fig. 3c), and linear-decoded variational autoencoder (LDVAE)45, a variation of scVI with a linear decoder (Extended Data Fig. 6a). Overall, we found that additional domain knowledge distilled into expiMap compensates for the lower model capacity compared with nonlinear models enabling it to achieve competitive performance (Supplementary Note 5). This is aligned with recent results2,20 demonstrating the improved performance of deep learning-based models by integrating domain knowledge into modelling.

### Learning new GPs

Leveraging domain knowledge is crucial for the rapid and interpretable analysis of new query datasets within the context of a reference atlas. However, domain knowledge is not always comprehensive, complete and up to date for a novel phenomenon (for example, a new disease). Thus, the ability to learn new GPs to analyse query data containing new variations, such as new states or cell populations, is pivotal. We address this by allowing expiMap to learn novel GPs associated with the query data that exist in the knowledge base but are not detected previously in the reference model, as well as de novo programs that are not described in the knowledge base (Methods).

We further confirmed the independence of newly learned GPs (Supplementary Note 6 and Supplementary Table 6). Finally, we showed our model is robust in learning new GPs and existing GPs across different data subsampling scenarios and model hyperparameters (for detailed analysis, see Supplementary Note 6 and Supplementary Tables 712). Overall, we demonstrated that expiMap can learn pre-defined GPs not in the reference GP matrix for populations present only in the query data during query training while having the ability to enrich the pre-defined GPs with new genes (see also Supplementary Note 7 and Extended Data Fig. 7f, g), not in the program. In addition, we demonstrated that expiMap is not restricted to pre-defined GPs and can learn de novo GPs without any user supervision or prior knowledge.

### Interpreting treatment responses of patients with COVID-19

To demonstrate the medical use of interpretable atlas querying, we focused on the cellular response to infection during coronavirus disease 2019 (COVID-19) and the effect of immunosuppressive interventions. We leveraged the integrated immune PBMC atlas to map IFN-β dataset (as in Fig. 2a) and a new dataset from two patients (P1 and P2) at different COVID stages (severe disease and during the remission process: D1, severe COVID-19 on day 1; D5 and D7, remission on days 5 and 7, respectively). Both patients were treated with tocilizumab, an immunosuppressive drug targeting the interleukin-6 receptor62. The integrated dataset (Fig. 5b,c) was re-annotated using canonical markers identifying 20 cell states from the myeloid and lymphoid compartments, including rare populations such as megakaryocytes and erythroid progenitors, as well as a population of CD10+ B cells (Fig. 5c). From the integrated embedding produced by expiMap, we could observe that some cellular states are associated with disease severity, which may be related to differences in the cellular response to tocilizumab.

Our analyses pointed us towards CD8+ T cells and monocytes (Fig. 5c) in both severe and remission stages that did not integrate into the healthy reference, unlike other populations from the same patients. We investigated this by performing a differential GP test between severe query cells and control cells to identify GPs that could explain this separation. We identified transcriptional programs of antiviral response at different clinical stages of COVID-19 and in specific PBMC cell types. Pathogen recognition receptor (PRR) RIG-I/MDA5 and GPCR pathways displayed differential behaviour in CD8+ T cells (Fig. 5d) and CD14+ monocytes (Fig. 5e) in severe COVID-19 (D1) and during remission (D5 and D7). RIG-I/MDA5 and GPCR pathways initiate the innate immune response and modulate the adaptive immune responses during viral infections63 and are reported to coordinate the inflammatory dynamics during COVID-19 (refs. 64,65). These findings suggest that a complex cellular communication circuit may be differentially activated in both patients and may be related to the differences in treatment response at the cellular level.

Next, we estimated underlying cellular communication pathways using CellChat66 and compared them at different clinical stages in our integrated Immune atlas. This analysis revealed that the annexin pathway displayed differential transcriptional behaviour in the severe and remission stages of P1 and P2, involving CD14+ and CD16+ monocytes, natural killer (NK) cells and CD8+ T cells (Fig. 5f and Supplementary Fig. 6). Annexins are structural proteins that participate in the regulation of inflammatory responses and homeostasis67 and have been associated with disease severity in COVID-19 (refs. 68,69). In this circuit, CD14+ and CD16+ monocytes show the potential to receive signals from NK cells and CD8+ T cells for P1D1. In P1D5, the annexin pathway switches completely to signalling between CD16+ monocytes and CD4+ T cells. In stark contrast, P2D1 is characterized by the annexin pathway focusing on CD14+ monocytes, which continues throughout the remission stage (P2D5), with the addition of CD16+ monocytes persisting towards D7 of remission (P2D7) (Supplementary Fig. 6).

Although expression levels of annexins have been described as biomarkers for the prediction of disease severity69, our analysis using expiMap is the first to describe the expression of ligand–receptor pairs from the annexin pathway at the cellular level with the potential to interact between monocytes (CD14+ and CD16+), NK cells and T cells in COVID. The differences observed between patients in the expression of annexin-related interaction circuits may be related to the capability of viral clearance in each patient62 and the early expression of FPR1 by CD16+ monocytes, which is associated with the early detection of pathogenic molecules and tissue damage70. Interestingly, our analysis shows the expression of IFNG for P2D7 by NK and CD8+ T cells (Extended Data Fig. 8), which may indicate a more complex antiviral response than in P1, independent of the symptomatic resolution attained by tocilizumab. Moreover, when contrasting our results with the annexin circuit in the data from IFN-stimulated cells, we observed that the inferred cell–cell interactions using the annexin pathway were dominated by the expression of ANXA1 in DCs rather than FPR1 in CD14+ monocytes (Extended Data Fig. 8). Our results do not illustrate the same circuit; however, this may indicate a lung-specific interaction operating in the lungs after the monocytes migrate to the affected lung tissue.

Although both these patients recovered after treatment with tocilizumab, clinical studies demonstrate that this behaviour is not consistent, and other factors, such as tocilizumab posology, may affect the clinical outcome71. At the cellular level, expiMap identifies transcriptional and cell–cell interaction circuits with the potential to be druggable, such as RIG-I/MDA5 and annexins, to help suppress cytokine storm syndrome in patients with COVID-19, which results in hospitalization.

### expiMap resolves disease heterogeneity in Pancreas

As a final use case, we asked whether expiMap could assist with interpretable cell type annotation and the analysis of cell state heterogeneity. We used expiMap to integrate three non-type 2 diabetes (T2D) pancreatic datasets72,73,74,75 (Methods and Supplementary Note 8) differing in multiple biological factors, including sex, age and stress status, using PanglaoDB marker sets to enable cell type identification76,77 and Reactome pathways to identify molecular processes78 differentially active between biological conditions. Before integration, we removed immune cells from the reference to assess whether new cell types in the query could be successfully integrated. We projected a new dataset (query) that included healthy and T2D cells into this reference (Fig. 6a,b). On the integrated embedding (Fig. 6b), a separation between studies is observed. This is expected due to biological differences between the integrated mouse models, such as disease state and age. We also performed integrations with scArches + scVI, Seurat V4 and Symphony (Extended Data Fig. 9a) and assessed the integration quality using scIB metrics (Extended Data Fig. 9b), showing that expiMap is one of the top-performing methods.

Next, we automatically transferred cell type annotations from reference to query (Fig. 6d, Supplementary Fig. 7 and Methods). Analysing expiMap-generated scores of pancreatic cell type-associated PanglaoDB GPs (Supplementary Fig. 7c) helped with the annotation of ambiguous cell clusters (Supplementary Fig. 7b). For example, expiMap scores helped to resolve potential doublets (for example, immune–endocrine doublets) and small cell populations (for example, acinar cells) that were marked as unknown or wrongly annotated (Fig. 6d and Supplementary Fig. 7b,c). As automated cell type annotation methods often produce unreliable results in challenging cell populations, such as doublets, rare cell types or transitional cell states, manual assessment of marker expression is still required. However, expression can be affected by batch effects76, while expiMap scores are directly comparable. Furthermore, when specific cell types are missing from the reference, the annotation transfer cannot be performed, such as for the immune cells that were present only in the query (Fig. 6a and Supplementary Fig. 7a). In such a case, expiMap enables GP-enrichment analysis to provide insights into cell types. Similarly, expiMap scores can resolve coarsely annotated cell types. We show that the GP cell type scores for the immune cell subpopulations (Fig. 6d) provide similar information as the manually curated markers (Supplementary Fig. 8). As online marker databases often contain multiple putative markers, often of insufficient quality, manual selection of markers becomes challenging79,80. Indeed, we tried to use the top PanglaoDB markers for B-cell annotation (Ebf1, Cd74 and Cd52 out of 110 markers). However, they lacked sufficient specificity and sensitivity, while expiMap score based on all PanglaoDB markers correctly annotated B-cell lineage, corresponding to the B-cell lineage marker Cd79a81 as well as non-activated B cell (Cd19 and Ms4a1) and activated plasma B cell (Jchain) markers (Supplementary Fig. 8). This can be explained by the prioritization of informative genes within expiMap for data-specific cell reconstruction to create a single batch-corrected GP score, helping to resolve ambiguities and challenges of automatic reference-based classifiers79. We also show that expiMap scores explain why the diabetes model and healthy beta cells do not overlap in integration, as indicated by differential activity of identity and maturation GPs (Fig. 6c, Extended Data Fig. 9 and Supplementary Note 8).

