Related papers: On data depth in infinite dimensional spaces

The simplicial depth, like other relevant multivariate statistical data depth functions, vanishes right outside the convex hull of the support of the distribution with respect to which the depth is computed. This is problematic when it is…

The proliferation of deep learning techniques led to a wide range of advanced analytics applications in important business areas such as predictive maintenance or product recommendation. However, as the effectiveness of advanced analytics…

Density-functional theory is a formally exact description of a many-body quantum system in terms of its density; in practice, however, approximations to the universal density functional are required. In this work, a model based on deep…

Deep sequence models are receiving significant interest in current machine learning research. By representing probability distributions that are fit to data using maximum likelihood estimation, such models can model data on general…

The ability to collect and analyze large amounts of data is a growing problem within the scientific community. The growing gap between data and users calls for innovative tools that address the challenges faced by big data volume, velocity…

Important information concerning a multivariate data set, such as clusters and modal regions, is contained in the derivatives of the probability density function. Despite this importance, nonparametric estimation of higher order derivatives…

In this short note we present several infinite dimensional theorems which generalize corresponding facts from the finite dimensional differential inclusions theory.

The utilization of statistical methods an their applications within the new field of study known as Topological Data Analysis has has tremendous potential for broadening our exploration and understanding of complex, high-dimensional data…

This article introduces a general statistical modeling principle called "Density Sharpening" and applies it to the analysis of discrete count data. The underlying foundation is based on a new theory of nonparametric approximation and…

We develop a novel exploratory tool for non-Euclidean object data based on data depth, extending the celebrated Tukey's depth for Euclidean data. The proposed metric halfspace depth, applicable to data objects in a general metric space,…

We extend the classical notion of the spherical depth in \mathbb{R}^k, to the important setup of data on a Riemannian manifold. We show that this notion of depth satisfies a set of desirable properties. For the empirical version of this…

Objective functions that optimize deep neural networks play a vital role in creating an enhanced feature representation of the input data. Although cross-entropy-based loss formulations have been extensively used in a variety of supervised…

Centrality descriptors are widely used to rank nodes according to specific concept(s) of importance. Despite the large number of centrality measures available nowadays, it is still poorly understood how to identify the node which can be…

The evolution of human intelligence led to the huge amount of data in the information space. Accessing and processing this data helps in finding solutions to applied problems based on finite-dimensional models. We argue, that formally, such…

The concept of illumination bodies studied in convex geometry is used to amend the halfspace depth for multivariate data. The proposed notion of illumination enables finer resolution of the sample points, naturally breaks ties in the…

We study the numerical integration problem for functions with infinitely many variables. The function spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms related to the ANOVA decomposition of the…

In inverse problems, one attempts to infer spatially variable functions from indirect measurements of a system. To practitioners of inverse problems, the concept of "information" is familiar when discussing key questions such as which parts…

Aggregation of large databases in a specific format is a frequently used process to make the data easily manageable. Interval-valued data is one of the data types that is generated by such an aggregation process. Using traditional methods…

We propose a new approach, called as functional deep neural network (FDNN), for classifying multi-dimensional functional data. Specifically, a deep neural network is trained based on the principle components of the training data which shall…

This paper is the second in the series Commutative Scaling of Width and Depth (WD) about commutativity of infinite width and depth limits in deep neural networks. Our aim is to understand the behaviour of neural functions (functions that…