2,344 research outputs found
Renormalization: an advanced overview
We present several approaches to renormalization in QFT: the multi-scale
analysis in perturbative renormalization, the functional methods \`a la
Wetterich equation, and the loop-vertex expansion in non-perturbative
renormalization. While each of these is quite well-established, they go beyond
standard QFT textbook material, and may be little-known to specialists of each
other approach. This review is aimed at bridging this gap.Comment: Review, 130 pages, 33 figures; v2: misprints corrected, refs. added,
minor improvements; v3: some changes to sect. 5, refs. adde
Local Kernels and the Geometric Structure of Data
We introduce a theory of local kernels, which generalize the kernels used in
the standard diffusion maps construction of nonparametric modeling. We prove
that evaluating a local kernel on a data set gives a discrete representation of
the generator of a continuous Markov process, which converges in the limit of
large data. We explicitly connect the drift and diffusion coefficients of the
process to the moments of the kernel. Moreover, when the kernel is symmetric,
the generator is the Laplace-Beltrami operator with respect to a geometry which
is influenced by the embedding geometry and the properties of the kernel. In
particular, this allows us to generate any Riemannian geometry by an
appropriate choice of local kernel. In this way, we continue a program of
Belkin, Niyogi, Coifman and others to reinterpret the current diverse
collection of kernel-based data analysis methods and place them in a geometric
framework. We show how to use this framework to design local kernels invariant
to various features of data. These data-driven local kernels can be used to
construct conformally invariant embeddings and reconstruct global
diffeomorphisms
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