To search for molecular changes between the healthy control and T2D-model beta cells from the STZ study, we used the expiMap Bayes test with the Reactome GPs (Supplementary Table 13). We demonstrate that there is only a small overlap between the genes of the enriched GPs (Fig. 6e), simplifying the interpretation. We observed differences in energy metabolism, unfolded protein response (UPR) and islet communication, as previously reported in the original study (Supplementary Note 9 and Supplementary Table 14). To identify whether the enriched GPs separate cells into multiple populations within samples, we analysed the distributions of Reactome GP scores. The score of interactions between lymphoid and non-immune cells (Fig. 6f) had a multimodal distribution within T2D-model beta cells treated with insulin, potentially indicating the presence of multiple cell states within individual samples. For scores from other enriched GPs, we did not observe such distinct multimodal patterns within individual samples.

One of the key dysfunction processes in T2D, also identified in our enrichment analysis, is the UPR, which results from pro-insulin synthesis rate that exceeds the protein processing capacity of cells, leading to beta-cell dysfunction and death82,83. Thus, we compared scores of enriched GPs associated with UPR and protein synthesis and processing across individual cells (Fig. 6g). As expiMap produces batch-corrected GP scores we could also perform cross-study comparison with reference. We observed a high correlation between the UPR and asparagine N-linked glycosylation GP scores (absolute correlation coefficient of 0.93) across all datasets with extreme GP scores in T2D-model cells (Fig. 6g). An increase in N-linked glycosylation had been previously implicated in diabetes, although the regulatory background is not clear84,85. We further support the implication of N-linked glycosylation in T2D and its potential association with immune response (Fig. 6h, Supplementary Fig. 9 and Supplementary Note 10). We also assessed how multiple genes contribute to GP scores and how GP rather than gene-level comparison reduces noise (Supplementary Note 10).

Finally, we applied expiMap on another Pancreas dataset capturing mouse endocrinogenesis86 to demonstrate the model’s applicability on continuous developmental processes (Supplementary Note 11 and Extended Data Fig. 10). Overall, our results demonstrate that the expiMap GP activity analysis captures a complex differentiation and perturbation process in Pancreas.

## Discussion

We introduced expiMap for interpretable single-cell reference mapping. Our model embeds domain knowledge in the form of GPs into the deep learning architectures used for reference mapping and can further complement these GPs with newly discovered unconstrained GPs for query datasets. The interpretability of the model allows the users to generate immediate inferences about the query once mapped to a reference within the context of GPs. This contrasts with the existing analysis pipelines, which involve multiple steps and, without end-to-end learning, necessarily aggregate processing errors from previous steps. Interestingly, in a comparison across five different organ atlases, we found that the constrained expiMap model did not lose expressiveness versus an unconstrained conditional variational autoencoder model; indeed, prior constraints appeared to improve reference mapping and de novo data integration performance, confirming the earlier concepts of adding ‘differentiable programs’20. Through various applications, we demonstrated the interpretability of the model.

Reference mapping with expiMap provides a new perspective on data integration and reference mapping. In scenarios with significant differences in the datasets, such as cross-species mapping, the query data might not be fully aligned in the reference owing to the substantial biological and technical differences dominating the overall representation obtained by existing methods. This phenomenon makes it challenging to distinguish shared and unique signals between datasets. expiMap enables the integration of datasets along the axes of variations explained by a single or multiple GPs, where the datasets share variations and are mixed. This mixing stems from the commonality of the datasets in those programs. Such insights could not be obtained by, for example, looking at the overall uniform manifold approximation and projection (UMAP), which would be influenced by all genes, might be misleading and could obscure such commonalities. As we demonstrated when mapping COVID-19 patient data, CD8+ T cells from patients with COVID-19 were separate from IFN-β-treated CD8+ T cells in the global representation obtained from all GPs in UMAP (Fig. 5b,c). At the same time, they are integrated within specific GPs, capturing shared signals in two different cell states (Fig. 5d). Overall, expiMap can provide more insights into data integration by contextualizing it within GPs.

Our model leverages domain knowledge to improve the interpretability of deep learning models useful for single-cell genomics. With increasing availability87,88 of curated domain knowledge, expiMap can be trained on multiple databases while pruning irrelevant information. However, selecting the relevant knowledge to include in the model can affect the model’s performance. As we demonstrated, including IFN-related knowledge can improve the performance in reference mapping (Fig. 2), while excluding it can lead to poor mapping of the query (Supplementary Fig. 4). Another limitation concerns the interpretation and validation of newly learned GPs that capture new variations in the query data. As we demonstrated, looking at distribution plots and visualizing the embedding can decipher the variations. However, the validation at the gene level requires further expert knowledge for each biological system. Another limitation is the modelling hierarchies in unsupervised settings, starting from single genes to GPs and to higher-level biological processes. Previous work, such as knowledge-primed neural networks89, P-net26 and visible neural networks90 employed hierarchical modelling, but in supervised settings, to predict tumour type or cell states. Using a similar strategy in an unsupervised model would add another layer of analysis to mapping data, not only to GPs but also to biological processes, and potentially improve the performance. A final limitation of general deep learning models may be data hunger. To determine the sensitivity of our model to dataset size, we trained models of increasing quality by incrementally including more training samples in the reference building task (Extended Data Fig. 6b). We observed that expiMap outperformed the linear baseline of a non-biologically informed linear decoder model (LDVAE) in a low-data regime. The more complex non-amortized scVI achieved the best results with increased number of training samples, while expiMap outperformed scVI and LDVAE. Overall, these results suggest that incorporating prior knowledge leads to more sample-efficient learning in the presence of fewer samples than non-biologically informed models with similar complexity (for example, LDVAE). Further, when more training samples are available to learn GP activities efficiently, expiMap performs with complex nonlinear models.

Although we demonstrated expiMap by using single-cell RNA sequencing data, the model is naturally extendable to multimodal91,92,93 datasets. Recent technological advances in single-cell biology allow the simultaneous capture of chromatin accessibility, gene expression and protein levels in single cells4. This makes it possible to learn the hierarchy of connected representations by distilling domain knowledge about regulatory elements, transcription and translation, covering multiple cellular processes into the representation learning methods. Another potentially exciting direction is the combination of the expiMap architecture with in vitro perturbation modelling approaches5,6,33 to model in vitro perturbations of GPs. Finally, given the availability of spatial transcriptomics data94, it is possible to adapt expiMap to include information about cell-to-cell communication95 in the learned representations to gain further insights into cellular communications and signalling.

Researchers in the field of single-cell genomics are moving towards using reference mapping to analyse new query datasets. We envision that expiMap will further advance the applicability of reference mapping methods by bringing a new layer of interpretability and mechanistic understanding to integrative single-cell data analysis facilitating biological hypothesis generation and discovery.

## Methods

### expiMap model

Our model builds upon the framework of (conditional) variational autoencoders34,96. The log-likelihood of the data for expiMap can be written as

$${{{\mathrm{log}}}}p_\theta \left( {{\mathbf {X}}|W,{\mathbf {C}}} \,\right)p\left( W \,\right) = {{{\mathrm{log}}}}{\displaystyle\mathop{\int}\limits_{\mathbf {Z}}} {p_\theta } \left( {\mathbf {X}}|{\mathbf {Z}},W,{\mathbf {C}}\, \right)p\left( {\mathbf {Z}} \,\right)p\left( W \,\right)d{\mathbf {Z}}$$
(1)
$$p_{\theta} \left( {\mathbf{X}|\mathbf{Z},W,\mathbf{C}}\, \right) = {{{\mathrm{NB}}}}\left( {g\left( {\left[ {\mathbf{Z},\mathbf{C}} \right]\left[ {W,L} \right]^{\mathrm{T}}} \right),\mathbf{CD}} \right),$$
(2)

where $$g\left( x \right) = {{{\mathrm{softmax}}}}\left( x \right) \times S$$ is a softmax function that is multiplied by the library size S of each cell. Alternatively, g(x) could also be a softplus or exponential function. Further, X is a random variable representing gene expression, C indicates conditions (for example, batch ID) and $$p_\theta \left( {\mathbf X\mid \mathbf Z,W,\mathbf C} \right)$$ is the output distribution, also called a decoder in the setting of variational autoencoders, used to model X given the latent variable Z.

$${{{\mathrm{NB}}}}\left( { \cdot , \cdot } \right)$$ in equation (2) denotes the mean and dispersion parametrized negative binomial distribution, $$\left[ { \cdot , \cdot } \right]$$ means a column-stacked matrix, W and L are matrix parameters for latent variables Z and one-hot encoded conditions C, respectively; and D is a matrix of condition-specific dispersion parameters for each gene. W is a n × m matrix with n corresponding to the number of genes and m corresponding to the number of GPs provided as an input.

