155 research outputs found
Supermetric search
Metric search is concerned with the efficient evaluation of queries in metric spaces. In general, a large space of objects is arranged in such a way that, when a further object is presented as a query, those objects most similar to the query can be efficiently found. Most mechanisms rely upon the triangle inequality property of the metric governing the space. The triangle inequality property is equivalent to a finite embedding property, which states that any three points of the space can be isometrically embedded in two-dimensional Euclidean space. In this paper, we examine a class of semimetric space that is finitely four-embeddable in three-dimensional Euclidean space. In mathematics this property has been extensively studied and is generally known as the four-point property. All spaces with the four-point property are metric spaces, but they also have some stronger geometric guarantees. We coin the term supermetric space as, in terms of metric search, they are significantly more tractable. Supermetric spaces include all those governed by Euclidean, Cosine, Jensen–Shannon and Triangular distances, and are thus commonly used within many domains. In previous work we have given a generic mathematical basis for the supermetric property and shown how it can improve indexing performance for a given exact search structure. Here we present a full investigation into its use within a variety of different hyperplane partition indexing structures, and go on to show some more of its flexibility by examining a search structure whose partition and exclusion conditions are tailored, at each node, to suit the individual reference points and data set present there. Among the results given, we show a new best performance for exact search using a well-known benchmark
Signature of the Simplicial Supermetric
We investigate the signature of the Lund-Regge metric on spaces of simplicial
three-geometries which are important in some formulations of quantum gravity.
Tetrahedra can be joined together to make a three-dimensional piecewise linear
manifold. A metric on this manifold is specified by assigning a flat metric to
the interior of the tetrahedra and values to their squared edge-lengths. The
subset of the space of squared edge-lengths obeying triangle and analogous
inequalities is simplicial configuration space. We derive the Lund-Regge metric
on simplicial configuration space and show how it provides the shortest
distance between simplicial three-geometries among all choices of gauge inside
the simplices for defining this metric (Regge gauge freedom). We show
analytically that there is always at least one physical timelike direction in
simplicial configuration space and provide a lower bound on the number of
spacelike directions. We show that in the neighborhood of points in this space
corresponding to flat metrics there are spacelike directions corresponding to
gauge freedom in assigning the edge-lengths. We evaluate the signature
numerically for the simplicial configuration spaces based on some simple
triangulations of the three-sphere (S^3) and three-torus (T^3). For the surface
of a four-simplex triangulation of S^3 we find one timelike direction and all
the rest spacelike over all of the simplicial configuration space. For the
triangulation of T^3 around flat space we find degeneracies in the simplicial
supermetric as well as a few gauge modes corresponding to a positive
eigenvalue. Moreover, we have determined that some of the negative eigenvalues
are physical, i.e. the corresponding eigenvectors are not generators of
diffeomorphisms. We compare our results with the known properties of continuum
superspace.Comment: 24 pages, RevTeX, 4 eps Figures. Submitted to Classical Quantum
Gravit
A unified geometric framework for boundary charges and dressings: non-Abelian theory and matter
Boundaries in gauge theories are a delicate issue. Arbitrary boundary choices
enter the calculation of charges via Noether's second theorem, obstructing the
assignment of unambiguous physical charges to local gauge symmetries. Replacing
the arbitrary boundary choice with new degrees of freedom suggests itself. But,
concretely, such boundary degrees of freedom are spurious---i.e. they are not
part of the original field content of the theory---and have to disappear upon
gluing. How should we fit them into what we know about field-theory? We resolve
these issues in a unified and geometric manner, by introducing a connection
1-form, , in the field-space of Yang-Mills theory. Using this geometric
tool, a modified version of symplectic geometry---here called `horizontal'---is
possible. Independently of boundary conditions, this formalism bestows to each
region a physical notion of charge: the horizontal Noether charge. The
horizontal gauge charges always vanish, while global charges still arise for
reducible configurations characterized by global symmetries. The field-content
itself is used as a reference frame to distinguish `gauge' and `physical'; no
new degrees of freedom, such as group-valued edge modes, are required.
