An algebraic approach to multiresolution analysis

Foote, Richard

doi:10.1090/s0002-9947-05-03656-1

Cited by 6 publications

(9 citation statements)

References 23 publications

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“…A dendrogram like that shown in Figure 6 is invariant as a representation or structuring of a data set relative to rotation (alternatively, here: permutation) of left and right child nodes. These rotation (or permutation) symmetries are defined by the wreath product group (see [20,21,18] for an introduction and applications in signal and image processing), and can be used with any m-ary tree, although we will treat the binary or 2-way case here. For the group actions, with respect to which we will seek invariance, we consider independent cyclic shifts of the subnodes of a given node (hence, at each level).…”

Section: Wreath Product Group Corresponding To a Hierarchical Clusteringmentioning

confidence: 99%

Hierarchical Clustering for Finding Symmetries and Other Patterns in Massive, High Dimensional Datasets

Murtagh

Contreras

2012

Intelligent Systems Reference Library

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Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. "Structure" can be understood as symmetry and a range of symmetries are expressed by hierarchy. Such symmetries directly point to invariants, that pinpoint intrinsic properties of the data and of the background empirical domain of interest. We review many aspects of hierarchy here, including ultrametric topology, generalized ultrametric, linkages with lattices and other discrete algebraic structures and with p-adic number representations. By focusing on symmetries in data we have a powerful means of structuring and analyzing massive, high dimensional data stores. We illustrate the powerfulness of hierarchical clustering in case studies in chemistry and finance, and we provide pointers to other published case studies.

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Section: Wreath Product Group Corresponding To a Hierarchical Clusteringmentioning

confidence: 99%

Hierarchical Clustering for Finding Symmetries and Other Patterns in Massive, High Dimensional Datasets

Murtagh

Contreras

2012

Intelligent Systems Reference Library

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“…A dendrogram like that shown in Figure 5 is invariant as a representation or structuring of a data set relative to rotation (alternatively, here: permutation) of left and right child nodes. These rotation (or permutation) symmetries are defined by the wreath product group (see [21,22,19] for an introduction and applications in signal and image processing), and can be used with any m-ary tree, although we will treat the binary or 2-way case here. For the group actions, with respect to which we will seek invariance, we consider independent cyclic shifts of the subnodes of a given node (hence, at each level).…”

Section: Wreath Product Group Corresponding To a Hierarchical Clusteringmentioning

confidence: 99%

Symmetry in data mining and analysis: A unifying view based on hierarchy

Murtagh

2009

Proc. Steklov Inst. Math.

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Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. The data sets themselves are explicitly linked as a form of representation to an observational or otherwise empirical domain of interest. "Structure" has long been understood as symmetry which can take many forms with respect to any transformation, including point, translational, rotational, and many others. Symmetries directly point to invariants, that pinpoint intrinsic properties of the data and of the background empirical domain of interest. As our data models change so too do our perspectives on analyzing data. The structures in data surveyed here are based on hierarchy, represented as p-adic numbers or an ultrametric topology.

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“…The panoply of wavelet bases and efficient wavelet transforms offers enticing new possibilities for finding optimal theoretical and computational compromises between these competing dynamics, particular in multi-dimensional systems where the Ising model has only been effectively investigated by Monte Carlo methods. Additional theoretical insight might also be garnered by seeking to relate the action of the Renormalization (semi-)Group on certain (Sobolev) spaces of functions to affine group actions on these spaces, as alluded to in [5]. Such connections may further illuminate the relation between the RG and wavelet approaches.…”

Section: Renormalization Groups and Continuous Waveletsmentioning

confidence: 99%

Looking through wavelets to view the Ising problem

Mirchandani

Evans

Snapp

et al. 2009

2009 IEEE/SP 15th Workshop on Statistical Signal Processing

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In the theoretical study of phase transitions and critical phenomena, renormalization groups (RG) provide a course-grain representation of a many particle system and aid in the determination of the critical exponents associated with phase transitions. Monte Carlo and Molecular Dynamic simulations are often the simulation tools used for verification. However, the rich structure and multiscale properties of wavelets have found limited use in these applications. To promote exploration here, we apply and investigate wavelets to the solution of the simplest of molecular models, the Ising model and evaluate the most fundamental of objects in statistical mechanics, the partition function, now based on a wavelet-modifed Hamiltonian. As in [1] we iteratively compute the waveletbased coarse grain 2-D finite-size Ising model. Thermodynamic properties are calculated to predict, given the finite size effect, potential for phase transition. Exact calculations are confirmed using Monte Carlo simulations. Based on this investigation and corresponding insight gained, we propose a wavelet-based investigation in three open areas: Wavelet Bases, Critical Exponents and Phase Transitions, Renormalization Groups and Continuous Wavelets.

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An algebraic approach to multiresolution analysis

Cited by 6 publications

References 23 publications

Hierarchical Clustering for Finding Symmetries and Other Patterns in Massive, High Dimensional Datasets

Hierarchical Clustering for Finding Symmetries and Other Patterns in Massive, High Dimensional Datasets

Symmetry in data mining and analysis: A unifying view based on hierarchy

Looking through wavelets to view the Ising problem

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