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We establish a general theory for projective dimensions of the logarithmic derivation modules of hyperplane arrangements. That includes the addition-deletion and restriction theorem, Yoshinaga-type result, and the division theorem for…
We adopt a measure-theoretic perspective on the Riemannian approximation scheme proving a sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds. We show that the zero-order term in the limit is a singular measure…
Dimension reduction is widely regarded as an effective way for decreasing the computation, storage and communication loads of data-driven intelligent systems, leading to a growing demand for statistical methods that allow analysis (e.g.,…
We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in $\mathbb{R}^2$. These equations can be seen as a sub-Riemannian counterpart…
Dropout Regularization, serving to reduce variance, is nearly ubiquitous in Deep Learning models. We explore the relationship between the dropout rate and model complexity by training 2,000 neural networks configured with random…
Climate projections using data driven machine learning models acting as emulators, is one of the prevailing areas of research to enable policy makers make informed decisions. Use of machine learning emulators as surrogates for…
To have a superior generalization, a deep learning neural network often involves a large size of training sample. With increase of hidden layers in order to increase learning ability, neural network has potential degradation in accuracy.…
The advent of modern data collection and processing techniques has seen the size, scale, and complexity of data grow exponentially. A seminal step in leveraging these rich datasets for downstream inference is understanding the…
A union of an arrangement of affine hyperplanes $H$ in $R^d$ is the real algebraic variety associated to the principal ideal generated by the polynomial $p_{H}$ given as the product of the degree one polynomials which define the hyperplanes…
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…
Statistical depth measures the centrality of a point with respect to a given distribution or data cloud. It provides a natural center-outward ordering of multivariate data points and yields a systematic nonparametric multivariate analysis…
Point clouds have been recognized as a crucial data structure for 3D content and are essential in a number of applications such as virtual and mixed reality, autonomous driving, cultural heritage, etc. In this paper, we propose a set of…
Existing work on understanding deep learning often employs measures that compress all data-dependent information into a few numbers. In this work, we adopt a perspective based on the role of individual examples. We introduce a measure of…
Motivated by two case studies using primary care records from the Clinical Practice Research Datalink, we describe statistical methods that facilitate the analysis of tall data, with very large numbers of observations. Our focus is on…
We provide new theoretical results in the field of inverse regression methods for dimension reduction. Our approach is based on the study of some empirical processes that lie close to a certain dimension reduction subspace, called the…
The correct estimation of the head pose is a problem of the great importance for many applications. For instance, it is an enabling technology in automotive for driver attention monitoring. In this paper, we tackle the pose estimation…
Depth estimation is a fundamental task in computer vision with diverse applications. Recent advancements in deep learning have led to powerful depth foundation models (DFMs), yet their evaluation remains challenging due to inconsistencies…
In this paper, we present a study of a kernel-based consensual aggregation on randomly projected high-dimensional features of predictions for regression. The aggregation scheme is composed of two steps: the high-dimensional features of…
Subspace clustering methods have been widely studied recently. When the inputs are 2-dimensional (2D) data, existing subspace clustering methods usually convert them into vectors, which severely damages inherent structures and relationships…
Geometric data analysis and learning has emerged as a distinct and rapidly developing research area, increasingly recognized for its effectiveness across diverse applications. At the heart of this field lies curvature, a powerful and…