201,939 research outputs found
Noncommutative spaces and matrix embeddings on flat R^{2n+1}
We conjecture an embedding operator which assigns, to any 2n+1 hermitian
matrices, a 2n-dimensional hypersurface in flat (2n + 1)-dimensional Euclidean
space. This corresponds to precisely defining a fuzzy D(2n)-brane corresponding
to N D0-branes. Points on the emergent hypersurface correspond to zero
eigenstates of the embedding operator, which have an interpretation as coherent
states underlying the emergent noncommutative geometry. Using this
correspondence, all physical properties of the emergent D(2n)-brane can be
computed. We apply our conjecture to noncommutative flat and spherical spaces.
As a by-product, we obtain a construction of a rotationally symmetric flat
noncommutative space in 4 dimensions.Comment: 14 pages, no figures. v2: added references and a clarificatio
Non-standard embedding and five-branes in heterotic M-Theory
We construct vacua of M-theory on S^1/Z_2 associated with Calabi-Yau
three-folds. These vacua are appropriate for compactification to N=1
supersymmetry theories in both four and five dimensions. We allow for general
E_8 x E_8 gauge bundles and for the presence of five-branes. The five-branes
span the four-dimensional uncompactified space and are wrapped on holomorphic
curves in the Calabi-Yau space. Properties of these vacua, as well as of the
resulting low-energy theories, are discussed. We find that the low-energy gauge
group is enlarged by gauge fields that originate on the five-brane
world-volumes. In addition, the five-branes increase the types of new E_8 x E_8
breaking patterns allowed by the non-standard embedding. Characteristic
features of the low-energy theory, such as the threshold corrections to the
gauge kinetic functions, are significantly modified due to the presence of the
five-branes, as compared to the case of standard or non-standard embeddings
without five-branes.Comment: 34 pages, Latex 2e with amsmath, typos removed, factors corrected,
refs improve
A Comparison of Tests for Embeddings
It is possible to compare results for the classical tests for embeddings of chaotic data with the results of a recently proposed test. The classical tests, which depend on real numbers (fractal dimensions, Lyapunov exponents) averaged over an attractor, are compared with a topological test that depends on integers. The comparison can only be done for mappings into three dimensions. We find that the classical tests fail to predict when a mapping is an embedding and when it is not. We point out the reasons for this failure, which are not restricted to three dimensions
Conditional t-SNE: Complementary t-SNE embeddings through factoring out prior information
Dimensionality reduction and manifold learning methods such as t-Distributed
Stochastic Neighbor Embedding (t-SNE) are routinely used to map
high-dimensional data into a 2-dimensional space to visualize and explore the
data. However, two dimensions are typically insufficient to capture all
structure in the data, the salient structure is often already known, and it is
not obvious how to extract the remaining information in a similarly effective
manner. To fill this gap, we introduce \emph{conditional t-SNE} (ct-SNE), a
generalization of t-SNE that discounts prior information from the embedding in
the form of labels. To achieve this, we propose a conditioned version of the
t-SNE objective, obtaining a single, integrated, and elegant method. ct-SNE has
one extra parameter over t-SNE; we investigate its effects and show how to
efficiently optimize the objective. Factoring out prior knowledge allows
complementary structure to be captured in the embedding, providing new
insights. Qualitative and quantitative empirical results on synthetic and
(large) real data show ct-SNE is effective and achieves its goal
Approximated and User Steerable tSNE for Progressive Visual Analytics
Progressive Visual Analytics aims at improving the interactivity in existing
analytics techniques by means of visualization as well as interaction with
intermediate results. One key method for data analysis is dimensionality
reduction, for example, to produce 2D embeddings that can be visualized and
analyzed efficiently. t-Distributed Stochastic Neighbor Embedding (tSNE) is a
well-suited technique for the visualization of several high-dimensional data.
tSNE can create meaningful intermediate results but suffers from a slow
initialization that constrains its application in Progressive Visual Analytics.
We introduce a controllable tSNE approximation (A-tSNE), which trades off speed
and accuracy, to enable interactive data exploration. We offer real-time
visualization techniques, including a density-based solution and a Magic Lens
to inspect the degree of approximation. With this feedback, the user can decide
on local refinements and steer the approximation level during the analysis. We
demonstrate our technique with several datasets, in a real-world research
scenario and for the real-time analysis of high-dimensional streams to
illustrate its effectiveness for interactive data analysis
A Unified Approach to Attractor Reconstruction
In the analysis of complex, nonlinear time series, scientists in a variety of
disciplines have relied on a time delayed embedding of their data, i.e.
attractor reconstruction. The process has focused primarily on heuristic and
empirical arguments for selection of the key embedding parameters, delay and
embedding dimension. This approach has left several long-standing, but common
problems unresolved in which the standard approaches produce inferior results
or give no guidance at all. We view the current reconstruction process as
unnecessarily broken into separate problems. We propose an alternative approach
that views the problem of choosing all embedding parameters as being one and
the same problem addressable using a single statistical test formulated
directly from the reconstruction theorems. This allows for varying time delays
appropriate to the data and simultaneously helps decide on embedding dimension.
A second new statistic, undersampling, acts as a check against overly long time
delays and overly large embedding dimension. Our approach is more flexible than
those currently used, but is more directly connected with the mathematical
requirements of embedding. In addition, the statistics developed guide the user
by allowing optimization and warning when embedding parameters are chosen
beyond what the data can support. We demonstrate our approach on uni- and
multivariate data, data possessing multiple time scales, and chaotic data. This
unified approach resolves all the main issues in attractor reconstruction.Comment: 22 pages, revised version as submitted to CHAOS. Manuscript is
currently under review. 4 Figures, 31 reference
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