20 research outputs found
A general theorem of existence of quasi absolutely minimal Lipschitz extensions
In this paper we consider a wide class of generalized Lipschitz extension
problems and the corresponding problem of finding absolutely minimal Lipschitz
extensions. We prove that if a minimal Lipschitz extension exists, then under
certain other mild conditions, a quasi absolutely minimal Lipschitz extension
must exist as well. Here we use the qualifier "quasi" to indicate that the
extending function in question nearly satisfies the conditions of being an
absolutely minimal Lipschitz extension, up to several factors that can be made
arbitrarily small.Comment: 33 pages. v3: Correction to Example 2.4.3. Specifically,
alpha-H\"older continuous functions, for alpha strictly less than one, do not
satisfy (P3). Thus one cannot conclude that quasi-AMLEs exist in this case.
Please note that the error remains in the published version of the paper in
Mathematische Annalen. v2: Several minor corrections and edits, a new
appendix (Appendix A
Bi-stochastic kernels via asymmetric affinity functions
In this short letter we present the construction of a bi-stochastic kernel p
for an arbitrary data set X that is derived from an asymmetric affinity
function {\alpha}. The affinity function {\alpha} measures the similarity
between points in X and some reference set Y. Unlike other methods that
construct bi-stochastic kernels via some convergent iteration process or
through solving an optimization problem, the construction presented here is
quite simple. Furthermore, it can be viewed through the lens of out of sample
extensions, making it useful for massive data sets.Comment: 5 pages. v2: Expanded upon the first paragraph of subsection 2.1. v3:
Minor changes and edits. v4: Edited comments and added DO
Diffusion maps for changing data
Graph Laplacians and related nonlinear mappings into low dimensional spaces
have been shown to be powerful tools for organizing high dimensional data. Here
we consider a data set X in which the graph associated with it changes
depending on some set of parameters. We analyze this type of data in terms of
the diffusion distance and the corresponding diffusion map. As the data changes
over the parameter space, the low dimensional embedding changes as well. We
give a way to go between these embeddings, and furthermore, map them all into a
common space, allowing one to track the evolution of X in its intrinsic
geometry. A global diffusion distance is also defined, which gives a measure of
the global behavior of the data over the parameter space. Approximation
theorems in terms of randomly sampled data are presented, as are potential
applications.Comment: 38 pages. 9 figures. To appear in Applied and Computational Harmonic
Analysis. v2: Several minor changes beyond just typos. v3: Minor typo
corrected, added DO
The S-parameter in Holographic Technicolor Models
We study the S parameter, considering especially its sign, in models of
electroweak symmetry breaking (EWSB) in extra dimensions, with fermions
localized near the UV brane. Such models are conjectured to be dual to 4D
strong dynamics triggering EWSB. The motivation for such a study is that a
negative value of S can significantly ameliorate the constraints from
electroweak precision data on these models, allowing lower mass scales (TeV or
below) for the new particles and leading to easier discovery at the LHC. We
first extend an earlier proof of S>0 for EWSB by boundary conditions in
arbitrary metric to the case of general kinetic functions for the gauge fields
or arbitrary kinetic mixing. We then consider EWSB in the bulk by a Higgs VEV
showing that S is positive for arbitrary metric and Higgs profile, assuming
that the effects from higher-dimensional operators in the 5D theory are
sub-leading and can therefore be neglected. For the specific case of AdS_5 with
a power law Higgs profile, we also show that S ~ + O(1), including effects of
possible kinetic mixing from higher-dimensional operator (of NDA size) in the
theory. Therefore, our work strongly suggests that S is positive in
calculable models in extra dimensions.Comment: 21 pages, 2 figures. v2: references adde
Kymatio: Scattering Transforms in Python
The wavelet scattering transform is an invariant signal representation
suitable for many signal processing and machine learning applications. We
present the Kymatio software package, an easy-to-use, high-performance Python
implementation of the scattering transform in 1D, 2D, and 3D that is compatible
with modern deep learning frameworks. All transforms may be executed on a GPU
(in addition to CPU), offering a considerable speed up over CPU
implementations. The package also has a small memory footprint, resulting
inefficient memory usage. The source code, documentation, and examples are
available undera BSD license at https://www.kymat.io
Toward a Systematic Holographic QCD: A Braneless Approach
Recently a holographic model of hadrons motivated by AdS/CFT has been
proposed to fit the low energy data of mesons. We point out that the infrared
physics can be developed in a more systematic manner by exploiting backreaction
of the nonperturbative condensates. We show that these condensates can
naturally provide the IR cutoff corresponding to confinement, thus removing
some of the ambiguities from the original formulation of the model. We also
show how asymptotic freedom can be incorporated into the theory, and the
substantial effect it has on the glueball spectrum and gluon condensate of the
theory. A simple reinterpretation of the holographic scale results in a
non-perturbative running for alpha_s which remains finite for all energies. We
also find the leading effects of adding the higher condensate into the theory.
The difficulties for such models to reproduce the proper Regge physics lead us
to speculate about extensions of our model incorporating tachyon condensation.Comment: 27 pages, LaTe
Coarse Graining of Data via Inhomogeneous Diffusion Condensation
Big data often has emergent structure that exists at multiple levels of
abstraction, which are useful for characterizing complex interactions and
dynamics of the observations. Here, we consider multiple levels of abstraction
via a multiresolution geometry of data points at different granularities. To
construct this geometry we define a time-inhomogeneous diffusion process that
effectively condenses data points together to uncover nested groupings at
larger and larger granularities. This inhomogeneous process creates a deep
cascade of intrinsic low pass filters on the data affinity graph that are
applied in sequence to gradually eliminate local variability while adjusting
the learned data geometry to increasingly coarser resolutions. We provide
visualizations to exhibit our method as a continuously-hierarchical clustering
with directions of eliminated variation highlighted at each step. The utility
of our algorithm is demonstrated via neuronal data condensation, where the
constructed multiresolution data geometry uncovers the organization, grouping,
and connectivity between neurons.Comment: 14 pages, 7 figure