Different choices of reference fields give different 's, which are
cousins of gauge-fixing like the Higgs-unitary and Coulomb gauges. But the
formalism extends well beyond gauge-fixings, for instance by avoiding the
Gribov problem. For one choice of , would-be Goldstone modes arising
from the condensation of matter degrees of freedom play precisely the role of
the known group-valued edge modes, but here they arise as preferred coordinates
in field space, rather than new fields. For another choice, in the Abelian
case, recovers the Dirac dressing of the electron.Comment: 71 pages, 3 appendices, 9 figures. Summary of the results at the
beginning of the paper. v2: numerous improvements in the presentation, and
introduction of new references, taking colleague feedback into accoun
Stabilized Quantum Gravity: Stochastic Interpretation and Numerical Simulation
Following the reasoning of Claudson and Halpern, it is shown that
"fifth-time" stabilized quantum gravity is equivalent to Langevin evolution
(i.e. stochastic quantization) between fixed non-singular, but otherwise
arbitrary, initial and final states. The simple restriction to a fixed final
state at is sufficient to stabilize the theory. This
equivalence fixes the integration measure, and suggests a particular
operator-ordering, for the fifth-time action of quantum gravity. Results of a
numerical simulation of stabilized, latticized Einstein-Cartan theory on some
small lattices are reported. In the range of cosmological constant \l
investigated, it is found that: 1) the system is always in the broken phase
; and 2) the negative free energy is large, possibly singular,
in the vincinity of \l = 0. The second finding may be relevant to the
cosmological constant problem.Comment: 22 pages, 3 figures (now included as a postscript file
The lowest modes around Gaussian solutions of tensor models and the general relativity
In the previous paper, the number distribution of the low-lying spectra
around Gaussian solutions representing various dimensional fuzzy tori of a
tensor model was numerically shown to be in accordance with the general
relativity on tori. In this paper, I perform more detailed numerical analysis
of the properties of the modes for two-dimensional fuzzy tori, and obtain
conclusive evidences for the agreement. Under a proposed correspondence between
the rank-three tensor in tensor models and the metric tensor in the general
relativity, conclusive agreement is obtained between the profiles of the
low-lying modes in a tensor model and the metric modes transverse to the
general coordinate transformation. Moreover, the low-lying modes are shown to
be well on a massless trajectory with quartic momentum dependence in the tensor
model. This is in agreement with that the lowest momentum dependence of metric
fluctuations in the general relativity will come from the R^2-term, since the
R-term is topological in two dimensions. These evidences support the idea that
the low-lying low-momentum dynamics around the Gaussian solutions of tensor
models is described by the general relativity. I also propose a renormalization
procedure for tensor models. A classical application of the procedure makes the
patterns of the low-lying spectra drastically clearer, and suggests also the
existence of massive trajectories.Comment: 31 pages, 8 figures, Added references, minor corrections, a
misleading figure replace
On QCD String Theory and AdS Dynamics
The AdS/CFT correspondence of elementary string theory has been recently
suggested as a ``microscopic'' approach to QCD string theory in various
dimensions. We use the microscopic theory to show that the ultraviolet regime
on the string world-sheet is mapped to the ultraviolet effects in QCD. In the
case of QCD_2, a world-sheet path integral representation of QCD strings is
known, in terms of a topological rigid string theory whose world-sheet
supersymmetry is reminiscent of Parisi-Sourlas supersymmetry. We conjecture
that the supersymmetric rigid string theory is dual to the elementary Type IIB
string theory in the singular AdS background that corresponds to the large-N
limit of QCD_2. We also generalize the rigid string with world-sheet
Parisi-Sourlas supersymmetry to dimensions greater than two, and argue that the
theory is asymptotically free, a non-zero string tension is generated
dynamically through dimensional transmutation, and the theory is topological
only asymptotically in the ultraviolet.Comment: 29pp. v2: typos corrected, final version to appear in JHE
High-dimensional simplexes for supermetric search
In a metric space, triangle inequality implies that, for any three objects, a triangle with edge lengths corresponding to their pairwise distances can be formed. The n-point property is a generalisation of this where, for any (n+1) objects in the space, there exists an n-dimensional simplex whose edge lengths correspond to the distances among the objects. In general, metric spaces do not have this property; however in 1953, Blumenthal showed that any semi-metric space which is isometrically embeddable in a Hilbert space also has the n-point property. We have previously called such spaces supermetric spaces, and have shown that many metric spaces are also supermetric, including Euclidean, Cosine, Jensen-Shannon and Triangular spaces of any dimension. Here we show how such simplexes can be constructed from only their edge lengths, and we show how the geometry of the simplexes can be used to determine lower and upper bounds on unknown distances within the original space. By increasing the number of dimensions, these bounds converge to the true distance. Finally we show that for any Hilbert-embeddable space, it is possible to construct Euclidean spaces of arbitrary dimensions, from which these lower and upper bounds of the original space can be determined. These spaces may be much cheaper to query than the original. For similarity search, the engineering tradeoffs are good: we show significant reductions in data size and metric cost with little loss of accuracy, leading to a significant overall improvement in exact search performance
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