Optimal-Transport Analysis of Single-Cell Gene Expression Identifies Developmental Trajectories in Reprogramming.

Understanding the molecular programs that guide differentiation during development is a major challenge. Here, we introduce Waddington-OT, an approach for studying developmental time courses to infer ancestor-descendant fates and model the regulatory programs that underlie them. We apply the method to reconstruct the landscape of reprogramming from 315,000 single-cell RNA sequencing (scRNA-seq) profiles, collected at half-day intervals across 18 days. The results reveal a wider range of developmental programs than previously characterized. Cells gradually adopt either a terminal stromal state or a mesenchymal-to-epithelial transition state. The latter gives rise to populations related to pluripotent, extra-embryonic, and neural cells, with each harboring multiple finer subpopulations. The analysis predicts transcription factors and paracrine signals that affect fates and experiments validate that the TF Obox6 and the cytokine GDF9 enhance reprogramming efficiency. Our approach sheds light on the process and outcome of reprogramming and provides a framework applicable to diverse temporal processes in biology.

SPANN: annotating single-cell resolution spatial transcriptome data with scRNA-seq data.

MOTIVATION: The rapid development of spatial transcriptome technologies has enabled researchers to acquire single-cell-level spatial data at an affordable price. However, computational analysis tools, such as annotation tools, tailored for these data are still lacking. Recently, many computational frameworks have emerged to integrate single-cell RNA sequencing (scRNA-seq) and spatial transcriptomics datasets. While some frameworks can utilize well-annotated scRNA-seq data to annotate spatial expression patterns, they overlook critical aspects. First, existing tools do not explicitly consider cell type mapping when aligning the two modalities. Second, current frameworks lack the capability to detect novel cells, which remains a key interest for biologists. RESULTS: To address these problems, we propose an annotation method for spatial transcriptome data called SPANN. The main tasks of SPANN are to transfer cell-type labels from well-annotated scRNA-seq data to newly generated single-cell resolution spatial transcriptome data and discover novel cells from spatial data. The major innovations of SPANN come from two aspects: SPANN automatically detects novel cells from unseen cell types while maintaining high annotation accuracy over known cell types. SPANN finds a mapping between spatial transcriptome samples and RNA data prototypes and thus conducts cell-type-level alignment. Comprehensive experiments using datasets from various spatial platforms demonstrate SPANN's capabilities in annotating known cell types and discovering novel cell states within complex tissue contexts. AVAILABILITY: The source code of SPANN can be accessed at https://github.com/ddb-qiwang/SPANN-torch. CONTACT: dengmh@math.pku.edu.cn.

scDOT: enhancing single-cell RNA-Seq data annotation and uncovering novel cell types through multi-reference integration.

The proliferation of single-cell RNA-seq data has greatly enhanced our ability to comprehend the intricate nature of diverse tissues. However, accurately annotating cell types in such data, especially when handling multiple reference datasets and identifying novel cell types, remains a significant challenge. To address these issues, we introduce Single Cell annotation based on Distance metric learning and Optimal Transport (scDOT), an innovative cell-type annotation method adept at integrating multiple reference datasets and uncovering previously unseen cell types. scDOT introduces two key innovations. First, by incorporating distance metric learning and optimal transport, it presents a novel optimization framework. This framework effectively learns the predictive power of each reference dataset for new query data and simultaneously establishes a probabilistic mapping between cells in the query data and reference-defined cell types. Secondly, scDOT develops an interpretable scoring system based on the acquired probabilistic mapping, enabling the precise identification of previously unseen cell types within the data. To rigorously assess scDOT's capabilities, we systematically evaluate its performance using two diverse collections of benchmark datasets encompassing various tissues, sequencing technologies and diverse cell types. Our experimental results consistently affirm the superior performance of scDOT in cell-type annotation and the identification of previously unseen cell types. These advancements provide researchers with a potent tool for precise cell-type annotation, ultimately enriching our understanding of complex biological tissues.

