1,393 research outputs found
The Data Big Bang and the Expanding Digital Universe: High-Dimensional, Complex and Massive Data Sets in an Inflationary Epoch
Recent and forthcoming advances in instrumentation, and giant new surveys,
are creating astronomical data sets that are not amenable to the methods of
analysis familiar to astronomers. Traditional methods are often inadequate not
merely because of the size in bytes of the data sets, but also because of the
complexity of modern data sets. Mathematical limitations of familiar algorithms
and techniques in dealing with such data sets create a critical need for new
paradigms for the representation, analysis and scientific visualization (as
opposed to illustrative visualization) of heterogeneous, multiresolution data
across application domains. Some of the problems presented by the new data sets
have been addressed by other disciplines such as applied mathematics,
statistics and machine learning and have been utilized by other sciences such
as space-based geosciences. Unfortunately, valuable results pertaining to these
problems are mostly to be found only in publications outside of astronomy. Here
we offer brief overviews of a number of concepts, techniques and developments,
some "old" and some new. These are generally unknown to most of the
astronomical community, but are vital to the analysis and visualization of
complex datasets and images. In order for astronomers to take advantage of the
richness and complexity of the new era of data, and to be able to identify,
adopt, and apply new solutions, the astronomical community needs a certain
degree of awareness and understanding of the new concepts. One of the goals of
this paper is to help bridge the gap between applied mathematics, artificial
intelligence and computer science on the one side and astronomy on the other.Comment: 24 pages, 8 Figures, 1 Table. Accepted for publication: "Advances in
Astronomy, special issue "Robotic Astronomy
Standard Model in multiscale theories and observational constraints
We construct and analyze the Standard Model of electroweak and strong
interactions in multiscale spacetimes with (i) weighted derivatives and (ii)
-derivatives. Both theories can be formulated in two different frames,
called fractional and integer picture. By definition, the fractional picture is
where physical predictions should be made. (i) In the theory with weighted
derivatives, it is shown that gauge invariance and the requirement of having
constant masses in all reference frames make the Standard Model in the integer
picture indistinguishable from the ordinary one. Experiments involving only
weak and strong forces are insensitive to a change of spacetime dimensionality
also in the fractional picture, and only the electromagnetic and gravitational
sectors can break the degeneracy. For the simplest multiscale measures with
only one characteristic time, length and energy scale , and
, we compute the Lamb shift in the hydrogen atom and constrain the
multiscale correction to the ordinary result, getting the absolute upper bound
. For the natural choice of the
fractional exponent in the measure, this bound is strengthened to
, corresponding to and
. Stronger bounds are obtained from the measurement of the
fine-structure constant. (ii) In the theory with -derivatives, considering
the muon decay rate and the Lamb shift in light atoms, we obtain the
independent absolute upper bounds and
. For , the Lamb shift alone yields
.Comment: 25 pages. v2: authors' metadata corrected; v3: references added, new
material added including a comparison with varying-couplings and effective
field theories, a section on predictivity and falsifiability of multiscale
theories, a discussion on classical CPT, expanded conclusions, and new QED
constraints from the fine-structure constant; v3: minor typos corrected to
match the published versio
Multiscale Geometric Methods for Data Sets II: Geometric Multi-Resolution Analysis
Data sets are often modeled as point clouds in , for large. It is
often assumed that the data has some interesting low-dimensional structure, for
example that of a -dimensional manifold , with much smaller than .
When is simply a linear subspace, one may exploit this assumption for
encoding efficiently the data by projecting onto a dictionary of vectors in
(for example found by SVD), at a cost for data points. When
is nonlinear, there are no "explicit" constructions of dictionaries that
achieve a similar efficiency: typically one uses either random dictionaries, or
dictionaries obtained by black-box optimization. In this paper we construct
data-dependent multi-scale dictionaries that aim at efficient encoding and
manipulating of the data. Their construction is fast, and so are the algorithms
that map data points to dictionary coefficients and vice versa. In addition,
data points are guaranteed to have a sparse representation in terms of the
dictionary. We think of dictionaries as the analogue of wavelets, but for
approximating point clouds rather than functions.Comment: Re-formatted using AMS styl
- …