Related papers: On data depth in infinite dimensional spaces

Dimensionality reduction is a common method for analyzing and visualizing high-dimensional data across domains. Dimensionality-reduction algorithms involve complex optimizations and the reduced dimensions computed by these algorithms…

This paper answers the question of which functional depth to use to construct a boxplot for functional data. It shows that integrated depths, e.g., the popular modified band depth, do not result in well-defined boxplots. Instead, we argue…

Topological data analysis (TDA) provides insight into data shape. The summaries obtained by these methods are principled global descriptions of multi-dimensional data whilst exhibiting stable properties such as robustness to deformation and…

The advent of modern technology, permitting the measurement of thousands of characteristics simultaneously, has given rise to floods of data characterized by many large or even huge datasets. This new paradigm presents extraordinary…

The intrinsic dimensionality refers to the ``true'' dimensionality of the data, as opposed to the dimensionality of the data representation. For example, when attributes are highly correlated, the intrinsic dimensionality can be much lower…

In many problems in machine learning and operations research, we need to optimize a function whose input is a random variable or a probability density function, i.e. to solve optimization problems in an infinite dimensional space. On the…

In this work, we propose a convenient framework for infinite-dimensional analysis (including both real and complex analysis in infinite dimensions), in which differentiation (in some weak sense) and integration operations can be easily…

Entropy is useful in statistical problems as a measure of irreversibility, randomness, mixing, dispersion, and number of microstates. However, there remains ambiguity over the precise mathematical formulation of entropy, generalized beyond…

With the advancement of data science, the collection of increasingly complex datasets has become commonplace. In such datasets, the data dimension can be extremely high, and the underlying data generation process can be unknown and highly…

Real world-datasets characterized by discrete features are ubiquitous: from categorical surveys to clinical questionnaires, from unweighted networks to DNA sequences. Nevertheless, the most common unsupervised dimensional reduction methods…

In spite of its fundamental importance, inference has not been an inherent function of multidimensional models and analytical applications. These models are mainly aimed at numeric (quantitative) analysis where the notions of inference and…

Derivative-free algorithms seek the minimum of a given function based only on function values queried at appropriate points. Although these methods are widely used in practice, their performance is known to worsen as the problem dimension…

Statistical depths provide a fundamental generalization of quantiles and medians to data in higher dimensions. This paper proposes a new type of globally defined statistical depth, based upon control theory and eikonal equations, which…

Large-scale, high-quality data are considered an essential factor for the successful application of many deep learning techniques. Meanwhile, numerous real-world deep learning tasks still have to contend with the lack of sufficient amounts…

Entropy is a measure of heterogeneity widely used in applied sciences, often when data are collected over space. Recently, a number of approaches has been proposed to include spatial information in entropy. The aim of entropy is to…

Classical multivariate statistics measures the outlyingness of a point by its Mahalanobis distance from the mean, which is based on the mean and the covariance matrix of the data. A multivariate depth function is a function which, given a…

Due to the abundance of 2D product images from the Internet, developing efficient and scalable algorithms to recover the missing depth information is central to many applications. Recent works have addressed the single-view depth estimation…

Visualizing high dimensional data by projecting them into two or three dimensional space is one of the most effective ways to intuitively understand the data's underlying characteristics, for example their class neighborhood structure.…

A number of fundamental quantities in statistical signal processing and information theory can be expressed as integral functions of two probability density functions. Such quantities are called density functionals as they map density…

This work seeks to tackle the inherent complexity of dataspaces by introducing a novel data structure that can represent datasets across multiple levels of abstraction, ranging from local to global. We propose the concept of a multilevel…