Matching single cells across modalities with contrastive learning and optimal transport.

Understanding the interactions between the biomolecules that govern cellular behaviors remains an emergent question in biology. Recent advances in single-cell technologies have enabled the simultaneous quantification of multiple biomolecules in the same cell, opening new avenues for understanding cellular complexity and heterogeneity. Still, the resulting multimodal single-cell datasets present unique challenges arising from the high dimensionality and multiple sources of acquisition noise. Computational methods able to match cells across different modalities offer an appealing alternative towards this goal. In this work, we propose MatchCLOT, a novel method for modality matching inspired by recent promising developments in contrastive learning and optimal transport. MatchCLOT uses contrastive learning to learn a common representation between two modalities and applies entropic optimal transport as an approximate maximum weight bipartite matching algorithm. Our model obtains state-of-the-art performance on two curated benchmarking datasets and an independent test dataset, improving the top scoring method by 26.1% while preserving the underlying biological structure of the multimodal data. Importantly, MatchCLOT offers high gains in computational time and memory that, in contrast to existing methods, allows it to scale well with the number of cells. As single-cell datasets become increasingly large, MatchCLOT offers an accurate and efficient solution to the problem of modality matching.

Detection of PatIent-Level distances from single cell genomics and pathomics data with Optimal Transport (PILOT).

Although clinical applications represent the next challenge in single-cell genomics and digital pathology, we still lack computational methods to analyze single-cell or pathomics data to find sample-level trajectories or clusters associated with diseases. This remains challenging as single-cell/pathomics data are multi-scale, i.e., a sample is represented by clusters of cells/structures, and samples cannot be easily compared with each other. Here we propose PatIent Level analysis with Optimal Transport (PILOT). PILOT uses optimal transport to compute the Wasserstein distance between two individual single-cell samples. This allows us to perform unsupervised analysis at the sample level and uncover trajectories or cellular clusters associated with disease progression. We evaluate PILOT and competing approaches in single-cell genomics or pathomics studies involving various human diseases with up to 600 samples/patients and millions of cells or tissue structures. Our results demonstrate that PILOT detects disease-associated samples from large and complex single-cell or pathomics data. Moreover, PILOT provides a statistical approach to find changes in cell populations, gene expression, and tissue structures related to the trajectories or clusters supporting interpretation of predictions.

Optimal transport and control of active drops.

Understanding the complex patterns in space-time exhibited by active systems has been the subject of much interest in recent times. Complementing this forward problem is the inverse problem of controlling active matter. Here, we use optimal control theory to pose the problem of transporting a slender drop of an active fluid and determine the dynamical profile of the active stresses to move it with minimal viscous dissipation. By parametrizing the position and size of the drop using a low-order description based on lubrication theory, we uncover a natural "gather-move-spread" strategy that leads to an optimal bound on the maximum achievable displacement of the drop relative to its size. In the continuum setting, the competition between passive surface tension and active controls generates richer behavior with futile oscillations and complex drop morphologies that trade internal dissipation against the transport cost to select optimal strategies. Our work combines active hydrodynamics and optimal control in a tractable and interpretable framework and begins to pave the way for the spatiotemporal manipulation of active matter.

Unpaired deep learning for pharmacokinetic parameter estimation from dynamic contrast-enhanced MRI without AIF measurements.

DCE-MRI provides information about vascular permeability and tissue perfusion through the acquisition of pharmacokinetic parameters. However, traditional methods for estimating these pharmacokinetic parameters involve fitting tracer kinetic models, which often suffer from computational complexity and low accuracy due to noisy arterial input function (AIF) measurements. Although some deep learning approaches have been proposed to tackle these challenges, most existing methods rely on supervised learning that requires paired input DCE-MRI and labeled pharmacokinetic parameter maps. This dependency on labeled data introduces significant time and resource constraints and potential noise in the labels, making supervised learning methods often impractical. To address these limitations, we present a novel unpaired deep learning method for estimating pharmacokinetic parameters and the AIF using a physics-driven CycleGAN approach. Our proposed CycleGAN framework is designed based on the underlying physics model, resulting in a simpler architecture with a single generator and discriminator pair. Crucially, our experimental results indicate that our method does not necessitate separate AIF measurements and produces more reliable pharmacokinetic parameters than other techniques.

