429 research outputs found
Coxeter Groups and Wavelet Sets
A traditional wavelet is a special case of a vector in a separable Hilbert
space that generates a basis under the action of a system of unitary operators
defined in terms of translation and dilation operations. A
Coxeter/fractal-surface wavelet is obtained by defining fractal surfaces on
foldable figures, which tesselate the embedding space by reflections in their
bounding hyperplanes instead of by translations along a lattice. Although both
theories look different at their onset, there exist connections and
communalities which are exhibited in this semi-expository paper. In particular,
there is a natural notion of a dilation-reflection wavelet set. We prove that
dilation-reflection wavelet sets exist for arbitrary expansive matrix
dilations, paralleling the traditional dilation-translation wavelet theory.
There are certain measurable sets which can serve simultaneously as
dilation-translation wavelet sets and dilation-reflection wavelet sets,
although the orthonormal structures generated in the two theories are
considerably different
Reflection positive stochastic processes indexed by lie groups
Reflection positivity originates from one of the Osterwalder-Schrader axioms for constructive quantum field theory. It serves as a bridge between euclidean and relativistic quantum field theory. In mathematics, more specifically, in representation theory, it is related to the Cartan duality of symmetric Lie groups (Lie groups with an involution) and results in a transformation of a unitary representation of a symmetric Lie group to a unitary representation of its Cartan dual. In this article we continue our investigation of representation theoretic aspects of reflection positivity by discussing reflection positive Markov processes indexed by Lie groups, measures on path spaces, and invariant gaussian measures in spaces of distribution vectors. This provides new constructions of reflection positive unitary representations
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page
Reflection Positivity---A Representation Theoretic Perspective
Refection Positivity is a central theme at the crossroads of Lie group
representations, euclidean and abstract harmonic analysis, constructive quantum
field theory, and stochastic processes. This book provides the first
presentation of the representation theoretic aspects of Refection Positivity
and discusses its connections to those different fields on a level suitable for
doctoral students and researchers in related fields.Comment: This is a preliminary version of a book on Reflection Positivit
Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket
We provide a definition of integral, along paths in the Sierpinski gasket K,
for differential smooth 1-forms associated to the standard Dirichlet form K. We
show how this tool can be used to study the potential theory on K. In
particular, we prove: i) a de Rham reconstruction of a 1-form from its periods
around lacunas in K; ii) a Hodge decomposition of 1-forms with respect to the
Hilbertian energy norm; iii) the existence of potentials of smooth 1-forms on a
suitable covering space of K. We finally show that this framework provides
versions of the de Rham duality theorem for the fractal K.Comment: Some proofs have been clarified, reference to previous literature is
now more accurate, 33 pages, 6 figure
The fundamental pro-groupoid of an affine 2-scheme
A natural question in the theory of Tannakian categories is: What if you
don't remember \Forget? Working over an arbitrary commutative ring , we
prove that an answer to this question is given by the functor represented by
the \'etale fundamental groupoid \pi_1(\spec(R)), i.e.\ the separable
absolute Galois group of when it is a field. This gives a new definition
for \'etale \pi_1(\spec(R)) in terms of the category of -modules rather
than the category of \'etale covers. More generally, we introduce a new notion
of "commutative 2-ring" that includes both Grothendieck topoi and symmetric
monoidal categories of modules, and define a notion of for the
corresponding "affine 2-schemes." These results help to simplify and clarify
some of the peculiarities of the \'etale fundamental group. For example,
\'etale fundamental groups are not "true" groups but only profinite groups, and
one cannot hope to recover more: the "Tannakian" functor represented by the
\'etale fundamental group of a scheme preserves finite products but not all
products.Comment: 46 pages + bibliography. Diagrams drawn in Tik
Resolving Prime Modules: The Structure of Pseudo-cographs and Galled-Tree Explainable Graphs
The modular decomposition of a graph is a natural construction to capture
key features of in terms of a labeled tree whose vertices are
labeled as "series" (), "parallel" () or "prime". However, full
information of is provided by its modular decomposition tree only,
if is a cograph, i.e., does not contain prime modules. In this case,
explains , i.e., if and only if the lowest common
ancestor of and has label "". Pseudo-cographs,
or, more general, GaTEx graphs are graphs that can be explained by labeled
galled-trees, i.e., labeled networks that are obtained from the modular
decomposition tree of by replacing the prime vertices in by
simple labeled cycles. GaTEx graphs can be recognized and labeled galled-trees
that explain these graphs can be constructed in linear time.
In this contribution, we provide a novel characterization of GaTEx graphs in
terms of a set of 25 forbidden induced subgraphs.
This characterization, in turn, allows us to show that GaTEx graphs are closely
related to many other well-known graph classes such as -sparse and
-reducible graphs, weakly-chordal graphs, perfect graphs with perfect
order, comparability and permutation graphs, murky graphs as well as interval
graphs, Meyniel graphs or very strongly-perfect and brittle graphs. Moreover,
we show that every GaTEx graph as twin-width at most 1.Comment: 18 pages, 3 figure
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