29,649 research outputs found
Landscapes of data sets and functoriality of persistent homology
The aim of this article is to describe a new perspective on functoriality of
persistent homology and explain its intrinsic symmetry that is often
overlooked. A data set for us is a finite collection of functions, called
measurements, with a finite domain. Such a data set might contain internal
symmetries which are effectively captured by the action of a set of the domain
endomorphisms. Different choices of the set of endomorphisms encode different
symmetries of the data set. We describe various category structures on such
enriched data sets and prove some of their properties such as decompositions
and morphism formations. We also describe a data structure, based on coloured
directed graphs, which is convenient to encode the mentioned enrichment. We
show that persistent homology preserves only some aspects of these landscapes
of enriched data sets however not all. In other words persistent homology is
not a functor on the entire category of enriched data sets. Nevertheless we
show that persistent homology is functorial locally. We use the concept of
equivariant operators to capture some of the information missed by persistent
homology
On Discrete Symmetries in su(2) and su(3) Affine Theories and Related Graphs
We classify the possible finite symmetries of conformal field theories with
an affine Lie algebra su(2) and su(3), and discuss the results from the
perspective of the graphs associated with the modular invariants. The
highlights of the analysis are first, that the symmetries we found in either
case are matched by the graph data in a perfect way in the case of su(2), but
in a looser way for su(3), and second, that some of the graphs lead naturally
to projective representations, both in su(2) and in su(3).Comment: 34 pages, 4 eps figure
Non periodic Ishibashi states: the su(2) and su(3) affine theories
We consider the su(2) and su(3) affine theories on a cylinder, from the point
of view of their discrete internal symmetries. To this end, we adapt the usual
treatment of boundary conditions leading to the Cardy equation to take the
symmetry group into account. In this context, the role of the Ishibashi states
from all (non periodic) bulk sectors is emphasized. This formalism is then
applied to the su(2) and su(3) models, for which we determine the action of the
symmetry group on the boundary conditions, and we compute the twisted partition
functions. Most if not all data relevant to the symmetry properties of a
specific model are hidden in the graphs associated with its partition function,
and their subgraphs. A synoptic table is provided that summarizes the many
connections between the graphs and the symmetry data that are to be expected in
general.Comment: 19 pages, 3 figure
Towards fully covariant machine learning
Any representation of data involves arbitrary investigator choices. Because
those choices are external to the data-generating process, each choice leads to
an exact symmetry, corresponding to the group of transformations that takes one
possible representation to another. These are the passive symmetries; they
include coordinate freedom, gauge symmetry, and units covariance, all of which
have led to important results in physics. In machine learning, the most visible
passive symmetry is the relabeling or permutation symmetry of graphs. Our goal
is to understand the implications for machine learning of the many passive
symmetries in play. We discuss dos and don'ts for machine learning practice if
passive symmetries are to be respected. We discuss links to causal modeling,
and argue that the implementation of passive symmetries is particularly
valuable when the goal of the learning problem is to generalize out of sample.
This paper is conceptual: It translates among the languages of physics,
mathematics, and machine-learning. We believe that consideration and
implementation of passive symmetries might help machine learning in the same
ways that it transformed physics in the twentieth century.Comment: substantial revision from v1; submitted to TML
Data types with symmetries and polynomial functors over groupoids
Polynomial functors are useful in the theory of data types, where they are
often called containers. They are also useful in algebra, combinatorics,
topology, and higher category theory, and in this broader perspective the
polynomial aspect is often prominent and justifies the terminology. For
example, Tambara's theorem states that the category of finite polynomial
functors is the Lawvere theory for commutative semirings. In this talk I will
explain how an upgrade of the theory from sets to groupoids is useful to deal
with data types with symmetries, and provides a common generalisation of and a
clean unifying framework for quotient containers (cf. Abbott et al.), species
and analytic functors (Joyal 1985), as well as the stuff types of Baez-Dolan.
The multi-variate setting also includes relations and spans, multispans, and
stuff operators. An attractive feature of this theory is that with the correct
homotopical approach - homotopy slices, homotopy pullbacks, homotopy colimits,
etc. - the groupoid case looks exactly like the set case. After some standard
examples, I will illustrate the notion of data-types-with-symmetries with
examples from quantum field theory, where the symmetries of complicated tree
structures of graphs play a crucial role, and can be handled elegantly using
polynomial functors over groupoids. (These examples, although beyond species,
are purely combinatorial and can be appreciated without background in quantum
field theory.) Locally cartesian closed 2-categories provide semantics for
2-truncated intensional type theory. For a fullfledged type theory, locally
cartesian closed \infty-categories seem to be needed. The theory of these is
being developed by D.Gepner and the author as a setting for homotopical
species, and several of the results exposed in this talk are just truncations
of \infty-results obtained in joint work with Gepner. Details will appear
elsewhere.Comment: This is the final version of my conference paper presented at the
28th Conference on the Mathematical Foundations of Programming Semantics
(Bath, June 2012); to appear in the Electronic Notes in Theoretical Computer
Science. 16p
Statistics of Feynman amplitudes in -theory
The amplitude of subdivergence-free logarithmically divergent Feynman graphs
in -theory in 4 spacetime dimensions is given by a single number, the
Feynman period. We numerically compute the periods of 1.3 million
non-isomorphic completed graphs, this represents more than 31 million graphs
contributing to the beta function. Our data set includes all primitive graphs
up to 13 loops, and non-complete samples up to 18 loops, with an accuracy of
ca. 4 significant digits.
We implement all known symmetries of the period in a new computer program and
count them up to 14 loops. We discover some combinations of symmetries that had
been overlooked earlier, resulting in an overall slightly lower count of
independent graphs than previously assumed.
Using the numerical data, we examine the distribution of Feynman periods. We
confirm the leading asymptotic growth of the average period with growing loop
order. At high loop order, a limiting distribution is reached for the
amplitudes near the mean. We construct two different models to approximate this
distribution. A small class of graphs, most notably the zigzags, grows
significantly faster than the mean and causes the limiting distribution to have
divergent moments even when normalized to unit mean. We examine the relation
between the period and various properties of the underlying graphs. We confirm
the strong correlation with the Hepp bound, the Martin invariant, and the
number of 6-edge cuts. We find that, on average, the amplitude of planar graphs
is significantly larger than that of non-planar graphs, irrespective of
symmetry.
We estimate the primitive contribution to the 18-loop beta function of the
-symmetric theory. We confirm that primitive graphs constitute a large
part of the known asymptotics of the beta function in MS. However, we can not
determine if they are, asymptotically, the only leading contribution.Comment: 59 pages, 70 figures, 17 table
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