Wasserstein regression with empirical measures and density estimation for sparse data.

The problem of modeling the relationship between univariate distributions and one or more explanatory variables lately has found increasing interest. Existing approaches proceed by substituting proxy estimated distributions for the typically unknown response distributions. These estimates are obtained from available data but are problematic when for some of the distributions only few data are available. Such situations are common in practice and cannot be addressed with currently available approaches, especially when one aims at density estimates. We show how this and other problems associated with density estimation such as tuning parameter selection and bias issues can be side-stepped when covariates are available. We also introduce a novel version of distribution-response regression that is based on empirical measures. By avoiding the preprocessing step of recovering complete individual response distributions, the proposed approach is applicable when the sample size available for each distribution varies and especially when it is small for some of the distributions but large for others. In this case, one can still obtain consistent distribution estimates even for distributions with only few data by gaining strength across the entire sample of distributions, while traditional approaches where distributions or densities are estimated individually fail, since sparsely sampled densities cannot be consistently estimated. The proposed model is demonstrated to outperform existing approaches through simulations and Environmental Influences on Child Health Outcomes data.

Alternating the population and control neural networks to solve high-dimensional stochastic mean-field games.

We present APAC-Net, an alternating population and agent control neural network for solving stochastic mean-field games (MFGs). Our algorithm is geared toward high-dimensional instances of MFGs that are not approachable with existing solution methods. We achieve this in two steps. First, we take advantage of the underlying variational primal-dual structure that MFGs exhibit and phrase it as a convex-concave saddle-point problem. Second, we parameterize the value and density functions by two neural networks, respectively. By phrasing the problem in this manner, solving the MFG can be interpreted as a special case of training a generative adversarial network (GAN). We show the potential of our method on up to 100-dimensional MFG problems.

Constrained Reweighting of Distributions: An Optimal Transport Approach.

We commonly encounter the problem of identifying an optimally weight-adjusted version of the empirical distribution of observed data, adhering to predefined constraints on the weights. Such constraints often manifest as restrictions on the moments, tail behavior, shapes, number of modes, etc., of the resulting weight-adjusted empirical distribution. In this article, we substantially enhance the flexibility of such a methodology by introducing a nonparametrically imbued distributional constraint on the weights and developing a general framework leveraging the maximum entropy principle and tools from optimal transport. The key idea is to ensure that the maximum entropy weight-adjusted empirical distribution of the observed data is close to a pre-specified probability distribution in terms of the optimal transport metric, while allowing for subtle departures. The proposed scheme for the re-weighting of observations subject to constraints is reminiscent of the empirical likelihood and related ideas, but offers greater flexibility in applications where parametric distribution-guided constraints arise naturally. The versatility of the proposed framework is demonstrated in the context of three disparate applications where data re-weighting is warranted to satisfy side constraints on the optimization problem at the heart of the statistical task-namely, portfolio allocation, semi-parametric inference for complex surveys, and ensuring algorithmic fairness in machine learning algorithms.

A Gibbs Posterior Framework for Fair Clustering.