The prior p(w) in equation (1) is defined as:

$$\begin{array}{l} {{{\mathrm{log}}}}p\left( {W_{:,\,j}} \right) = {{{\mathrm{log}}}}\mathop {\int}\limits_{\tau ^2} p \left( {W_{:,j}|\tau ^2} \right)p\left( {\tau ^2|\alpha } \right)d\tau ^2 = - \alpha \left\| {W_{:,j}} \right\|_2,\\{{{\mathrm{log}}}}p\left( W \,\right) = - \alpha{\displaystyle{ \mathop {\sum}\limits_j}} {\left\| {W_{:,\,j}} \right\|_2} \\p\left( {W_{:,j}|\tau ^2} \right) = {{{\mathcal{N}}}}\left( {0,\tau ^2I} \right),p\left( {\tau ^2|\alpha } \right) = {{{\mathrm{Gamma}}}}\left( {\frac{{n + 1}}{2},\frac{{\alpha ^2}}{2}} \right)\end{array}$$

The constants were omitted because they do not affect the optimization. We use a hierarchical Bayesian prior on the columns of W with the parameter τ2 integrated out as in oi-VAE97, resulting in the lasso regularization term. The lasso regularization allows the model to de-activate the GPs that do not contribute to the reconstruction loss in the model. α is a hyperparameter specifying the strength of the group lasso regularization.

The evidence lower bound (ELBO) is a part of our total loss to train the model. During the model training, the posterior distribution $$p_{\theta} \left( {\mathbf Z|\mathbf X,\mathbf C} \right)$$ is approximated by the variational distribution $$q_\phi \left( {\mathbf Z|\mathbf X,\mathbf C} \right)$$, which includes a deep neural network parameterized with ϕ; it is also called an encoder. The ELBO can be written as:

$$\begin{array}{l} {\mathrm{log}}{\displaystyle{\mathop {\int}\limits_{\mathbf Z}}} {p_{\theta} } ( {\mathbf{X}|\mathbf{Z},W,\mathbf{C}} )p(\mathbf{Z})p(W)d \mathbf{Z} \ge {\displaystyle{\mathop {\int}\limits_{\mathbf Z}}}{q_{\phi} } ( {\mathbf{Z}|\mathbf{X},\mathbf{C}} ){\mathrm{log}}\frac{{p_{\theta} ( {\mathbf{ X}|\mathbf{Z},W,\mathbf{C}} )p( \mathbf{Z} )p( W\, )}}{{q_{\phi} ( {\mathbf{Z}|\mathbf{X},\mathbf{C}} )}}d\mathbf{Z}\\={\Bbb E}_{q_{\phi} ( {\mathbf{Z}|\mathbf{X},\mathbf{C}} )}\left[ {{{\mathrm{log}}}}p_{\theta} \left( {\mathbf{ X}|\mathbf{Z},\mathbf{W},\mathbf{C}} \right) \right] - {\Bbb K}{\Bbb L}\left( {q_{\phi} \left( {\mathbf{Z}|\mathbf{X},\mathbf{C}} \right)\parallel p\left( \mathbf{Z}\, \right)} \right)+ {{{\mathrm{log}}}}p\left( W\, \right)\\={{{\mathrm{ELBO}}}}\left( {\theta ,\phi ,W} \right)\end{array}$$
(3)

where θ and ϕ are parameters of the decoder and the encoder, respectively.

### GP matrix

We use tab-delimited text files where the rows represent gene sets as an input to construct masks for W (see the previous section). The first column is reserved for the name of the gene sets and the other columns should contain the names of genes. Gene matrix transposed files (.gmt file format) could be directly used in our API as an input.

A database could be also passed to the model in the form of a binary matrix B with columns corresponding to GPs and rows corresponding to genes, with $$B_{i,j} = 1$$ if the ith gene is in the jth GP and 0 otherwise. Such a matrix is actually always constructed from the files described above before passing to the model. We refer to matrix B as the GP matrix.

### Defining hard/soft gene membership

The decoder network in equation (2) consists of a linear layer $$\mathbf H = \left[ {\mathbf Z,\mathbf C} \right]\left[ {W,L} \right]^T$$, in which the output is then transformed to a negative binomial means by the nonlinear function g(H). The GP matrix B specifies GPs and the gene memberships for these programs. The matrix B is used as a mask for the matrix of the decoder weights W, where the parameters for inactive genes in each GP are set to zero and do not change during training if the hard mask is used.

$$W_{i,j} = \left\{ {\begin{array}{*{20}{l}} {0\,{{{\mathrm{if}}}}\,B_{i,j} = 0,} \hfill \\ {w_{i,j}\,{{{\mathrm{otherwise}}}}} \hfill \end{array}} \right.$$
(4)

In the case of a soft mask, we add a regularization term that forces gene weights for genes that are not originally part of a GP to become zero, but also allows them to become active (non-zero) if they contribute to the reconstruction:

$$R_\gamma \left( W \right) = \gamma {\displaystyle{\mathop {\sum}\limits_j}} {\left\| {W_{:,\,j} \odot M_{:,\,j}} \right\|_1}$$
(5)
$$M_{i,j} = \left\{ {\begin{array}{*{20}{l}} {1\,{{{\mathrm{if}}}}\,B_{i,j} = 0,} \hfill \\ {0\,{{{\mathrm{otherwise}}}}} \hfill \end{array}} \right.$$
(6)

Some columns of M can be set to a vector of ones $$M_{:,k} = \vec 1$$ by setting $$B_{:,k} = \vec 0$$ to allow the introduction of sparse GPs.

Both variants (hard and soft masks) force the elements of Z to correspond to the GPs encoded in W.

### Learning new GPs

To allow new GPs to be learned, the model can be extended with additional nodes in reference training or query projection. For this, the last layer of the encoder is expanded with additional nodes connected to the existing nodes from the previous layer and producing the new vector Znew; in the decoder, the additional matrix Wnew is concatenated to W (now denoted by Wold), resulting in:

$$p\left( {\mathbf{X}|\mathbf{Z}_{{{{\mathrm{old}}}}},\mathbf {Z}_{{{{\mathrm{new}}}}},W_{{{{\mathrm{old}}}}},W_{{{{\mathrm{new}}}}},\mathbf{C}} \right)={{{\mathrm{NB}}}}\left( {g\left( {\left[ {\mathbf{Z}_{{{{\mathrm{old}}}}},\mathbf{ Z}_{{{{\mathrm{new}}}}},\mathbf{C}} \right]\left[ {W_{{{{\mathrm{old}}}}},W_{{{{\mathrm{new}}}}},L} \right]^T} \right),\mathbf CD} \right)$$

In addition, L1 regularization is added to Wnew, which is equivalent to the Laplace before this matrix. In addition, for each element of the vector Znew the sample estimate of HSIC between the element and the other elements of Zold and Znew is added as a regularization term to the loss22. Also Wnew can be constrained with hard gene membership or regularized with soft gene membership (see the previous section) as Wold using an additional GP database. In this case we do not use HSIC regularization for these new constrained nodes.

### Training

We use the stochastic proximal gradient descent to optimize the ELBO loss (equation (3)) with additional regularization terms. We also multiply the Kullback–Leibler (KL) divergence in the ELBO loss by the regularization coefficient β. Excluding the group lasso $$R_\alpha \left( W \right) = - {{{\mathrm{log}}}}p\left( W \right)$$ and soft mask term Rγ(W) that appear in the proximal update step (discussed further), the loss function of the model can be written as:

$$\begin{array}{*{20}{l}} {F\left( {\theta ,\phi ,W} \right)} & = & {\frac{1}{N}{\displaystyle{\mathop {\sum}\limits_{i}^{N}}} {{\Bbb E}_{q_{\phi} \left( {\mathbf{ Z}_i|\mathbf{X}_i,\mathbf{C}} \right)}} \left[ { - {{{\mathrm{log}}}}p_{\theta} \left( {\mathbf{X}_i|\mathbf{Z}_i,W,\mathbf C} \right)} \right] + \beta {\Bbb K}{\Bbb L}\left( {q_{\phi} \left( {\mathbf{Z}_i|\mathbf{X}_i,\mathbf{C}} \right)\parallel p\left( {\mathbf{Z}_i} \right)} \right)} \hfill \\ {} \hfill & {} \hfill & { + \nu R_{\phi} ^{{{{\mathrm{HSIC}}}}}\left( {\mathbf{Z}_{{{{\mathrm{old}}}}},\mathbf{Z}_{{{{\mathrm{new}}}}}} \right)} \hfill \end{array}$$
(7)

where $$R_{\phi} ^{{{{\mathrm{HSIC}}}}}\left( {\mathbf{Z}_{{{{\mathrm{old}}}}},\mathbf{ Z}_{{{{\mathrm{new}}}}}} \right)$$ is a sample estimate of the HSIC regularization term. In addition, Znew and Zold are the old (existing in reference model) and new (learned in query training) unconstrained programs, respectively.

