Inference for empirical Wasserstein distances on finite spaces

Hypothesis Test and Confidence Analysis with Wasserstein Distance on General Dimension

We develop a general framework for statistical inference with the 1-Wasserstein distance. Recently, the Wasserstein distance has attracted considerable attention and has been widely applied to various machine learning tasks because of its excellent properties. However, hypothesis tests and a confidence analysis for it have not been established in a general multivariate setting. This is because the limit distribution of the empirical distribution with the Wasserstein distance is unavailable without strong restriction. To address this problem, in this study, we develop a novel nonasymptotic gaussian approximation for the empirical 1-Wasserstein distance. Using the approximation method, we develop a hypothesis test and confidence analysis for the empirical 1-Wasserstein distance. We also provide a theoretical guarantee and an efficient algorithm for the proposed approximation. Our experiments validate its performance numerically.

Colocalization for super-resolution microscopy via optimal transport

Super-resolution fluorescence microscopy is a widely used technique in cell biology. Stimulated emission depletion (STED) microscopy enables the recording of multiple-color images with subdiffraction resolution. The enhanced resolution leads to new challenges regarding colocalization analysis of macromolecule distributions. We demonstrate that well-established methods for the analysis of colocalization in diffraction-limited datasets and for coordinate-stochastic nanoscopy are not equally well suited for the analysis of high-resolution STED images. We propose optimal transport colocalization, which measures the minimal transporting cost below a given spatial scale to match two protein intensity distributions. Its validity on simulated data as well as on dual-color STED recordings of yeast and mammalian cells is demonstrated. We also extend the optimal transport colocalization methodology to coordinate-stochastic nanoscopy.

Optimal Estimation of Wasserstein Distance on A Tree with An Application to Microbiome Studies

The weighted UniFrac distance, a plug-in estimator of the Wasserstein distance of read counts on a tree, has been widely used to measure the microbial community difference in microbiome studies. Our investigation however shows that such a plug-in estimator, although intuitive and commonly used in practice, suffers from potential bias. Motivated by this finding, we study the problem of optimal estimation of the Wasserstein distance between two distributions on a tree from the sampled data in the high-dimensional setting. The minimax rate of convergence is established. To overcome the bias problem, we introduce a new estimator, referred to as the moment-screening estimator on a tree (MET), by using implicit best polynomial approximation that incorporates the tree structure. The new estimator is computationally efficient and is shown to be minimax rate-optimal. Numerical studies using both simulated and real biological datasets demonstrate the practical merits of MET, including reduced biases and statistically more significant differences in microbiome between the inactive Crohn's disease patients and the normal controls.

Special function methods for bursty models of transcription

We explore a Markov model used in the analysis of gene expression, involving the bursty production of pre-mRNA, its conversion to mature mRNA, and its consequent degradation. We demonstrate that the integration used to compute the solution of the stochastic system can be approximated by the evaluation of special functions. Furthermore, the form of the special function solution generalizes to a broader class of burst distributions. In light of the broader goal of biophysical parameter inference from transcriptomics data, we apply the method to simulated data, demonstrating effective control of precision and runtime. Finally, we propose and validate a non-Bayesian approach for parameter estimation based on the characteristic function of the target joint distribution of pre-mRNA and mRNA.

Photoacoustic imaging to assess pixel-based sO2 distributions in experimental prostate tumors

A protocol for photoacoustic imaging (PAI) has been developed to assess pixel-based oxygen saturation (sO2) distributions of experimental tumor models. The protocol was applied to evaluate the dependence of PAI results on measurement settings, reproducibility of PAI, and for the characterization of the oxygenation status of experimental prostate tumor sublines (Dunning R3327-H, -HI, -AT1) implanted subcutaneously in male Copenhagen rats. The three-dimensional (3-D) PA data employing two wavelengths were used to estimate sO2 distributions. If the PA signal was sufficiently strong, the distributions were independent from signal gain, threshold, and positioning of animals. Reproducibility of sO2 distributions with respect to shape and median values was demonstrated over several days. The three tumor sublines were characterized by the shapes of their sO2 distributions and their temporal response after external changes of the oxygen supply (100% O2 or air breathing and clamping of tumor-supplying artery). The established protocol showed to be suitable for detecting temporal changes in tumor oxygenation as well as differences in oxygenation between tumor sublines. PA results were in accordance with histology for hypoxia, perfusion, and vasculature. The presented protocol for the assessment of pixel-based sO2 distributions provides more detailed information as compared to conventional region-of-interest-based analysis of PAI, especially with respect to the detection of temporal changes and tumor heterogeneity.