The rise of machine learning-driven decision-making has sparked a growing emphasis on algorithmic fairness. Within the realm of clustering, the notion of balance is utilized as a criterion for attaining fairness, which characterizes a clustering mechanism as fair when the resulting clusters maintain a consistent proportion of observations representing individuals from distinct groups delineated by protected attributes. Building on this idea, the literature has rapidly incorporated a myriad of extensions, devising fair versions of the existing frequentist clustering algorithms, e.g., k-means, k-medioids, etc., that aim at minimizing specific loss functions. These approaches lack uncertainty quantification associated with the optimal clustering configuration and only provide clustering boundaries without quantifying the probabilities associated with each observation belonging to the different clusters. In this article, we intend to offer a novel probabilistic formulation of the fair clustering problem that facilitates valid uncertainty quantification even under mild model misspecifications, without incurring substantial computational overhead. Mixture model-based fair clustering frameworks facilitate automatic uncertainty quantification, but tend to showcase brittleness under model misspecification and involve significant computational challenges. To circumnavigate such issues, we propose a generalized Bayesian fair clustering framework that inherently enjoys decision-theoretic interpretation. Moreover, we devise efficient computational algorithms that crucially leverage techniques from the existing literature on optimal transport and clustering based on loss functions. The gain from the proposed technology is showcased via numerical experiments and real data examples.

Cellstitch: 3D cellular anisotropic image segmentation via optimal transport.

BACKGROUND: Spatial mapping of transcriptional states provides valuable biological insights into cellular functions and interactions in the context of the tissue. Accurate 3D cell segmentation is a critical step in the analysis of this data towards understanding diseases and normal development in situ. Current approaches designed to automate 3D segmentation include stitching masks along one dimension, training a 3D neural network architecture from scratch, and reconstructing a 3D volume from 2D segmentations on all dimensions. However, the applicability of existing methods is hampered by inaccurate segmentations along the non-stitching dimensions, the lack of high-quality diverse 3D training data, and inhomogeneity of image resolution along orthogonal directions due to acquisition constraints; as a result, they have not been widely used in practice. METHODS: To address these challenges, we formulate the problem of finding cell correspondence across layers with a novel optimal transport (OT) approach. We propose CellStitch, a flexible pipeline that segments cells from 3D images without requiring large amounts of 3D training data. We further extend our method to interpolate internal slices from highly anisotropic cell images to recover isotropic cell morphology. RESULTS: We evaluated the performance of CellStitch through eight 3D plant microscopic datasets with diverse anisotropic levels and cell shapes. CellStitch substantially outperforms the state-of-the art methods on anisotropic images, and achieves comparable segmentation quality against competing methods in isotropic setting. We benchmarked and reported 3D segmentation results of all the methods with instance-level precision, recall and average precision (AP) metrics. CONCLUSIONS: The proposed OT-based 3D segmentation pipeline outperformed the existing state-of-the-art methods on different datasets with nonzero anisotropy, providing high fidelity recovery of 3D cell morphology from microscopic images.

Associating somatic mutation with clinical outcomes through kernel regression and optimal transport.

Somatic mutations in cancer patients are inherently sparse and potentially high dimensional. Cancer patients may share the same set of deregulated biological processes perturbed by different sets of somatically mutated genes. Therefore, when assessing the associations between somatic mutations and clinical outcomes, gene-by-gene analysis is often under-powered because it does not capture the complex disease mechanisms shared across cancer patients. Rather than testing genes one by one, an intuitive approach is to aggregate somatic mutation data of multiple genes to assess their joint association with clinical outcomes. The challenge is how to aggregate such information. Building on the optimal transport method, we propose a principled approach to estimate the similarity of somatic mutation profiles of multiple genes between tumor samples, while accounting for gene-gene similarities defined by gene annotations or empirical mutational patterns. Using such similarities, we can assess the associations between somatic mutations and clinical outcomes by kernel regression. We have applied our method to analyze somatic mutation data of 17 cancer types and identified at least five cancer types, where somatic mutations are associated with overall survival, progression-free interval, or cytolytic activity.

Autoregressive optimal transport models.