Then, to minimize the objective function we use the update scheme

$$\begin{array}{*{20}{l}} {\theta ^{\left( {t + 1} \right)}} \hfill & = \hfill & {\theta ^{\left( t \right)} - \eta \nabla _\theta {\hat {F}}\left( {\theta ,\phi ,W} \right)} \hfill \\ {\phi ^{\left( {t + 1} \right)}} \hfill & = \hfill & {\phi ^{\left( t \right)} - \eta \nabla _\phi {\hat {F}}\left( {\theta ,\phi ,W} \right)} \hfill \\ {W^{\left( {t + 1} \right)}} \hfill & = \hfill & {\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\eta R_\alpha + \eta R_\gamma } \end{array}\left( {W^{\left( t \right)} - \eta \nabla _W{\hat {F}}\left( {\theta ,\phi ,W} \right)} \right)} \hfill \end{array}$$
(8)

where $${\hat {F}}\left( {\theta ,\phi ,W} \right)$$ denotes an estimate of the function (equation (7)) over a mini-batch of samples (as in the standard stochastic gradient descent algorithm), t is the step in the gradient descent algorithm and η is the learning rate.

$$\left( {R_\alpha + R_\gamma } \right)\left( W\, \right) = \alpha{\displaystyle{ \mathop {\sum}\limits_j}} {\left\| {W_{:,\,j}} \right\|_2} + \gamma {\displaystyle{\mathop {\sum}\limits_j}} {\left\| {W_{:,\,j} \odot M_{:,\,j}} \right\|_1}$$

is the lasso and soft mask regularization term and its proximal operator is

$$\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\eta R_\alpha + \eta R_\gamma } \end{array}\left( V\, \right) = {{{\mathrm{arg}}\,{\mathrm{min}}}}_L\left[ {\frac{1}{2}\Vert {L - V}{\Vert}_F^2 + \eta \alpha {\displaystyle{\mathop{\sum}\limits_{j}}} {\Vert {L_{:,\,j}} {\Vert}_2} + \eta \gamma {\displaystyle{\mathop {\sum}\limits_{j}}} {\Vert {L_{:,\,j} \odot M_{:,\,j}} {\Vert}_1} } \right]$$
(9)

The hard mask variant implies γ = 0. The proximal operator above has a closed-form expression (see the next section for the derivation), so it is easy to apply it after the stochastic gradient descent update. The gradient for the expectation terms is obtained with the reparametrization trick, as is common in the VAE framework96.

### Proximal operators for expiMap

To derive the closed form of the proximal operator described in the previous section, we need two theorems.

### Theorem 1

(Proximal operator of separable functions.)

Suppose that $$f:E_1 \times E_2 \times \ldots \times E_m \to ( - \infty ,\infty ]$$ is given by

$$\begin{array}{*{20}{l}} {f\left( {\mathbf{x}_1,\mathbf{x}_2, \ldots ,\mathbf{x}_m} \right) ={\displaystyle{ \mathop {\sum}\limits_{i = 1}^m}}\, {f_i\left( {\mathbf{x}_i} \right)} } \hfill \\ {\mathbf{x}_i \in E_i,i = 1,2, \ldots ,m} \hfill \end{array}$$

Then for any $$x_1 \in E_1,x_2 \in E_2, \ldots ,x_m \in E_m$$,

$$\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ f \end{array}\left( {\mathbf{x}_1,\mathbf{x}_2, \ldots ,\mathbf{x}_m} \right) = \begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {f_1} \end{array}\left( {\mathbf{x}_1} \right) \times \begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {f_2} \end{array}\left( {\mathbf{x}_2} \right) \times \ldots \times \begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {f_m} \end{array}\left( {\mathbf{x}_m} \right)$$

where Ei denotes a vector space and × is a Cartesian product. The proof of this theorem can be found in ref. 98.

### Theorem 2

(Decomposition of the proximal operator.)

A sufficient condition for $$\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {f + g} \end{array} = \begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ f \end{array} \circ \begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ g \end{array}$$ is

$$\forall \mathbf{x} \in H\,\partial g\left( {\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ f \end{array}\left( \mathbf{x} \right)} \right) \supseteq \partial g\left( \mathbf{x} \right)$$

where f and g are closed (or, equivalently here, continuous), convex functions; H denotes a Hilbert space; and $$\partial g$$ stands for a subgradient of g. The proof of the theorem can be found in ref. 99.

We use the two theorems above to find the closed form of the proximal operator (equation (9)). The explicit form of the regularization function is:

$$R_{\alpha ,\gamma }\left( W \,\right) = \left( {R_\alpha + R_\gamma } \right)\left( W \right) = \alpha{\displaystyle{ \mathop {\sum}\limits_j}} {\left\| {W_{:,\,j}} \right\|_2} + \gamma{\displaystyle{ \mathop {\sum}\limits_j}} {\left\| {W_{:,\,j} \odot M_{:,\,j}} \right\|_1}$$
(10)

The sums in the regularization function are made over columns of W; thus, this function is clearly separable in columns, and the theorem 1 is applicable here. This means that we only need to calculate the proximal operator for a column, as we can find the full proximal operator as a Cartesian product of the proximal operators for different columns. This is the same as using its own proximal operator for each column of W separately.

The regularization summand for a separate column k of W can be written as

$$R_{\alpha ,\gamma }^k\left( W\, \right) = \left( {R_\alpha ^k + R_\gamma ^k} \right)\left( W\, \right) = \alpha \left\| {W_{:,\,k}} \right\|_2 + \gamma \left\| {W_{:,\,k} \odot M_{:,k}} \right\|_1$$
(11)

The regularization summand (equation (11)) has the form of a sum, so the theorem 2 has to be used.

For the group lasso part $$\alpha \parallel \cdot \parallel _2$$ the proximal operator can be immediately obtained (from ref. 98) as

$$\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\eta R_\alpha ^k} \end{array}\left( \mathbf{v} \right) = \left\{ {\begin{array}{*{20}{l}} {\mathbf{v} - \eta \alpha \frac{\mathbf{v}}{{\left\| \mathbf{v} \right\|_2}},} \hfill & {{{{\mathrm{if}}}}\,\left\| \mathbf{v} \right\|_2 > \eta \alpha } \hfill \\ {0,} \hfill & {{{{\mathrm{if}}}}\,\left\| \mathbf{v} \right\|_2 \le \eta \alpha } \hfill \end{array}} \right.$$
(12)

It should be noted that in the case when the mask’s column equals a vector of ones $$M_{:,\,k} = \vec 1$$ the proximal operator for the second summand in equation (11) $$\gamma \parallel \cdot \parallel _1$$ is just a proximal operator for a standard L1 regularization and can be written98 as

$$\begin{array}{*{20}{l}} {{{{\mathcal{T}}}}_\gamma \left( y \right)} = {\left\{ {\begin{array}{*{20}{l}} {y - \gamma ,} \hfill & {{{{\mathrm{if}}}}\,y \ge \gamma } \hfill \\ {0,} \hfill & {{{{\mathrm{if}}}}\,\left| y \right| < \gamma } \hfill \\ {y - \gamma ,} \hfill & {{{{\mathrm{if}}}}\,y \le \gamma } \hfill \end{array}} \right.} \hfill \\ {\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\gamma \parallel \cdot \parallel _1} \end{array}\left( \mathbf{v} \right)} = {{{{\mathcal{T}}}}_\gamma \left( {v_1} \right) \times {{{\mathcal{T}}}}_\gamma \left( {v_2} \right) \times \ldots \times {{{\mathcal{T}}}}_\gamma \left( {v_G} \right)} \hfill \end{array}$$
(13)

In addition, the subgradient $$\partial \left( {\gamma \parallel \mathbf{v}\parallel _1} \right)$$ (from ref. 98, rewritten) is

$$\begin{array}{*{20}{l}} {{{{\mathrm{sgn}}}}_\gamma \left( y \right)} \hfill & = \hfill & {\left\{ {\begin{array}{*{20}{l}} {\gamma ,} \hfill & {{{{\mathrm{if}}}}\,y > 0} \hfill \\ {\left[ { - \gamma ,\gamma } \right],} \hfill & {{{{\mathrm{if}}}}\,y = 0,} \hfill \\ { - \gamma ,} \hfill & {{{{\mathrm{if}}}}\,y < 0} \hfill \end{array}} \right.} \hfill \\ {\partial \left( {\gamma \parallel \mathbf{v}\parallel _1} \right)} \hfill & = \hfill & {{{{\mathrm{sgn}}}}_\gamma \left( {v_1} \right) \times {{{\mathrm{sgn}}}}_\gamma \left( {v_2} \right) \times \ldots \times {{{\mathrm{sgn}}}}_\gamma \left( {v_G} \right)} \hfill \end{array}$$
(14)

The proximal operator $$\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\alpha \parallel \cdot \parallel _2} \end{array}\left( \mathbf{v} \right)$$ is equal to equation (12) (without η). By direct calculation for $$\mathbf{v}^ \ast = \begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\alpha \parallel \cdot \parallel _2} \end{array}\left( \mathbf{v} \right)$$ the following holds $$\forall i = 1, \ldots ,G$$: if $$v_i = 0$$, then $$v_i^ \ast = 0$$; if $$v_i < 0$$, then $$v_i^ \ast \le 0$$; if $$v_i > 0$$, then $$v_i^ \ast \ge 0$$.

