Related papers: Error Probabilities for Halfspace Depth
The manifold of empirical mean values of statistical data ad infinitum has a geometric shape that depends on the probability measure that governs the generating model. Large deviation theory produces entropy functions that depend on both…
Directional data are constrained to lie on the unit sphere of~$\mathbb{R}^q$ for some~$q\geq 2$. To address the lack of a natural ordering for such data, depth functions have been defined on spheres. However, the depths available either…
Understanding geometric properties of natural language processing models' latent spaces allows the manipulation of these properties for improved performance on downstream tasks. One such property is the amount of data spread in a model's…
Given a probability measure $\mu $ on ${\mathbb R}^n$, Tukey's half-space depth is defined for any $x\in {\mathbb R}^n$ by $\varphi_{\mu }(x)=\inf\{\mu (H):H\in {\cal H}(x)\}$, where ${\cal H}(x)$ is the set of all half-spaces $H$ of…
Neural networks appear to have mysterious generalization properties when using parameter counting as a proxy for complexity. Indeed, neural networks often have many more parameters than there are data points, yet still provide good…
Functional data analysis has been a growing field of study in recent decades, and one fundamental task in functional data analysis is estimating the sample location. A notion called statistical depth has been extended from multivariate data…
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…
This article studies a general divide-and-conquer algorithm for approximating continuous one-dimensional probability distributions with finite mean. The article presents a numerical study that compares pre-existing approximation schemes…
Tukey's depth offers a powerful tool for nonparametric inference and estimation, but also encounters serious computational and methodological difficulties in modern statistical data analysis. This paper studies how to generalize and compute…
A theoretical framework is developed to describe the transformation that distributes probability density functions uniformly over space. In one dimension, the cumulative distribution can be used, but does not generalize to higher…
We present a new fast approximate algorithm for Tukey (halfspace) depth level sets and its implementation-ABCDepth. Given a $d$-dimensional data set for any $d\geq 1$, the algorithm is based on a representation of level sets as…
The real-life data have a complex and non-linear structure due to their nature. These non-linearities and the large number of features can usually cause problems such as the empty-space phenomenon and the well-known curse of dimensionality.…
A common belief in high-dimensional data analysis is that data are concentrated on a low-dimensional manifold. This motivates simultaneous dimension reduction and regression on manifolds. We provide an algorithm for learning gradients on…
A method providing optimal estimate of probability density functions (PDFs) from time series is proposed. It allows almost arbitrary resolution PDFs when applied to either, sampled analytic functions or digitized data from experiments. When…
We study the machine learning task for models with operators mapping between the Wasserstein space of probability measures and a space of functions, like e.g. in mean-field games/control problems. Two classes of neural networks, based on…
The main focus of this work is on providing a formal definition of statistical depth for functional data on the basis of six properties, recognising topological features such as continuity, smoothness and contiguity. Amongst our depth…
Many algorithms in machine learning and computational geometry require, as input, the intrinsic dimension of the manifold that supports the probability distribution of the data. This parameter is rarely known and therefore has to be…
Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…
Data cohesion, a recently introduced measure inspired by social interactions, uses distance comparisons to assess relative proximity. In this work, we provide a collection of results which can guide the development of cohesion-based methods…
Dimension reduction is a technique used to transform data from a high-dimensional space into a lower-dimensional space, aiming to retain as much of the original information as possible. This approach is crucial in many disciplines like…