Scalable and consistent embedding of probability measures into Hilbert spaces via measure quantization

This paper is focused on statistical learning from data that come as probability measures. In this setting, popular approaches consist in embedding such data into a Hilbert space with either Linearized Optimal Transport or Kernel Mean Embedding. However, the cost of computing such embeddings prohibits their direct use in large-scale settings. We study two methods based on measure quantization for approximating input probability measures with discrete measures of small-support size. The first one is based on optimal quantization of each input measure, while the second one relies on mean-measure quantization. We study the consistency of such approximations, and its implication for scalable embeddings of probability measures into a Hilbert space at a low computational cost. We finally illustrate our findings with various numerical experiments.

Low dimensional representation of multi-patient flow cytometry datasets using optimal transport for minimal residual disease detection in leukemia

Representing and quantifying Minimal Residual Disease (MRD) in Acute Myeloid Leukemia (AML), a type of cancer that affects the blood and bone marrow, is essential in the prognosis and follow-up of AML patients. As traditional cytological analysis cannot detect leukemia cells below 5\%, the analysis of flow cytometry dataset is expected to provide more reliable results. In this paper, we explore statistical learning methods based on optimal transport (OT) to achieve a relevant low-dimensional representation of multi-patient flow cytometry measurements (FCM) datasets considered as high-dimensional probability distributions. Using the framework of OT, we justify the use of the K-means algorithm for dimensionality reduction of multiple large-scale point clouds through mean measure quantization by merging all the data into a single point cloud. After this quantization step, the visualization of the intra and inter-patients FCM variability is carried out by embedding low-dimensional quantized probability measures into a linear space using either Wasserstein Principal Component Analysis (PCA) through linearized OT or log-ratio PCA of compositional data. Using a publicly available FCM dataset and a FCM dataset from Bordeaux University Hospital, we demonstrate the benefits of our approach over the popular kernel mean embedding technique for statistical learning from multiple high-dimensional probability distributions. We also highlight the usefulness of our methodology for low-dimensional projection and clustering patient measurements according to their level of MRD in AML from FCM. In particular, our OT-based approach allows a relevant and informative two-dimensional representation of the results of the FlowSom algorithm, a state-of-the-art method for the detection of MRD in AML using multi-patient FCM.