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We consider simplicial sets equipped with a notion of smallness, and observe that this slight "topological" extension of the "algebraic" simplicial language allows a concise reformulation of a number of classical notions in topology, e.g.…

The completely positive rank is an analogue of topological covering dimension, defined for nuclear C*-algebras via completely positive approximations. These may be thought of as simplicial approximations of the algebra, which leads to the…

We consider a natural way of extending the Lebesgue covering dimension to various classes of infinite dimensional separable metric spaces.

This paper focuses on the relation between computational learning theory and resource-bounded dimension. We intend to establish close connections between the learnability/nonlearnability of a concept class and its corresponding size in…

In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by…

We define a class of probability distributions that we call simplicial mixture models, inspired by simplicial complexes from algebraic topology. The parameters of these distributions represent their topology and we show that it is possible…

Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system - which involves a continuous self-map on a metric space. There are many notions of…

Conceptual Scaling is a useful standard tool in Formal Concept Analysis and beyond. Its mathematical theory, as elaborated in the last chapter of the FCA monograph, still has room for improvement. As it stands, even some of the basic…

Simplicial distributions provide a framework for studying quantum contextuality, a generalization of Bell's non-locality. Understanding extremal simplicial distributions is of fundamental importance with applications to quantum computing.…

Classical unsupervised learning methods like clustering and linear dimensionality reduction parametrize large-scale geometry when it is discrete or linear, while more modern methods from manifold learning find low dimensional representation…

The dimension of random simplicial complexes (defined as the maximal dimension among all faces) is a natural extreme value associated with the complex, and is closely related to other functionals defined by a maximum, such as the clique…

In this paper we present a mathematical framework on linking of embeddings of compact topological spaces into Euclidean spaces and separability of linked embeddings under a specific class of dimension reduction maps. As applications of the…

Teaching dimension is a learning theoretic quantity that specifies the minimum training set size to teach a target model to a learner. Previous studies on teaching dimension focused on version-space learners which maintain all hypotheses…

We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimensio. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that…

ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated…

The notion of a simplicial set originated in algebraic topology, and has also been utilized extensively in category theory, but until relatively recently was not used outside of those fields. However, with the increasing prominence of…

We generalize simplicial minisuperspace models associated with restricting the topology of the universe to be that of a cone over a closed connected combinatorial $3-$manifold by considering the presence of a massive scalar field. By…

Supervised dimensionality reduction has emerged as an important theme in the last decade. Despite the plethora of models and formulations, there is a lack of a simple model which aims to project the set of patterns into a space defined by…

We develop the theory of probabilistic variants of the one-category and diagonal topological complexity, which bound the classical LS-category and topological complexity from below. Unlike any other classical or probabilistic invariants,…

We consider the long-standing question of finding a parameter of a class of probability distributions that characterizes its PAC learnability. We provide a rather surprising answer - no such parameter exists. Our techniques allow us to show…