Related papers: Intrinsic Dimensionality
Pointwise tangential dimensions are introduced for metric spaces. Under regularity conditions, the upper, resp. lower, tangential dimensions of X at x can be defined as the supremum, resp. infimum, of box dimensions of the tangent sets, a…
A central aim of theoretical physics is to account for the structure of matter at the most elementary level as underlying the Standard Model of particle physics, and ideally also as a basis for a substantial dark sector, as distributed in…
We define the concept of self-similarity of an object by considering endomorphisms of the object as `similarity' maps. A variety of interesting examples of self-similar objects in geometry, algebra and arithmetic are introduced.…
A novel definition of the stimulus-specific information is presented, which is particularly useful when the stimuli constitute a continuous and metric set, as for example, position in space. The approach allows one to build the spatial…
The manifold hypothesis suggests that high-dimensional data often lie on or near a low-dimensional manifold. Estimating the dimension of this manifold is essential for leveraging its structure, yet existing work on dimension estimation is…
Hyperdimensional (HD) computing is a set of neurally inspired methods for obtaining high-dimensional, low-precision, distributed representations of data. These representations can be combined with simple, neurally plausible algorithms to…
Over the last decade, the term spatial computing has grown to have two different, though not entirely unrelated, definitions. The first definition of spatial computing stems from industry, where it refers primarily to new kinds of…
This is a brief discussion of the following features of the Universal Extra Dimension (UED) model: (i) Formulation, (ii) Indirect bounds, (iii) Collider search and the Inverse Problem, (iv) Astrophysical bounds, and (v) UED with two extra…
We present a model to measure the similarity in appearance between different materials, which correlates with human similarity judgments. We first create a database of 9,000 rendered images depicting objects with varying materials, shape…
Intrinsic core generalises the finite-dimensional notion of the relative interior to arbitrary (real) vector spaces. Our main goal is to provide a self-contained overview of the key results pertaining to the intrinsic core and to elucidate…
The goal of these notes is to give a brief explanation of how electric-magnetic duality in four dimensions is related to the existence of an unusual conformal field theory in six dimensions.
Similarity search (nearest neighbor search) is a problem of pursuing the data items whose distances to a query item are the smallest from a large database. Various methods have been developed to address this problem, and recently a lot of…
Ordinal data analysis is an interesting direction in machine learning. It mainly deals with data for which only the relationships `$<$', `$=$', `$>$' between pairs of points are known. We do an attempt of formalizing structures behind…
Basically information means selection within a domain (value or definition set) of possibilities. For objectifiable, comparable and precise information the domain should be the same for all. Therefore the global (online) definition of the…
This note provides a short guide to dimensional analysis in Lorentzian and general relativity and in differential geometry. It tries to revive Dorgelo and Schouten's notion of 'intrinsic' or 'absolute' dimension of a tensorial quantity. The…
The local intrinsic dimension (LID) of data is a fundamental quantity in signal processing and learning theory, but quantifying the LID of high-dimensional, complex data has been a historically challenging task. Recent works have discovered…
We discuss the dimensional characterization of the solutions space of a formally integrable system of partial differential equations and provide certain formulas for calculations of these dimensional quantities.
In recent years, the concepts of ``diversity'' and ``inclusion'' have attracted considerable attention across a range of fields, encompassing both social and biological disciplines. To fully understand these concepts, it is critical to not…
Modeling data as being sampled from a union of independent subspaces has been widely applied to a number of real world applications. However, dimensionality reduction approaches that theoretically preserve this independence assumption have…
This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear subspaces generated by the best…