23,705 research outputs found

    What are natural concepts? A design perspective

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    Conceptual spaces have become an increasingly popular modeling tool in cognitive psychology. The core idea of the conceptual spaces approach is that concepts can be represented as regions in similarity spaces. While it is generally acknowledged that not every region in such a space represents a natural concept, it is still an open question what distinguishes those regions that represent natural concepts from those that do not. The central claim of this paper is that natural concepts are represented by the cells of an optimally designed similarity space

    Topological transversals to a family of convex sets

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    Let F\mathcal F be a family of compact convex sets in Rd\mathbb R^d. We say that F\mathcal F has a \emph{topological ρ\rho-transversal of index (m,k)(m,k)} (ρ<m\rho<m, 0<k≤d−m0<k\leq d-m) if there are, homologically, as many transversal mm-planes to F\mathcal F as mm-planes containing a fixed ρ\rho-plane in Rm+k\mathbb R^{m+k}. Clearly, if F\mathcal F has a ρ\rho-transversal plane, then F\mathcal F has a topological ρ\rho-transversal of index (m,k),(m,k), for ρ<m\rho<m and k≤d−mk\leq d-m. The converse is not true in general. We prove that for a family F\mathcal F of ρ+k+1\rho+k+1 compact convex sets in Rd\mathbb R^d a topological ρ\rho-transversal of index (m,k)(m,k) implies an ordinary ρ\rho-transversal. We use this result, together with the multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann category of the Grassmannian, and different versions of the colorful Helly theorem by B\'ar\'any and Lov\'asz, to obtain some geometric consequences

    Learning Multiple Visual Tasks while Discovering their Structure

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    Multi-task learning is a natural approach for computer vision applications that require the simultaneous solution of several distinct but related problems, e.g. object detection, classification, tracking of multiple agents, or denoising, to name a few. The key idea is that exploring task relatedness (structure) can lead to improved performances. In this paper, we propose and study a novel sparse, non-parametric approach exploiting the theory of Reproducing Kernel Hilbert Spaces for vector-valued functions. We develop a suitable regularization framework which can be formulated as a convex optimization problem, and is provably solvable using an alternating minimization approach. Empirical tests show that the proposed method compares favorably to state of the art techniques and further allows to recover interpretable structures, a problem of interest in its own right.Comment: 19 pages, 3 figures, 3 table

    Equivariant dendroidal sets and simplicial operads

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    We establish a Quillen equivalence between the homotopy theories of equivariant Segal operads and equivariant simplicial operads with norm maps. Together with previous work, we further conclude that the homotopy coherent nerve is a right-Quillen equivalence from the model category of equivariant simplicial operads with norm maps to the model category structure for equivariant-∞\infty-operads in equivariant dendroidal sets.Comment: v3: Improvements to exposition and minor edits, in response to referee suggestion

    The rationality of vagueness

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    Prototypes, Poles, and Topological Tessellations of Conceptual Spaces

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    Abstract. The aim of this paper is to present a topological method for constructing discretizations (tessellations) of conceptual spaces. The method works for a class of topological spaces that the Russian mathematician Pavel Alexandroff defined more than 80 years ago. Alexandroff spaces, as they are called today, have many interesting properties that distinguish them from other topological spaces. In particular, they exhibit a 1-1 correspondence between their specialization orders and their topological structures. Recently, a special type of Alexandroff spaces was used by Ian Rumfitt to elucidate the logic of vague concepts in a new way. According to his approach, conceptual spaces such as the color spectrum give rise to classical systems of concepts that have the structure of atomic Boolean algebras. More precisely, concepts are represented as regular open regions of an underlying conceptual space endowed with a topological structure. Something is subsumed under a concept iff it is represented by an element of the conceptual space that is maximally close to the prototypical element p that defines that concept. This topological representation of concepts comes along with a representation of the familiar logical connectives of Aristotelian syllogistics in terms of natural settheoretical operations that characterize regular open interpretations of classical Boolean propositional logic. In the last 20 years, conceptual spaces have become a popular tool of dealing with a variety of problems in the fields of cognitive psychology, artificial intelligence, linguistics and philosophy, mainly due to the work of Peter Gärdenfors and his collaborators. By using prototypes and metrics of similarity spaces, one obtains geometrical discretizations of conceptual spaces by so-called Voronoi tessellations. These tessellations are extensionally equivalent to topological tessellations that can be constructed for Alexandroff spaces. Thereby, Rumfitt’s and Gärdenfors’s constructions turn out to be special cases of an approach that works for a more general class of spaces, namely, for weakly scattered Alexandroff spaces. This class of spaces provides a convenient framework for conceptual spaces as used in epistemology and related disciplines in general. Alexandroff spaces are useful for elucidating problems related to the logic of vague concepts, in particular they offer a solution of the Sorites paradox (Rumfitt). Further, they provide a semantics for the logic of clearness (Bobzien) that overcomes certain problems of the concept of higher2 order vagueness. Moreover, these spaces help find a natural place for classical syllogistics in the framework of conceptual spaces. The crucial role of order theory for Alexandroff spaces can be used to refine the all-or-nothing distinction between prototypical and nonprototypical stimuli in favor of a more fine-grained gradual distinction between more-orless prototypical elements of conceptual spaces. The greater conceptual flexibility of the topological approach helps avoid some inherent inadequacies of the geometrical approach, for instance, the so-called “thickness problem” (Douven et al.) and problems of selecting a unique metric for similarity spaces. Finally, it is shown that only the Alexandroff account can deal with an issue that is gaining more and more importance for the theory of conceptual spaces, namely, the role that digital conceptual spaces play in the area of artificial intelligence, computer science and related disciplines. Keywords: Conceptual Spaces, Polar Spaces, Alexandroff Spaces, Prototypes, Topological Tessellations, Voronoi Tessellations, Digital Topology
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