Series of univariate distributions indexed by equally spaced time points are ubiquitous in applications and their analysis constitutes one of the challenges of the emerging field of distributional data analysis. To quantify such distributional time series, we propose a class of intrinsic autoregressive models that operate in the space of optimal transport maps. The autoregressive transport models that we introduce here are based on regressing optimal transport maps on each other, where predictors can be transport maps from an overall barycenter to a current distribution or transport maps between past consecutive distributions of the distributional time series. Autoregressive transport models and their associated distributional regression models specify the link between predictor and response transport maps by moving along geodesics in Wasserstein space. These models emerge as natural extensions of the classical autoregressive models in Euclidean space. Unique stationary solutions of autoregressive transport models are shown to exist under a geometric moment contraction condition of Wu & Shao [(2004) Limit theorems for iterated random functions. Journal of Applied Probability 41, 425-436)], using properties of iterated random functions. We also discuss an extension to a varying coefficient model for first-order autoregressive transport models. In addition to simulations, the proposed models are illustrated with distributional time series of house prices across U.S. counties and annual summer temperature distributions.

Scalable Gromov-Wasserstein Based Comparison of Biological Time Series.

A time series is an extremely abundant data type arising in many areas of scientific research, including the biological sciences. Any method that compares time series data relies on a pairwise distance between trajectories, and the choice of distance measure determines the accuracy and speed of the time series comparison. This paper introduces an optimal transport type distance for comparing time series trajectories that are allowed to lie in spaces of different dimensions and/or with differing numbers of points possibly unequally spaced along each trajectory. The construction is based on a modified Gromov-Wasserstein distance optimization program, reducing the problem to a Wasserstein distance on the real line. The resulting program has a closed-form solution and can be computed quickly due to the scalability of the one-dimensional Wasserstein distance. We discuss theoretical properties of this distance measure, and empirically demonstrate the performance of the proposed distance on several datasets with a range of characteristics commonly found in biologically relevant data. We also use our proposed distance to demonstrate that averaging oscillatory time series trajectories using the recently proposed Fused Gromov-Wasserstein barycenter retains more characteristics in the averaged trajectory when compared to traditional averaging, which demonstrates the applicability of Fused Gromov-Wasserstein barycenters for biological time series. Fast and user friendly software for computing the proposed distance and related applications is provided. The proposed distance allows fast and meaningful comparison of biological time series and can be efficiently used in a wide range of applications.

Functional network analysis reveals an immune tolerance mechanism in cancer.

We present a technique to construct a simplification of a feature network which can be used for interactive data exploration, biological hypothesis generation, and the detection of communities or modules of cofunctional features. These are modules of features that are not necessarily correlated, but nevertheless exhibit common function in their network context as measured by similarity of relationships with neighboring features. In the case of genetic networks, traditional pathway analyses tend to assume that, ideally, all genes in a module exhibit very similar function, independent of relationships with other genes. The proposed technique explicitly relaxes this assumption by employing the comparison of relational profiles. For example, two genes which always activate a third gene are grouped together even if they never do so concurrently. They have common, but not identical, function. The comparison is driven by an average of a certain computationally efficient comparison metric between Gaussian mixture models. The method has its basis in the local connection structure of the network and the collection of joint distributions of the data associated with nodal neighborhoods. It is benchmarked on networks with known community structures. As the main application, we analyzed the gene regulatory network in lung adenocarcinoma, finding a cofunctional module of genes including the pregnancy-specific glycoproteins (PSGs). About 20% of patients with lung, breast, uterus, and colon cancer in The Cancer Genome Atlas (TCGA) have an elevated PSG+ signature, with associated poor group prognosis. In conjunction with previous results relating PSGs to tolerance in the immune system, these findings implicate the PSGs in a potential immune tolerance mechanism of cancers.

A machine learning framework for solving high-dimensional mean field game and mean field control problems.