This basically means that $${{{\mathrm{sgn}}}}_\gamma \left( {v_i^ \ast } \right) \supseteq {{{\mathrm{sgn}}}}_\gamma \left( {v_i} \right)$$. It immediately follows from the form of the subgradient (equation (14)) that

$$\begin{array}{*{20}{l}} {\mathbf{v}^ \ast = \begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\alpha \parallel \cdot \parallel _2} \end{array}\left( \mathbf{v} \right)} \hfill \\ {\partial \left( {\gamma \parallel \mathbf{v}^ \ast \parallel _1} \right) \supseteq \partial \left( {\gamma \parallel \mathbf{v}\parallel _1} \right)} \hfill \end{array}$$

Using this and the theorem 2 we can conclude that

$$\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\gamma \parallel \cdot \parallel _1 + \alpha \parallel \cdot \parallel _2} \end{array}\left( \mathbf{v} \right) = \begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\alpha \parallel \cdot \parallel _2} \end{array}\left( {\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\gamma \parallel \cdot \parallel _1} \end{array}\left( \mathbf{v} \right)} \right)$$
(15)

Therefore, the closed form of the proximal operator for the case $$M_{:,\,k} = \vec 1$$ is:

$$\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\eta R_{\alpha ,\gamma }^k} \end{array}\left( \mathbf{v} \right) = \begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\eta \alpha \parallel \cdot \parallel _2} \end{array}\left( {\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\eta \gamma \parallel \cdot \parallel _1} \end{array}\left( \mathbf{v} \right)} \right)$$
(16)

Moreover, the closed forms of $$\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\eta \alpha \parallel \cdot \parallel _2} \end{array}\left( \mathbf{v} \right)$$ and $$\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\eta \gamma \parallel \cdot \parallel _1} \end{array}\left( \mathbf{v} \,\right)$$ are given in equations (12) and (13), respectively.

For the case $$M_{:,\,k} \ne \vec 1$$, similar reasoning can be applied. First, the closed form of the proximal operator $$\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\gamma \parallel \, \cdot \, \odot \,M_{:,\,k}\parallel _1} \end{array}\left( \mathbf{v} \right)$$ for L1 norm of the vector of genes (gene weights in the factor) that are inactive in the annotation for the factor k can be written as

$$\begin{array}{l} {{{{\mathcal{A}}}}_\gamma ^{g,k}\left( y \right)} ={\left\{ {\begin{array}{lll} {y,}&{{{{\mathrm{if}}}}\,M_{g,k} = 0}\\ {{{{\mathcal{T}}}}_\gamma \left( y \right),}&{{{{\mathrm{if}}}}\,M_{g,k} = 1}\end{array}} \right.} \hfill \\ {\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\gamma \parallel \cdot \odot M_{:,\,k}\parallel _1} \end{array}\left( \mathbf{v} \right)}= {{{{\mathcal{A}}}}_\gamma ^{1,k}\left( {v_1} \right) \times \ldots \times {{{\mathcal{A}}}}_\gamma ^{G,k}\left( {v_G} \right)} \hfill \end{array}$$
(17)

where $${{{\mathcal{T}}}}_\gamma \left( y \right)$$ is the same as in equation (13).

The subgradient $$\partial \left( {\gamma \parallel \,\mathbf{v}\, \odot \,M_{:,\,k}\parallel _1} \right)$$ can be written as

$$\begin{array}{l} {{{{\mathcal{S}}}}_\gamma ^{g,k}\left( y \right)}={\left\{ {\begin{array}{*{20}{l}} {0,} \hfill & {{{{\mathrm{if}}}}\,M_{g,k} = 0} \hfill \\ {{{{\mathrm{sgn}}}}_\gamma \left( y \right),} \hfill & {{{{\mathrm{if}}}}\,M_{g,k} = 1} \hfill \end{array}} \right.} \hfill \\ {\partial \left( {\gamma \parallel \mathbf{v} \odot M_{:,\,k}\parallel _1} \right)}={{{{\mathcal{S}}}}_\gamma ^{1,k}\left( {v_1} \right) \times \ldots \times {{{\mathcal{S}}}}_{\lambda _1}^{G,k}\left( {v_G} \right)} \hfill \end{array}$$
(18)

where $${{{\mathrm{sgn}}}}_\gamma \left( y \right)$$ is the same as in equation (14).

Using the same reasoning as in the derivation of the proximal operator for sparse unannotated factors, we see that

$$\begin{array}{*{20}{l}} {\mathbf{v}^ \ast = \begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\alpha \parallel \cdot \parallel _2} \end{array}\left( \mathbf{v} \right)} \hfill \\ {\partial \left( {\gamma \parallel \mathbf{v}^ \ast \, \odot \,M_{:,\,k}\parallel _1} \right) \supseteq \partial \left( {\gamma \parallel \mathbf{v}\, \odot \,M_{:,\,k}\parallel _1} \right)} \hfill \end{array}$$

This means that we again can use the theorem 2 and obtain the closed form of the proximal operator (with the learning rate η) for the column k of W, which corresponds to the annotated factor k

$$\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\eta R_{\alpha ,\gamma }^k} \end{array}\left( \mathbf{v} \right) = \begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\eta \alpha \parallel \cdot \parallel _2} \end{array}\left( {\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\eta \gamma \parallel \cdot \, \odot \,M_{:,\,k}\parallel _1} \end{array}\left( \mathbf{v} \right)} \right)$$
(19)

In addition, the closed forms of $$\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\eta \alpha \parallel \cdot \parallel _2} \end{array}\left( \mathbf{v} \right)$$ and $$\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\eta \gamma \parallel \, \cdot \, \odot \,M_{:,\,k}\parallel _1} \end{array}\left( \mathbf{v} \right)$$ are given in equations (12) and (17), respectively.

Theorem 1 allows calculation of the output of the joint proximal operator $$\begin{array}{*{20}{c}} {{{{\mathrm{Prox}}}}} \\ {\eta R_\alpha + \eta R_\gamma } \end{array}\left( \cdot \right)$$ in equation (8) by applying the proximal operators (equations (12), (16) or (19)) on each column of the input of the joint operator independently.

### Reference mapping

The projection of a query dataset to a reference dataset is performed using the single-cell architectural surgery (scArches) approach15. After training a conditional VAE model for multiple batches of the reference dataset, the trained weights are transferred to a new model with additional conditional nodes used to map new query batches to the reference. Further, additional nodes for new learnable GPs can be added at this stage (see the learning new GP section). During the training of the expanded model for query projection, only the conditional weights connecting new batches and the weights for new GPs (if any) in both encoder and decoder are tuned; the rest of the weights are frozen. Projecting with scArches preserves the latent representation of the reference and projects the query data to the same latent space while correcting for batch effects between the query and data.

### Differential testing for GPs

To test the hypothesis $$H_0:Z_{i,a} > Z_{i,b}$$ versus $$H_1:Z_{i,a} \le Z_{i,b}$$, where $$Z_{i,a},Z_{i,b}$$ are the dimension i of the latent variables for the cells from the groups a and b respectively, we use the logarithm of the Bayes factor:

$${{{\mathrm{log}}}}K = {{{\mathrm{log}}}}\frac{{p\left( {H_0} \right)}}{{p\left( {H_1} \right)}} = {{{\mathrm{log}}}}\frac{{p\left( {H_0} \right)}}{{1 - p\left( {H_0} \right)}}$$
(20)

where $$p\left( {H_0} \right)$$ and $$p\left( {H_1} \right)$$ are the probabilities of the hypotheses H0 and H1, respectively.

We can compute $$P\left( {H_0} \right)$$ as:

$$\begin{array}{l}p\left( {H_0} \right) = p\left( {Z_{1,i} > Z_{2,i}|G_1 = a,G_2 = b} \right)\\\qquad\quad = {\Bbb E}_{p\left( {\mathbf X_1,\mathbf C_1|G_1 = a} \right)p\left( {\mathbf X_2,\mathbf C_2|G_2 = b} \right)}\left[ {p\left( {Z_{1,i} > Z_{2,i}|\mathbf X_1,\mathbf X_2,\mathbf C_1,\mathbf C_2} \right)} \right]\end{array}$$
(21)

where G1 and G2 denote the independent group variables for X1 and X2, respectively.

