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The quest for simplification in physics drives the exploration of concise mathematical representations for complex systems. This Dissertation focuses on the concept of dimensionality reduction as a means to obtain low-dimensional…
Entropy estimation is of practical importance in information theory and statistical science. Many existing entropy estimators suffer from fast growing estimation bias with respect to dimensionality, rendering them unsuitable for…
We study the problem of high-dimensional robust mean estimation in the presence of a constant fraction of adversarial outliers. A recent line of work has provided sophisticated polynomial-time algorithms for this problem with…
Machine learning tools have empowered a qualitatively new way to perform differential cross section measurements whereby the data are unbinned, possibly in many dimensions. Unbinned measurements can enable, improve, or at least simplify…
Robust estimators, like the median of a point set, are important for data analysis in the presence of outliers. We study robust estimators for locationally uncertain points with discrete distributions. That is, each point in a data set has…
This paper presents an extension and an elaboration of the theory of differential similarity, which was originally proposed in arXiv:1401.2411 [cs.LG]. The goal is to develop an algorithm for clustering and coding that combines a geometric…
As the most fundamental problem in statistics, robust location estimation has many prominent solutions, such as the trimmed mean, Winsorized mean, Hodges Lehmann estimator, Huber M estimator, and median of means. Recent studies suggest that…
It is widely believed that natural image data exhibits low-dimensional structure despite the high dimensionality of conventional pixel representations. This idea underlies a common intuition for the remarkable success of deep learning in…
Data depth is a powerful nonparametric tool originally proposed to rank multivariate data from center outward. In this context, one of the most archetypical depth notions is Tukey's halfspace depth. In the last few decades notions of depth…
We propose a novel probabilistic dimensionality reduction framework that can naturally integrate the generative model and the locality information of data. Based on this framework, we present a new model, which is able to learn a smooth…
Correcting for detector effects in experimental data, particularly through unfolding, is critical for enabling precision measurements in high-energy physics. However, traditional unfolding methods face challenges in scalability,…
Outlier recognition is a fundamental problem in data analysis and has attracted a great deal of attention in the past decades. However, most existing methods still suffer from several issues such as high time and space complexities or…
The Intrinsic Dimension (ID) is a key concept in unsupervised learning and feature selection, as it is a lower bound to the number of variables which are necessary to describe a system. However, in almost any real-world dataset the ID…
We propose a robust inferential procedure for assessing uncertainties of parameter estimation in high-dimensional linear models, where the dimension $p$ can grow exponentially fast with the sample size $n$. Our method combines the…
Multiple importance sampling estimators are widely used for computing intractable constants due to its reliability and robustness. The celebrated balance heuristic estimator belongs to this class of methods and has proved very successful in…
The aim of this paper is to present a new estimation procedure that can be applied in many statistical frameworks including density and regression and which leads to both robust and optimal (or nearly optimal) estimators. In density…
In novelty detection, the goal is to decide if a new data point should be categorized as an inlier or an outlier, given a training dataset that primarily captures the inlier distribution. Recent approaches typically use deep encoder and…
The problem of estimating the number of sources and their angles of arrival from a single antenna array observation has been an active area of research in the signal processing community for the last few decades. When the number of sources…
We present differentially private algorithms for high-dimensional mean estimation. Previous private estimators on distributions over $\mathbb{R}^d$ suffer from a curse of dimensionality, as they require $\Omega(d^{1/2})$ samples to achieve…
The concepts of spread and spread dimension of a metric space were introduced by Willerton in the context of quantifying biodiversity of ecosystems. This paper develops practical applications of spread dimension in the context of machine…