43,932 research outputs found
Interpretable statistics for complex modelling: quantile and topological learning
As the complexity of our data increased exponentially in the last decades, so has our
need for interpretable features. This thesis revolves around two paradigms to approach
this quest for insights.
In the first part we focus on parametric models, where the problem of interpretability
can be seen as a “parametrization selection”. We introduce a quantile-centric
parametrization and we show the advantages of our proposal in the context of regression,
where it allows to bridge the gap between classical generalized linear (mixed)
models and increasingly popular quantile methods.
The second part of the thesis, concerned with topological learning, tackles the
problem from a non-parametric perspective. As topology can be thought of as a way
of characterizing data in terms of their connectivity structure, it allows to represent
complex and possibly high dimensional through few features, such as the number of
connected components, loops and voids. We illustrate how the emerging branch of
statistics devoted to recovering topological structures in the data, Topological Data
Analysis, can be exploited both for exploratory and inferential purposes with a special
emphasis on kernels that preserve the topological information in the data.
Finally, we show with an application how these two approaches can borrow strength
from one another in the identification and description of brain activity through fMRI
data from the ABIDE project
Magnetic skyrmions and their lattices in triplet superconductors
Complete topological classification of solutions in SO(3) symmetric
Ginzburg-Landau free energy has been performed and a new class of solutions in
weak external magnetic field carrying two units of magnetic flux has been
identified. These solutions, magnetic skyrmions, do not have singular core like
Abrikosov vortices and at low magnetic field become lighter for strongly type
II superconductors. As a consequence, the lower critical magnetic field Hc1 is
reduced by a factor of log(kappa). Magnetic skyrmions repel each other as 1/r
at distances much larger then magnetic penetration depth forming relatively
robust triangular lattice. Magnetic induction near Hc1 increases gradually as
(H-Hc1)^2. This agrees very well with experiments on heavy fermion
superconductor UPt3. Newly discovered Ru based compounds Sr2RuO4 and
Sr2YRu(1-x)Cu(x)O6 are other possible candidates to possess skyrmion lattices.
Deviations from exact SO(3) symmetry are also studied.Comment: 23 pages, 10 eps figure
Classification of engineered topological superconductors
I perform a complete classification of 2d, quasi-1d and 1d topological
superconductors which originate from the suitable combination of inhomogeneous
Rashba spin-orbit coupling, magnetism and superconductivity. My analysis
reveals alternative types of topological superconducting platforms for which
Majorana fermions are accessible. Specifically, I observe that for quasi-1d
systems with Rashba spin-orbit coupling and time-reversal violating
superconductivity, as for instance due to a finite Josephson current flow,
Majorana fermions can emerge even in the absence of magnetism. Furthermore, for
the classification I also consider situations where additional "hidden"
symmetries emerge, with a significant impact on the topological properties of
the system. The latter, generally originate from a combination of space group
and complex conjugation operations that separately do not leave the Hamiltonian
invariant. Finally, I suggest alternative directions in topological quantum
computing for systems with additional unitary symmetries.Comment: To appear in New Journal of Physics for the Focus on Majorana
Fermions in Condensed Matter; Final version 19 pages, 6 figures: a new
section was added concerning the appearance of MFs in two coupled Rasba
semiconducting wires with proximity induced superconductivity and a finite
supercurrent flow, without the application of a magnetic field. Generally
improved discussion and references adde
Persistent Homology and String Vacua
We use methods from topological data analysis to study the topological
features of certain distributions of string vacua. Topological data analysis is
a multi-scale approach used to analyze the topological features of a dataset by
identifying which homological characteristics persist over a long range of
scales. We apply these techniques in several contexts. We analyze N=2 vacua by
focusing on certain distributions of Calabi-Yau varieties and Landau-Ginzburg
models. We then turn to flux compactifications and discuss how we can use
topological data analysis to extract physical informations. Finally we apply
these techniques to certain phenomenologically realistic heterotic models. We
discuss the possibility of characterizing string vacua using the topological
properties of their distributions.Comment: 32 pages, 12 pdf figure
Green's Function Method for Line Defects and Gapless Modes in Topological Insulators : Beyond Semiclassical Approach
Defects which appear in heterostructure junctions involving topological
insulators are sources of gapless modes governing the low energy properties of
the systems, as recently elucidated by Teo and Kane [Physical Review B82,
115120 (2010)]. A standard approach for the calculation of topological
invariants associated with defects is to deal with the spatial inhomogeneity
raised by defects within a semiclassical approximation. In this paper, we
propose a full quantum formulation for the topological invariants
characterizing line defects in three-dimensional insulators with no symmetry by
using the Green's function method. On the basis of the full quantum treatment,
we demonstrate the existence of a nontrivial topological invariant in the
topological insulator-ferromagnet tri-junction systems, for which a
semiclassical approximation fails to describe the topological phase. Also, our
approach enables us to study effects of electron-electron interactions and
impurity scattering on topological insulators with spatial inhomogeneity which
gives rise to the Axion electrodynamics responses.Comment: 15 pages, 3 figure
Towards Emotion Recognition: A Persistent Entropy Application
Emotion recognition and classification is a very active area of research. In
this paper, we present a first approach to emotion classification using
persistent entropy and support vector machines. A topology-based model is
applied to obtain a single real number from each raw signal. These data are
used as input of a support vector machine to classify signals into 8 different
emotions (calm, happy, sad, angry, fearful, disgust and surprised)
Long range order in gauge theories. Deformed QCD as a toy model
We study a number of different ingredients, related to long range order
observed in lattice QCD simulations, using a simple "deformed QCD" model. This
model is a weakly coupled gauge theory, which however has all the relevant
crucial elements allowing us to study difficult and nontrivial problems which
are known to be present in real strongly coupled QCD. In the present study, we
want to understand the physics of long range order in form of coherent low
dimensional vacuum configurations observed in Monte Carlo lattice simulations.
We demonstrate the presence of double-layer domain wall structures in the
deformed QCD, and study their interaction with localized topological monopoles.
Furthermore, we show that there is in fact an attractive interaction between
the two, such that the monopole favors a position within the domain wall.Comment: 10 pages, 5 figure
- …