The probability $$p\left( {Z_{1,i} > Z_{2,i}|\mathbf X_1,\mathbf X_2,\mathbf C_1,\mathbf C_2} \right)$$ inside the expectation in equation (21) can be estimated with the approximate posteriors $$q_\phi \left( {Z_{1,i}|\mathbf X,\mathbf C} \right)$$ and $$q_\phi \left( {Z_{2,i}|\mathbf X,\mathbf C} \right)$$, as follows:

$$p\left( {Z_{1,i} > Z_{2,i}|\mathbf X_1,\mathbf X_2,\mathbf C_1,\mathbf C_2} \right) \approx {\Bbb E}_{q_\phi \left( {Z_{1,i}|\mathbf X_1,\mathbf C_1} \right)q_\phi \left( {Z_{2,i}|\mathbf X_2,\mathbf C_2} \right)}\left[ {I\left( {Z_{1,i} > Z_{2,i}} \right)} \right]$$
(22)

where the expectation could be approximated by sampling or calculated from the closed form. When $$q_\phi \left( {Z_i|\mathbf X,\mathbf C} \right)$$ is Gaussian, we can calculate the expectation by

$$\begin{array}{l}{\Bbb E}_{q_\phi \left( {Z_{1,i}|\mathbf X_1,\mathbf C_1} \right)q_\phi \left( {Z_{2,i}|\mathbf X_2,\mathbf C_2} \right)}\left[ {I\left( {Z_{1,i} > Z_{2,i}} \right)} \right] = \frac{1}{2}{{{\mathrm{erfc}}}}\left( { - \frac{{\mu _i\left( {\mathbf X_1,\mathbf C_1} \right) - \mu _i\left( {\mathbf X_2,\mathbf C_2} \right)}}{{\sqrt {2\left( {\sigma _i^2\left( {\mathbf X_1,\mathbf C_1} \right) + \sigma _i^2\left( {\mathbf X_2,\mathbf C_2} \right)} \right)} }}} \right)\end{array}$$
(23)

The probabilities (equation (22)) can be averaged over many cells from both groups as in scVI40, to obtain the approximate value for equation (21).

Through the examples in the paper, we refer to the results obtained as ‘expiMap test results’ and at the threshold $$\left| {{{{\mathrm{log}}}}K} \right| \ge 2.3$$ as the ‘enriched results’, and call such GPs ‘differential GPs’ in the comparison of interest in this work.

### Gene importance score

Gene importance score for a gene in a GP is the absolute value of the decoder weight for the gene in the GP. Each column of the weight matrix W in the decoder (2) corresponds to a GP and each row corresponds to a gene. Because of the linearity of the decoder, a change in the latent score of the ith GP Zi affects the reconstruction of gene counts more for those genes with higher absolute values of the weights in W:,i. Consequently, we can rank genes in each GP by the absolute values of their weights in W. This ranking reflects the relative importance of a given GP for each gene; a higher ranking means that this gene is affected more by the GP.

### Latent scores directions

The signs of latent scores of GPs do not necessary correspond to up- or downregulation of these GPs. However, in some cases, it is possible to determine whether an increase in a latent score corresponds primarily to an increase or decrease in the expression of genes of a corresponding GP. This can be determined by analysing the decoder gene weights in the column corresponding to the GP (as described in the previous section). If most of the gene weights in the column of W corresponding to the GP are positive, then the higher positive latent score implies upregulation; in the opposite case of mostly negative weights, a lower negative score also means upregulation.

For the jth GP, the direction Dj of predominant upregulation (negative or positive) can be calculated heuristically by several methods. We use two methods:

$$\begin{array}{*{20}{l}} {{{{\mathrm{sum:}}}}D_j} \hfill & = \hfill & {{{{\mathrm{sign}}}}\left({\displaystyle{ {\mathop {\sum }\limits_i W_{i,j}}}} \right)} \hfill \\ {{{{\mathrm{counts:}}}}D_j} \hfill & = \hfill & {{{{\mathrm{sign}}}}\left({\displaystyle{ {\mathop {\sum }\limits_i }}{{{\mathrm{sign}}}}\left( {W_{i,j}} \right)} \right)} \hfill \end{array}$$

Then, we can multiply the latent score of the GP by this direction $$\tilde Z_j = Z_j \times D_j$$, so that a higher positive value of $$\tilde Z_j$$ always corresponds to predominant upregulation of the GP and a lower negative value to downregulation. These normalized scores can then be used for plotting or testing.

### Metrics for integration and evaluation

Integrations were evaluated with methods implemented in scIB. We evaluated biological conservation through graph cLISI, normalized mutual information (NMI), adjusted Rand index (ARI) and average silhouette width (ASW) for cell type; and batch correction through principal component regression, ASW for batch, kBET, graph connectivity and graph iLISI. All metrics are further described in the scIB paper60. The overall score was computed as the average of all scores.

We assessed the dominance of genes in a GP for Extended Data Fig. 3e by normalized entropy. The normalized entropy is calculated by dividing the entropy of the distribution of gene importance scores of the GP by the entropy of the uniform discrete distribution of the same size. The distribution of gene importance scores is obtained by dividing each score by the total sum of the scores. The normalized entropy scale is from 0 (absolutely concentrated) to 1 (uniformly spread weights).

### Choice of hyperparameters for expiMap training

#### Reference training and integration

The main hyperparameter that affects the quality of integration for the reference training is alpha_kl, the value of which is multiplied by the kl divergence term in the total loss. If the visualized latent space looks like a single blob after the reference training, we recommend to decrease the value of alpha_kl. If the visualized latent space shows bad integration quality, we recommend to increase the value of alpha_kl. The good default value in most cases is alpha_kl = 0.5. The required strength of group lasso regularization (alpha) depends on the number of used GPs and the size of the dataset. For 300–500 GPs, we recommend to use alpha = 0.7 and increase for larger numbers of GPs.

#### Reference mapping

We recommend to use 200 epochs and early_stopping = True for the query to reference mapping. Smaller datasets tend to require more epochs of training to map the query into the reference well. If you observe that the query is not integrated into the reference, we recommend to try longer training for the query mapping.

When using new unconstrained GPs, we recommend to start with ten of them or more. This ensures that all new significant sources of variation in the query would be covered by the new GPs. We also recommend to use L1 regularization for the new GPs (the gamma_ext parameter), it will make them sparser, and thus more interpretable, and also can de-activate redundant new GPs completely, which is important when the number of new unconstrained GPs is high.

If you use new constrained GPs with soft masks, it is important to monitor share of de-activated inactive genes of the soft masks in the constrained GPs. Set print_stats = True during the training, and check that at the end of the training process ‘Share of de-activated inactive genes in extension terms’ log show a number higher than 0.9. If this number is lower, it means that some of the constrained GPs lost their specialization given by the soft mask and added a lot of irrelevant genes. If this happens, it is better to increase the alpha_l1 parameter and retrain the model.

### Non-amortized scVI

We compared the integration performance of expiMap with scVI and non-amortized scVI. Non-amortized scVI is a VAE model similar to scVI, where the neural network encoder was replaced by a per cell vector of parameters for each cell in a dataset.

For each cell i there are vectors $$\mathbf \mu _i \in {\Bbb R}^Z$$ and $$\mathbf \sigma _i^2 \in {\Bbb R}^Z$$ with the size of the latent space. The jth latent variable for the cell i is obtained by sampling independently from the Gaussian distribution $$Z_{i,j} \sim {{{\mathcal{N}}}}\left( {\mu _{i,\,j},\sigma _{i,\,j}^2} \right)$$. The decoder is the same as in the standard scVI model. The non-amortized scVI model is trained by minimizing the negative ELBO in batches with a gradient descent algorithm as a standard VAE model.

### GSEA using limma-fry

Read counts were normalized using the trimmed mean of M-values (TMM)100 with singleton pairing implemented in edgeR101 to account for sparsity in the single cell RNAseq data. The fry test (Fast Approximation to ROAST)59 in limma58 R/Bioconductor package was applied to log counts per million (logCPM) values obtained by voom transformation102 to test for the enrichment of the gene set terms in the Reactome pathway database78. The Reactome database was obtained from the Molecular Signature Database (MSigDB)103,104.

### Datasets and pre-processing

All the cell type labels and metadata were obtained from original publications unless specifically stated below.