Mean field games (MFG) and mean field control (MFC) are critical classes of multiagent models for the efficient analysis of massive populations of interacting agents. Their areas of application span topics in economics, finance, game theory, industrial engineering, crowd motion, and more. In this paper, we provide a flexible machine learning framework for the numerical solution of potential MFG and MFC models. State-of-the-art numerical methods for solving such problems utilize spatial discretization that leads to a curse of dimensionality. We approximately solve high-dimensional problems by combining Lagrangian and Eulerian viewpoints and leveraging recent advances from machine learning. More precisely, we work with a Lagrangian formulation of the problem and enforce the underlying Hamilton-Jacobi-Bellman (HJB) equation that is derived from the Eulerian formulation. Finally, a tailored neural network parameterization of the MFG/MFC solution helps us avoid any spatial discretization. Our numerical results include the approximate solution of 100-dimensional instances of optimal transport and crowd motion problems on a standard work station and a validation using a Eulerian solver in two dimensions. These results open the door to much-anticipated applications of MFG and MFC models that are beyond reach with existing numerical methods.

Efficient Discretization of Optimal Transport.

Obtaining solutions to optimal transportation (OT) problems is typically intractable when marginal spaces are continuous. Recent research has focused on approximating continuous solutions with discretization methods based on i.i.d. sampling, and this has shown convergence as the sample size increases. However, obtaining OT solutions with large sample sizes requires intensive computation effort, which can be prohibitive in practice. In this paper, we propose an algorithm for calculating discretizations with a given number of weighted points for marginal distributions by minimizing the (entropy-regularized) Wasserstein distance and providing bounds on the performance. The results suggest that our plans are comparable to those obtained with much larger numbers of i.i.d. samples and are more efficient than existing alternatives. Moreover, we propose a local, parallelizable version of such discretizations for applications, which we demonstrate by approximating adorable images.

Joint User Association and Deployment Optimization for Energy-Efficient Heterogeneous UAV-Enabled MEC Networks.

Unmanned aerial vehicles (UAVs) providing additional on-demand communication and computing services have become a promising technology. However, the limited energy supply of UAVs, which constrains their service duration, has emerged as an obstacle in UAV-enabled networks. In this context, a novel task offloading framework is proposed in UAV-enabled mobile edge computing (MEC) networks. Specifically, heterogeneous UAVs with different communication and computing capabilities are considered and the energy consumption of UAVs is minimized via jointly optimizing user association and UAV deployment. The optimal transport theory is introduced to analyze the user association sub-problem, and the UAV deployment for each sub-region is determined by a dragonfly algorithm (DA). Simulation results show that the energy consumption performance is significantly improved by the proposed algorithm.

A Graph-Space Optimal Transport Approach Based on Kaniadakis κ-Gaussian Distribution for Inverse Problems Related to Wave Propagation.

Data-centric inverse problems are a process of inferring physical attributes from indirect measurements. Full-waveform inversion (FWI) is a non-linear inverse problem that attempts to obtain a quantitative physical model by comparing the wave equation solution with observed data, optimizing an objective function. However, the FWI is strenuously dependent on a robust objective function, especially for dealing with cycle-skipping issues and non-Gaussian noises in the dataset. In this work, we present an objective function based on the Kaniadakis κ-Gaussian distribution and the optimal transport (OT) theory to mitigate non-Gaussian noise effects and phase ambiguity concerns that cause cycle skipping. We construct the κ-objective function using the probabilistic maximum likelihood procedure and include it within a well-posed version of the original OT formulation, known as the Kantorovich-Rubinstein metric. We represent the data in the graph space to satisfy the probability axioms required by the Kantorovich-Rubinstein framework. We call our proposal the κ-Graph-Space Optimal Transport FWI (κ-GSOT-FWI). The results suggest that the κ-GSOT-FWI is an effective procedure to circumvent the effects of non-Gaussian noise and cycle-skipping problems. They also show that the Kaniadakis κ-statistics significantly improve the FWI objective function convergence, resulting in higher-resolution models than classical techniques, especially when κ=0.6.