Linearized Optimal Transport for Analysis of High-Dimensional Point-Cloud and Single-Cell Data

TL;DR

The paper tackles the challenge of comparing irregular, high-dimensional single-cell point clouds across individuals by adopting Linear Optimal Transport (LOT), which embeds each distribution relative to a fixed reference $oldsymbol{\nu}$ via the transport map $T_\boldsymbol{\nu}^{\mu}$. This yields fixed-length vectors $z \in \mathbb{R}^p$ that preserve transport geometry, enabling linear learning, interpretability, and synthesis. The authors demonstrate two core capabilities: interpretable classification of COVID-19 patient states using a linear SVM in LOT space with classifier weights mappable back to markers and spatial regions, and data generation for patient-derived organoids through linear operations in LOT space; they also derive barycenters representing averaged cellular profiles. They validate the approach on COVID-19 immune profiling and PDO data, achieving high discrimination (e.g., AUC $= 0.95$, accuracy $= 0.90$) and revealing biologically meaningful, interpretable signatures via spectral co-clustering. Overall, LOT provides a cohesive framework that unites predictive performance, interpretability, and generative modeling for high-dimensional, heterogeneous biological distributions, with broad applicability to understanding immune variation and treatment effects in single-cell data.

Abstract

Single-cell technologies generate high-dimensional point clouds of cells, enabling detailed characterization of complex patient states and treatment responses. Yet each patient is represented by an irregular point cloud rather than a simple vector, making it difficult to directly quantify and compare biological differences between individuals. Nonlinear methods such as kernels and neural networks achieve predictive accuracy but act as black boxes, offering little biological interpretability. To address these limitations, we adapt the Linear Optimal Transport (LOT) framework to this setting, embedding irregular point clouds into a fixed-dimensional Euclidean space while preserving distributional structure. This embedding provides a principled linear representation that preserves optimal transport geometry while enabling downstream analysis. It also forms a registration between any two patients, enabling direct comparison of their cellular distributions. Within this space, LOT enables: (i) \textbf{accurate and interpretable classification} of COVID-19 patient states, where classifier weights map back to specific markers and spatial regions driving predictions; and (ii) \textbf{synthetic data generation} for patient-derived organoids, exploiting the linearity of the LOT embedding. LOT barycenters yield averaged cellular profiles representing combined conditions or samples, supporting drug interaction testing. Together, these results establish LOT as a unified framework that bridges predictive performance, interpretability, and generative modeling. By transforming heterogeneous point clouds into structured embeddings directly traceable to the original data, LOT opens new opportunities for understanding immune variation and treatment effects in high-dimensional biological systems.

Figures (7)

• Figure 1: Cluster activation signatures contrasting COVID-positive and healthy groups. Each line plot shows standardized marker-level activations ($Z$-scores) for one co-cluster. Top: Cluster 1 shows broad up-regulation of T-cell lineage and inflammatory markers in COVID-positive patients, consistent with heightened immune activation. Bottom: Cluster 5 shows highly similar marker profiles between groups, but with consistently lower levels in COVID-positive patients, supporting its interpretation as a resting or naïve cluster. Together, these signatures highlight how distinct spatial-feature clusters capture biologically interpretable differences in immune response between conditions.
• Figure 2: PHATE on sample-level LOT embeddings, colored by patient. Left: raw LOT embeddings. Right: after per–patient DMSO control subtraction.
• Figure 3: Eigenvalue spectrum of $(1/(B-1))\,(\Delta - \bar{\Delta})(\Delta - \bar{\Delta})^\top$ (column-centered; $B=\langle\text{value}\rangle$). The bulk follows the Marčenko--Pastur distribution, with two dominant outliers indicating a structured immune response pattern.
• Figure 4: Projection of patient-level $\Delta$ vectors into the first two singular directions. Each line segment extends from the origin to a patient replicate's $\Delta$ vector, colored by patient ID. The strong alignment along the negative $\Delta$-SVD1 axis highlights a coherent treatment-induced deviation shared across patients.
• Figure : (a) 10 row × 8 column clusters