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Jointly embedding multiple single-cell omics measurements

Jie Liu, Yuanhao HuangiD, Ritambhara SinghiD, Jean‐Philippe VertiD, William Stafford NobleiD

Abstract

Many single-cell sequencing technologies are now available, but it is still difficult to apply multiple sequencing technologies to the same single cell. In this paper, we propose an unsupervised manifold alignment algorithm, MMD-MA, for integrating multiple measurements carried out on disjoint aliquots of a given population of cells. Effectively, MMD-MA performs an in silico co-assay by embedding cells measured in different ways into a learned latent space. In the MMD-MA algorithm, single-cell data points from multiple domains are aligned by optimizing an objective function with three components: (1) a maximum mean discrepancy (MMD) term to encourage the differently measured points to have similar distributions in the latent space, (2) a distortion term to preserve the structure of the data between the input space and the latent space, and (3) a penalty term to avoid collapse to a trivial solution. Notably, MMD-MA does not require any correspondence information across data modalities, either between the cells or between the features. Furthermore, MMD-MAs weak distributional requirements for the domains to be aligned allow the algorithm to integrate heterogeneous types of single cell measures, such as gene expression, DNA accessibility, chromatin organization, methylation, and imaging data. We demonstrate the utility of MMD-MA in simulation experiments and using a real data set involving single-cell gene expression and methylation data.

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In-depth summary

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Claude Opus 5.5 · xhigh effort · 7 Nov 2026 · from the full text of version 2 (21 May 2019) · CC BY 4.0

MMD-MA embeds cells measured by different single-cell assays into a shared latent space without cell or feature correspondence, recovering true pairings in simulations and a 61-cell expression/methylation co-assay.

Some assay pairs, such as scATAC-seq and scHi-C, both cleave genomic DNA and so cannot be run on one cell; computationally integrating separately measured cells is the alternative to a co-assay. Unlike GUMA, which assumes one-to-one cell matching, and MAGAN, which needs some known correspondence, MMD-MA aligns embedded distributions through a kernel MMD term and requires no correspondence between cells or features.

Key results

  1. SuggestedAcross three simulated manifolds, including one where each domain gets its own nonlinear (Swiss roll) mapping, MMD-MA recovered the hidden point pairing without being given it and outperformed JLMA.

    300 points per manifold, projected to 1000 and 2000 features with Gaussian noise (σ = 0.05); latent p = 5 deliberately misspecified. The average fraction of samples closer than the true match fell monotonically toward zero, and MMD-MA beat JLMA (k = 5 and k = 6) in all three simulations. · 4.1 Three simulations; Figure legends

  2. SuggestedOn a 61-cell co-assay of gene expression and methylation with the cell pairing hidden, MMD-MA placed most cells next to their true partner from the other modality.

    2486 genes kept after removing missing values; latent p = 5. The average fraction of samples closer than the true match was 0.49 in separate PCA projections and 0.024 after MMD-MA, consistent across learning rates; >50% of cells ended up closest to their true counterpart. · 4.2 Real world application results; Figure legends

  3. ShownMMD-MA ran in under one minute and was considerably faster than JLMA.

    Table 1 lists run times measured on a 3.00GHz Intel Xeon Gold 6136 processor for the three simulations and the real data set, with JLMA run at k = 5 or k = 6 for comparison. The exact values are not given in the text. · 4.1 Three simulations; Tables

  4. SuggestedAlignment quality was largely insensitive to the trade-off weights, the MMD kernel bandwidth and an overstated latent dimension.

    p = 5 was used throughout, although the true dimension is 2 (simulations 1 and 2) or 3 (simulation 3). Eight λ1/λ2 combinations per data set gave nearly the same performance, some by different convergence paths; varying σ also left performance nearly unchanged (Supplementary Figure S1). · 4.3 MMD-MA’s performance is robust to variations…; Figure legends; Tables

Methods, data and reproducibility

Each domain gets a kernel-parametrized map into a shared p-dimensional space. The maps are learned by minimizing a Gaussian-RBF MMD between the embedded point clouds, plus a distortion term (input versus embedded inner products) and a penalty that keeps the maps close to projections. Optimization is gradient descent with Adam, keeping the best of 100 random restarts.

  • Three simulated manifolds (2D branching; branching on a Swiss roll; 3D circular frustum), each projected into two domains of 1000 and 2000 features plus Gaussian noise · n = 300 data points per manifold
  • Published single-cell co-assay of gene expression and methylation rates (Methyl-Expr), used with the cell correspondence hidden · n = 61 cells; 2486 genes
Methods & tools
MMD-MA, JLMA, Adam, PCA, MDS
Data & accessions
http://noble.gs.washington.edu/proj/mmd-ma
Experimental conditions
A 3.00GHz Intel Xeon Gold 6136 processor; MMD-MA is reported to finish in under one minute (Table 1 covers the three simulations and the real data set).
Reproducibility gaps
The text gives no code repository or software versions; the project page is cited for the data sets and MMD-MA outputs. Exact λ1, λ2 and σ settings are in Table 2 and Supplementary Table S1, which the text does not reproduce.
Limitations (6)
  • The objective is non-convex, so only local minima can be reached; each data set is run from 100 random seeds and the solution with the lowest objective is kept. · stated by the authors
  • The MMD term matches the embedded distributions only globally, without the per-instance local geometry terms of JLMA and GUMA. Alignment also needs the domains to share well-structured latent variation; for example, two isotropic Gaussians could not be oriented relative to each other. · stated by the authors
  • Kernel matrices have to be held in memory. Scaling from hundreds or thousands of cells toward millions is left to future work, with random projection or Nystrom approximation suggested. · stated by the authors
  • GUMA could not be run with its authors' implementation, and MAGAN was excluded because it needs initial correspondence, so JLMA is the only baseline. · stated by the authors
  • Real-data validation rests on one 61-cell co-assay, and the text reports no baseline accuracy comparison on it. The latent dimension and kernel parameters are set by the user, and the λ weights were tuned by inspecting term magnitudes. · noted in this summary
  • Every test has equal cell numbers in the two domains and an exact one-to-one correspondence, so the motivating case (disjoint aliquots that differ in size or composition) is not directly evaluated. · noted in this summary

Who should read:Read this if you need to integrate single-cell modalities measured on separate cells, with no shared cells or features, and want a kernel-based alignment method that needs no correspondence.

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