#### Immune healthy atlas

The immune dataset includes samples from bone marrow cells and peripheral blood cells from different human samples. The bone marrow data were collected from Oetjen et al.48 and PBMC samples were obtained from 10x Genomics (https://support.10xgenomics.com/single-cell-gene-expression/datasets/3.0.0/pbmc_10k_v3), Freytag et al.49 and Sun et al.50. The detail of the retrieval path and the pre-processing can be found in Luecken et al.60 and Lotfollahi et al.15 We used the Reactome pathway database for annotations78 from MsigDB103,104; we also removed all pathways with fewer than 12 genes. The genes that were not in the GPs database were filtered out, reducing the total number of genes from approximately 11,000 to 3,690. Then 2,000 highly variable genes were selected for training.

#### PBMC IFN-β

This dataset contains cells from eight patients with Lupus treated with IFN-β or left untreated for 6 h (ref. 105). The pools from the IFN-β and control cells were mixed together and loaded to a 10x kit. The dataset was obtained from the Seurat tutorial (https://satijalab.org/seurat/articles/integration_introduction.html). We have used the same genes as in the reference (Immune atlas).

#### PBMC COVID-19

The dataset62 contains five peripheral blood samples from two patients with severe COVID-19 at three different timepoints, consisting of severe remission during treatment with tocilizumab. The blood samples were collected on day 1, within 12 h of tocilizumab treatment, and on day 5 for both patients. An additional blood sample was collected from patient 2 because the patient remained COVID-19 positive. The cell types were annotated using markers provided by the authors in the original study. We have used the same genes as in the reference (Immune atlas). The dataset is available on Gene Expression Omnibus (GEO); the accession number is GSE150861.

We used the integrated dataset to analyse cell–cell interactions using the CellChat package66. For this analysis, we used the non-integrated shared gene space between all the integrated datasets after removing those genes supported by fewer than five counts, for a total of 10,851 genes ready for analysis. We then ran CellChat on each subset using the curated database for interactions in human samples. The gene expression of each ligand–receptor pair was visualized using a dotplot generated using Scanpy 1.8.1 (ref. 61) and anndata 0.7.6 (ref. 106). The scripts for the analyses, as well as the package version used in the analysis, can be found in the ‘covid’ section of the repository.

#### Pancreas

The datasets are publicly available on GEO and further described in Supplementary Table 15. We removed low-quality cells (high mitochondrial fraction and low number of genes) using a study-specific threshold. For cell type annotation, we removed genes expressed in fewer than 20 cells in each study and normalized the expression in each study to 1 × 106 total counts, excluding highly expressed genes, and subsequently applied a log transformation. We merged datasets across studies using Ensembl IDs and retaining the genes expressed in all studies. We used merged data across studies, followed by the identification of highly variable genes, z-normalization and the computation of top PCA components. We clustered the data and plotted known pancreatic islet cell type markers to annotate cell types cluster-wise.

For integration, we separated the datasets into reference and query, as described in Supplementary Table 15. From the reference data, we removed immune cell types and their doublets. We removed genes expressed in fewer than 20 cells in the reference data. We used gene sets from PanglaoDB107 release from March 2020 and Reactome78 v4.0 and mapped them to mouse genes using Ensembl108 V103 orthologues. We used only gene sets with at least three genes and at most 200 genes. We excluded genes that were not present in these gene sets. With expiMap, we integrated the reference datasets using samples as batches and projected query samples. We also performed matched integrations with Seurat109, Symphony18 and scVI40. We evaluated different integrations, as described in the integration evaluation section. We used reference query split as batches and excluded non-healthy query samples as they were not expected to be integrated into the healthy reference owning to biological differences. For the downstream interpretation analysis, we used directed expiMap scores.

We used multiple methods to evaluate PanglaoDB cell type scores. We plotted the PanglaoDB cell type scores of expected pancreatic cell types on query UMAPs and visually compared the results to cell type annotation. We used the PanglaoDB gene set scores as features for the annotation transfer from reference to query with weighted k-nearest neighbour (kNN), as described in ref. 15. We evaluated the annotation transfer with F1 score and by visual evaluation of prediction accuracy and certainty on UMAP.

For gene-level analyses on integrated data we normalized expression with Scanpy using functions normalize_total and log1p.

#### Integration benchmark datasets

We leverage datasets from five different tissues including PBMCs (n = 161,764) (ref. 109), heart (n = 18,641) (ref. 11), lung (n = 65,662) (ref. 13), colon (n = 34,772) (ref. 110) and liver (n = 113,063) (ref. 111). All datasets, except heart, were obtained from the Sfaira database112, which includes cell type labels. Heart was obtained from the scVI package. For the expiMap training for each dataset, we used the Reactome pathway database, selected only pathways that contain more than 12 genes and filtered out all genes that are not present in any pathway, and then we selected 2,000 HVG for training. For the other models, we used the same lists of genes.

#### Mouse endocrinogenesis

We used the developmental dataset from mouse endocrinogenesis (n = 25,919) (ref. 86). The raw dataset is available at the GEO under accession number GSE132188. Cell type labels were obtained from an adata object provided by the authors of scVelo113. We used the Reactome pathway database version 7.5.1 for annotations78 from MsigDB103,104. Days 14.5 and 15.5 were used as a reference, and days 12.5 and 13.5 as a query. For the reference dataset, we removed all pathways with fewer than 13 genes. The genes that were not in the GPs database were filtered out, reducing the total number of genes from approximately 28,000 to 10,000. Then 4,000 highly variable genes were selected for training. For the query, we used the genes obtained after pre-processing the reference dataset. RNA velocities were calculated using scVelo.

### Methods for the query to reference benchmarks

scVI: we used the setup from the scarches tutorial for query to reference mapping with scVI (https://scarches.readthedocs.io/en/latest/scvi_surgery_pipeline.html).

Symphony: we used the parameters recommended in the repository (https://github.com/immunogenomics/symphony).

Seurat: we adapted the Seurat reference mapping tutorial (https://satijalab.org/seurat/articles/integration_mapping.html), but used supervised principal component analysis(sPCA) instead of PCA.

### Statistics and reproducibility

The details for pre-processing of the datasets used for the model training are provided in the section ‘Datasets and pre-processing’. If not indicated otherwise in that section or in the legends, no data were excluded from training and analysis. The hyperparameters chosen for model training for all experiments are listed in Supplementary Note 12: hyperparameters. The details of statistical tests employed for differential testing of GPs are provided in the sections ‘Differential testing for GPs’ and Supplementary Note 1: comparison with limma-fry. Metrics for integration used in the paper are described in the section ‘Metrics for integration and evaluation’. Robustness of query to reference mapping for different query dataset sizes is analysed in Supplementary Note 3: robustness of the model under different data query dataset sizes. Reproducibility and robustness of newly learned GPs are discussed in Supplementary Note 6: disentanglement and robustness of newly learned GPs.

### Protocol

A step-by-step protocol for installing the software, training the model and downstream analysis can be found on Nature Protocol Exchange114.

### Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.


text:
# Why use negacyclic convolutions for polynomial multiplication instead of regular convolutions?

When multiplying polynomials from $$\mathbb{Z}_q[X] / (X^n-1)$$, the discrete NTT is used because: $$f \cdot g = \mathsf{NTT}_n^{-1}\left( \mathsf{NTT}_n\left(f\right) * \mathsf{NTT}_n\left(g\right) \right)$$ However, in virtually all schemes I've seen the negacyclic convolution is used - the ring is $$\mathbb{Z}_q[X] / (X^n+1)$$ and a trick is used to compute $$\mathsf{NTT}_{2n}^{-1}\left( \mathsf{NTT}_{2n}\left(f\right) * \mathsf{NTT}_{2n}\left(g\right) \right)$$ using $$\mathsf{NTT}_n$$ because we have to multiply the polynomials in $$\mathbb{Z}_q[X] / (X^{2n}-1)$$.
My question is - why bother with $$\mathbb{Z}_q[X] / (X^n+1)$$ and not simply use $$\mathbb{Z}_q[X] / (X^n-1)$$, thus applying $$\mathsf{NTT}_n$$ in a straightforward way with no extra complications?

The underlying hard problem used to construct the cryptographic primitives requires us to use that ring of polynomials modulo $$X^n + 1$$.

For example, if the security of your scheme relies on RLWE, then you have to stick with the rings that make the RLWE secure, which means that you cannot use $$X^n - 1$$, as discussed in this answer.

You have the same situation for the Ring-SIS problem.

In general, you have to be careful when you instantiate any problem with $$X^n - 1$$ as the modulus, because one can evaluate the polynomials on $$1$$ to work over integers instead of polynomials. This is discussed, for example, in the section "Polynomial evaluation" of this paper.

You state (I have fixed your notation, residue classes are taken with a $$/$$ not $$\backslash)$$ the former is setminus which is not the same:

However, in virtually all schemes I've seen the negacyclic convolution is used - the ring is $$\mathbb{Z}_q[X] / (X^n+1)$$ and a trick is used to compute $$\mathsf{NTT}_{2n}^{-1}\left( \mathsf{NTT}_{2n}\left(f\right) * \mathsf{NTT}_{2n}\left(g\right) \right)$$ using $$\mathsf{NTT}_n$$ because we have to multiply the polynomials in $$\mathbb{Z}_q[X] / (X^{2n}-1)$$.

and ask the question:

why bother with $$\mathbb{Z}_q[X] / (X^n+1)$$ and not simply use $$\mathbb{Z}_q[X] \setminus (X^n-1)$$?

