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Metric data structures (distance oracles, distance labeling schemes, routing schemes) and low-distortion embeddings provide a powerful algorithmic methodology, which has been successfully applied for approximation algorithms \cite{llr},…

In Microscopic Particle Image Velocimetry ($\mu$PIV), velocity fields in microchannels are sampled over finite volumes within which the velocity fields themselves may vary significantly. In the past, this has limited measurements often to…

Measurement incompatibility--the impossibility of jointly measuring certain quantum observables--is a fundamental resource for quantum information processing. We develop a graph-theoretic framework for quantifying this resource for large…

Theoretical uncertainties in the predictions of relativistic mean-field models are estimated using a chi-square minimization procedure that is implemented by studying the small oscillations around the chi-square minimum. By diagonalizing…

We study the multivariate nonparametric change point detection problem, where the data are a sequence of independent $p$-dimensional random vectors whose distributions are piecewise-constant with Lipschitz densities changing at unknown…

Spaces with locally varying scale of measurement, like multidimensional structures with differently scaled dimensions, are pretty common in statistics and machine learning. Nevertheless, it is still understood as an open question how to…

An analysis of high-dimensional data can offer a detailed description of a system but is often challenged by the curse of dimensionality. General dimensionality reduction techniques can alleviate such difficulty by extracting a few…

Linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data, due to their simple geometric interpretations and typically attractive computational properties. These methods capture many data features of…

We characterize the asymptotic performance of a class of positive operator valued measurements (POVMs) where the only task is to make measurements on independent and identically distributed quantum states on finite-dimensional systems. The…

Partial measurements of relative position are a relatively common event during the observation of visual binary stars. However, these observations are typically discarded when estimating the orbit of a visual pair. In this article we…

A metric measure space is said to be Carnot-rectifiable if it can be covered up to a null set by countably many biLipschitz images of compact sets of a fixed Carnot group. In this paper, we give several characterisations of such notion of…

With the continued digitization of societal processes, we are seeing an explosion in available data. This is referred to as big data. In a research setting, three aspects of the data are often viewed as the main sources of challenges when…

We propose statistical procedures for detecting changes in the mean of spatial random fields observed on regular grids. The proposed framework provides a general approach to change detection in spatial processes. Extending a block-based…

We initiate a program of average smoothness analysis for efficiently learning real-valued functions on metric spaces. Rather than using the Lipschitz constant as the regularizer, we define a local slope at each point and gauge the function…

We introduce principal curves in Wasserstein space, and in general compact metric spaces. Our motivation for the Wasserstein case comes from optimal-transport-based trajectory inference, where a developing population of cells traces out a…

Deep metric learning (DML) is a cornerstone of many computer vision applications. It aims at learning a mapping from the input domain to an embedding space, where semantically similar objects are located nearby and dissimilar objects far…

Suppose $L$ simultaneous independent stochastic systems generate observations, where the observations from each system depend on the underlying parameter of that system. The observations are unlabeled (anonymized), in the sense that an…

We give the following characterization of rectifiable metric spaces. A metric space with positive lower Hausdorff density is rectifiable if and only if, for any subset $F$ and $f:F\to Y$, a Lipschitz map into a metric space with positive…

Let $(X,d)$ be a compact metric space. We consider the behavior of probability measures $\mu$ with the property that $$ \int_{X} d(x, y) d\mu(y) \qquad \mbox{is independent of}~x \in X.$$ It appears that such measures, when they exist,…

This paper deals with the problem of estimating the covariance matrix of a series of independent multivariate observations, in the case where the dimension of each observation is of the same order as the number of observations. Although…