相关论文: A local depth measure for general data
Topological measures and deficient topological measures generalize Borel measures and correspond to certain non-linear functionals. We study integration with respect to deficient topological measures on locally compact spaces. Such an…
The human visual system excels at detecting local blur of visual images, but the underlying mechanism is not well understood. Traditional views of blur such as reduction in energy at high frequencies and loss of phase coherence at localized…
We study the dimensional properties of Moran sets and Moran measures in doubling metric spaces. In particular, we consider local dimensions and $L^q$-dimensions. We generalize and extend several existing results in this area.
Depth perception is considered an invaluable source of information in the context of 3D mapping and various robotics applications. However, point cloud maps acquired using consumer-level light detection and ranging sensors (lidars) still…
Analysis and processing of data is a vital part of our modern society and requires vast amounts of computational resources. To reduce the computational burden, compressing and approximating data has become a central topic. We consider the…
Recent work has found that neural networks with stronger generalization tend to exhibit higher representational alignment with one another across architectures and training paradigms. In this work, we show that models with stronger…
Depth estimation, as a necessary clue to convert 2D images into the 3D space, has been applied in many machine vision areas. However, to achieve an entire surrounding 360-degree geometric sensing, traditional stereo matching algorithms for…
This paper presents a study of generalized polyhedral convexity under basic operations on multifunctions. We address the preservation of generalized polyhedral convexity under sums and compositions of multifunctions, the domains and ranges…
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…
Multidimensional scaling (MDS) is a family of methods that embed a given set of points into a simple, usually flat, domain. The points are assumed to be sampled from some metric space, and the mapping attempts to preserve the distances…
We show that if X is a Banach space without cotype, then every locally finite metric space embeds metrically into X.
Let $\mu$ be a Borel probability measure generated by a hyperbolic recurrent iterated function system defined on a nonempty compact subset of $\mathbb R^k$. We study the Hausdorff and the packing dimensions, and the quantization dimensions…
Functional depth is used for ranking functional observations from most outlying to most typical. The ranks produced by functional depth have been proposed as the basis for functional classifiers, rank tests, and data visualization…
Assume that $\mathfrak A$ is a real Banach space of finite dimension $n\geq2$. Consider any Borel probability measure $\nu$ supported on the unit ball $K$ of $\mathfrak A$. We show that \[\Delta(\nu)=\int_{x \in K}\int_{ y\in…
Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curvature concept that we use here captures the metric relations…
We introduce the notion of (almost isometric) local retracts in metric space as a natural non-linear version of the concepts of locally complemented and almost isometric ideals from Banach spaces. We prove that given two metric spaces…
A new depth-based clustering procedure for directional data is proposed. Such method is fully non-parametric and has the advantages to be flexible and applicable even in high dimensions when a suitable notion of depth is adopted. The…
In this paper, we consider the directional differentiability of metric projection and its properties in uniformly convex and uniformly smooth Bochner space Lp(S; X), in which (S, A, mu) is a positive measure space and X is a uniformly…
We study generalization in deep learning by appealing to complexity measures originally developed in approximation and information theory. While these concepts are challenged by the high-dimensional and data-defined nature of deep learning,…
Predicting depth is an essential component in understanding the 3D geometry of a scene. While for stereo images local correspondence suffices for estimation, finding depth relations from a single image is less straightforward, requiring…