77 research outputs found
Fast Fourier Transforms for the Rook Monoid
We define the notion of the Fourier transform for the rook monoid (also
called the symmetric inverse semigroup) and provide two efficient
divide-and-conquer algorithms (fast Fourier transforms, or FFTs) for computing
it. This paper marks the first extension of group FFTs to non-group semigroups
Separation of Variables and the Computation of Fourier Transforms on Finite Groups, II
We present a general diagrammatic approach to the construction of efficient
algorithms for computing the Fourier transform of a function on a finite group.
By extending work which connects Bratteli diagrams to the construction of Fast
Fourier Transform algorithms %\cite{sovi}, we make explicit use of the path
algebra connection to the construction of Gel'fand-Tsetlin bases and work in
the setting of quivers. We relate this framework to the construction of a {\em
configuration space} derived from a Bratteli diagram. In this setting the
complexity of an algorithm for computing a Fourier transform reduces to the
calculation of the dimension of the associated configuration space. Our methods
give improved upper bounds for computing the Fourier transform for the general
linear groups over finite fields, the classical Weyl groups, and homogeneous
spaces of finite groups, while also recovering the best known algorithms for
the symmetric group and compact Lie groups.Comment: 53 pages, 5 appendices, 34 figure
Orienteering in Knowledge Spaces: The Hyperbolic Geometry of Wikipedia Mathematics
In this paper we show how the coupling of the notion of a network with directions with the adaptation of the four-point probe from materials testing gives rise to a natural geometry on such networks. This four-point probe geometry shares many of the properties of hyperbolic geometry wherein the network directions take the place of the sphere at infinity, enabling a navigation of the network in terms of pairs of directions: the geodesic through a pair of points is oriented from one direction to another direction, the pair of which are uniquely determined. We illustrate this in the interesting example of the pages of Wikipedia devoted to Mathematics, or “The MathWiki.” The applicability of these ideas extends beyond Wikipedia to provide a natural framework for visual search and to prescribe a natural mode of navigation for any kind of “knowledge space” in which higher order concepts aggregate various instances of information. Other examples would include genre or author organization of cultural objects such as books, movies, documents or even merchandise in an online store
A unifying representation for a class of dependent random measures
We present a general construction for dependent random measures based on
thinning Poisson processes on an augmented space. The framework is not
restricted to dependent versions of a specific nonparametric model, but can be
applied to all models that can be represented using completely random measures.
Several existing dependent random measures can be seen as specific cases of
this framework. Interesting properties of the resulting measures are derived
and the efficacy of the framework is demonstrated by constructing a
covariate-dependent latent feature model and topic model that obtain superior
predictive performance
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