相关论文: Short-Distance Cutoffs in Curved Space

The often-asked question whether space-time is discrete or continuous may not be the right question to ask: Mathematically, it is possible that space-time possesses the differentiability properties of manifolds as well as the ultraviolet…

In information theory, the link between continuous information and discrete information is established through well-known sampling theorems. Sampling theory explains, for example, how frequency-filtered music signals are reconstructible…

There are competing schools of thought about the question of whether spacetime is fundamentally either continuous or discrete. Here, we consider the possibility that spacetime could be simultaneously continuous and discrete, in the same…

It has been shown that space-time coordinates can exhibit only very few types of short-distance structures, if described by linear operators: they can be continuous, discrete or "unsharp" in one of two ways. In the literature, various…

Discrete sampling theorem is formulated that refers to discrete signals specified by a finite number of their samples and band-limited in a domain of a certain orthogonal transform. Conditions of the recoverability of such signals from…

We assume that the points in volumes smaller than an elementary volume (which may have a Planck size) are indistinguishable in any physical experiment. This naturally leads to a picture of a discrete space with a finite number of degrees of…

Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a…

Deep Learning performs well when training data densely covers the experience space. For complex problems this makes data collection prohibitively expensive. We propose to intelligently select samples when constructing data sets in order to…

This survey addresses sampling discretization and its connections with other areas of mathematics. The survey concentrates on sampling discretization of norms of elements of finite-dimensional subspaces. We present here known results on…

In the era of rapidly increasing amounts of time series data, classification of variable objects has become the main objective of time-domain astronomy. Classification of irregularly sampled time series is particularly difficult because the…

Distance queries are a basic tool in data analysis. They are used for detection and localization of change for the purpose of anomaly detection, monitoring, or planning. Distance queries are particularly useful when data sets such as…

This paper provides an elementary, self-contained analysis of diffusion-based sampling methods for generative modeling. In contrast to existing approaches that rely on continuous-time processes and then discretize, our treatment works…

The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base…

We introduce and analyze a model for the transport of particles or energy in extended lattice systems. The dynamics of the model acts on a discrete phase space at discrete times but has nonetheless some of the characteristic properties of…

The set of space-time short-distance structures which can be described through linear operators is limited to a few basic cases. These are continua, lattices and a further short-distance structure which implies an ultraviolet cut-off. Under…

Distance matrices are matrices whose elements are the relative distances between points located on a certain manifold. In all cases considered here all their eigenvalues except one are non-positive. When the points are uncorrelated and…

A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…

Sampling theory has benefited from a surge of research in recent years, due in part to the intense research in wavelet theory and the connections made between the two fields. In this survey we present several extensions of the Shannon…

Starting from the hypothesis that both physics, in particular space-time and the physical vacuum, and the corresponding mathematics are discrete on the Planck scale we develop a certain framework in form of a '{\it cellular network}'…

Whether or not space-time is fundamentally discrete is of central importance for the development of the theory of quantum gravity. If the fundamental description of space-time is discrete, typically represented in terms of a graph or…