If we have the polynomial factorization $$x^{2n}-1=(x^n-1)(x^n+1)$$ then computations modulo $$x^{2n}-1$$ can be speeded up by convolving (via NTT) with respect to each one of the factors and then combining. So

1. We have a factorization that leads to a fast transform, so we take this route. The extreme case is complex FFT when we can factor all the way into linear factors $$x^n-1=\prod_{i=1}^n (\omega^i-1)$$ with $$\omega$$ a primitive $$n^{th}$$ root of unity.
2. The factorization is unique, you cannot use $$(x^n+1)$$ only since $$(x^n+1)^2=(x^{2n}+2 x^n+1)$$ and $$q$$ is in general not characteristic 2. If you meant actually just use $$x^{2n}-1$$ directly, you'd have no speedup.
• Sorry, my bad with the quotient notation. I think you misunderstood my question (or I have misunderstood your answer). I'm not asking how to multiply polynomials of degree $(2n-1)$ using NTTs of order $n$, and am not asking why it's impossible to multiply polynomials from $\mathbb{Z}_q[X]/(X^n+1)$ with NTTs. I'm asking why use polynomials modolu $X^n+1$ in the first place (which need the trick with $(X^{2n}-1)$), and not use polynomials modolu $X^n-1$ May 22 at 13:46
• if your length is $N=2n,$ i.e., even, this IS the factorisation you must use to begin with i.e., $x^{2n}-1=(x^n-1)(x^n+1)$ because any $2n^{th}$ root of unity in any field or ring is either order $2n$ (those which are -1 when raised to power $n$) or have order dividing $n$. May 23 at 13:29
• you start with $N$ as a given. May 23 at 13:29

format: plain


text:
# Tag Info

## Hot answers tagged front-end

8

There is an undocumented file in the installation directory named specialArgFunctions.tr: NotebookOpen @ FileNameJoin @ { $InstallationDirectory, "SystemFiles", "FrontEnd", "SystemResources" , "FunctionalFrequency", "specialArgFunctions.tr" } This file describes in detail how to attach completion actions to each parameter of listed functions. For ... 7 It turned out that the MathematicaClosingDialog.nb is a nice example for a graphical user interface that solely uses the FrontEnd MathKernel. On the downside, one has to deal with low-level code. First I made a copy of the original MathematicaClosingDialog.nb, renamed it as MathematicaClosingDialog_Default.nb, and then imported it into a notebook: nb = ... 7 This is a very common problem for people who work on data analysis. Here as a solution to the problem using LocatorPane and a few other functions and tricks. TooltipListPlot[data_, tipFunction_, listPlotOptions___] := DynamicModule[ {displayQ = False, yRange , xRange, pt, minX, maxX, minY, maxY, tip, threshold, tipPosition, nf, dataPoints, ... 6 To make multiple tables, you should create a style definition for it. The style definitions used in tutorials can be found in the file FileNameJoin[{$InstallationDirectory, "SystemFiles", "FrontEnd", "StyleSheets", "Wolfram", "Reference.nb"}] (Execute it on your machine to find the local file name.) The style you show is called "DefinitionBox3Col". ...

5

The option GraphicsBoxOptions -> {ImageSize -> 500} can be set for the plot in several ways. (The option PreserveImageOptions is the one that controls what happens when the output cell is replaced. By default, the current size in output cell is preserved.) Stylesheet One can create a new style, say, "LargeImageSize" by editing the stylesheet (menu ...

5

In the meantime, here is a way to repair the gridlines fixGridLines[plot : _Graphics | _Legended] := Module[{xmin, xmax, ymin, ymax}, {{xmin, xmax}, {ymin, ymax}} = Through[{Min, Max}[#]] & /@ Transpose@Cases[plot, {_Real, _Real}, Infinity]; With[{p = plot}, MakeBoxes[p, StandardForm]] /. pat : (GridLines -> ...

5

Try something like text = { {Style[" option name", Italic], Style["default value", Italic], ""}, {" BackSubstitution", "False", "whether the solutions \ngiven by Reduce..."}, {" Cubics", "False", "whether the...\n"}}; Column[{ Grid[text, Alignment -> Left, Background -> RGBColor[0.92, 0.88, 0.84], Dividers -> {None, {1 ...

4

Context Menu The content of context menus is defined in the ContextMenus.tr in the \$InstallationDirectory\SystemFiles\FrontEnd\TextResources folder. It is possible to modify this file within an editor (e.g. Notepad++). In order to get a new context menu item with the functionality you described: First make a backup copy of the original ContextMenus.tr, ...

4

This answer, which is a slight modification of my answer to a previous question, takes you most of the way there: Clear@notebookScreenshot notebookScreenshot[nb_] := Module[{left, top, size, opts = AbsoluteOptions@nb}, {left, top} = WindowMargins /. opts // Diagonal; size = WindowSize /. opts; Needs["GUIKit`"]; ...

3

You can change the Mathematica menu in the way you desire by editing the MenuSetup.tr file, and restarting Mathematica. I make this change each time I get a new version of Mathematica. Of course, this isn't ideal, because it involves making a permanent change to the system, but it does work - just follow the style of the rest of the file. Be careful to ...

2

Mathematica uses the voice set in System Preferences for it's built-in speech functions. It determines that voice when it is launched, so to get it to recognize a new voice preference you have quit and relaunch.

1

Under Windows, you select an expression then right-click on it and choose the Speak selection menu item from the contextual menu. The voice used is the one set in the system preferences. Using Speak or SpokenString give you some options, but they do not seem to relate to the Speak Selection Menu Item mentioned above.

1

This isn't exactly a screenshot but maybe it meets your needs? doc = CreateDocument[{ TextCell["Need help", "Section"], Cell[1/0], TextCell["Can you help?"]}, WindowTitle -> "MyNotebook"]; Rasterize@doc

1

I classify your question as "Serious" thus, here is my answer. In many cases I have found myself in situations where a "canonical solution" does not seem to exist for a given problem found while creating Mathematica solutions. In those cases I tend to create a reasonable enough solution. When I read your question I remembered that a couple of years ago I ...

Only top voted, non community-wiki answers of a minimum length are eligible


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### Imaging quantum entanglement

The main object of a paper published a couple of days ago on Science is to find an answer to the following question:
what kind of imaging process could reveal a Bell inequality?
The experimental set-up used a $\beta$-Barium Borate crystal pumped by a (quasi-cotinuous) laser. The pairs of entagled photons generated are subsequently separated on a beam splitter and propagate into two distinct optical systems like LIGO interferometer.
The results is the production of some images that shots the Bell inequality violation, like the following image:
Moreover, our demonstration shows that one can detect the signature of a Bell-type behavior within a single image acquired by an imaging setup. By demonstrating that quantum imaging can generate high-dimensional images illustrating the presence of Bell-type entanglement, we benchmark quantum imaging techniques against the most fundamental test of quantum mechanics.
Moreau, P. A., Toninelli, E., Gregory, T., Aspden, R. S., Morris, P. A., & Padgett, M. J. (2019). Imaging Bell-type nonlocal behavior. Science Advances, 5(7), eaaw2563. doi:10.1126/sciadv.aaw2563


text:
# Optimization of light structures: the vanishing mass conjecture

Abstract : We consider the shape optimization problem which consists in placing a given mass $m$ of elastic material in a design region so that the compliance is minimal. Having in mind optimal light structures, Our purpose is to show that the problem of finding the stiffest shape configuration simplifies as the total mass $m$ tends to zero: we propose an explicit relaxed formulation where the compliance appears after rescaling as a convex functional of the relative density of mass. This allows us to write necessary and sufficient optimality conditions for light structures following the Monge-Kantorovich approach developed recently in [5].
Keywords :
Document type :
Conference papers
Domain :

Cited literature [25 references]

https://hal-univ-tln.archives-ouvertes.fr/hal-01283428
Contributor : Guy Bouchitte Connect in order to contact the contributor
Submitted on : Sunday, January 5, 2020 - 10:04:05 AM
Last modification on : Monday, January 13, 2020 - 1:12:25 AM
Long-term archiving on: : Monday, April 6, 2020 - 6:25:30 PM

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### Identifiers

• HAL Id : hal-01283428, version 1

### Citation

Guy Bouchitté. Optimization of light structures: the vanishing mass conjecture. Homogenization and Multiple Scales (HMS2000), Luciano Carbone and Riccardo De Arcangelis., Jun 2001, Naples, Italy. pp.131--145. ⟨hal-01283428⟩

### Metrics

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