1,162 research outputs found
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Machine learning applications in search algorithms for gravitational waves from compact binary mergers
Gravitational waves from compact binary mergers are now routinely observed by Earth-bound detectors. These observations enable exciting new science, as they have opened a new window to the Universe.
However, extracting gravitational-wave signals from the noisy detector data is a challenging problem. The most sensitive search algorithms for compact binary mergers use matched filtering, an algorithm that compares the data with a set of expected template signals. As detectors are upgraded and more sophisticated signal models become available, the number of required templates will increase, which can make some sources computationally prohibitive to search for. The computational cost is of particular concern when low-latency alerts should be issued to maximize the time for electromagnetic follow-up observations. One potential solution to reduce computational requirements that has started to be explored in the last decade is machine learning. However, different proposed deep learning searches target varying parameter spaces and use metrics that are not always comparable to existing literature. Consequently, a clear picture of the capabilities of machine learning searches has been sorely missing.
In this thesis, we closely examine the sensitivity of various deep learning gravitational-wave search algorithms and introduce new methods to detect signals from binary black hole and binary neutron star mergers at previously untested statistical confidence levels. By using the sensitive distance as our core metric, we allow for a direct comparison of our algorithms to state-of-the-art search pipelines. As part of this thesis, we organized a global mock data challenge to create a benchmark for machine learning search algorithms targeting compact binaries. This way, the tools developed in this thesis are made available to the greater community by publishing them as open source software.
Our studies show that, depending on the parameter space, deep learning gravitational-wave search algorithms are already competitive with current production search pipelines. We also find that strategies developed for traditional searches can be effectively adapted to their machine learning counterparts. In regions where matched filtering becomes computationally expensive, available deep learning algorithms are also limited in their capability. We find reduced sensitivity to long duration signals compared to the excellent results for short-duration binary black hole signals
Bayesian inference for group-level cortical surface image-on-scalar-regression with Gaussian process priors
In regression-based analyses of group-level neuroimage data researchers
typically fit a series of marginal general linear models to image outcomes at
each spatially-referenced pixel. Spatial regularization of effects of interest
is usually induced indirectly by applying spatial smoothing to the data during
preprocessing. While this procedure often works well, resulting inference can
be poorly calibrated. Spatial modeling of effects of interest leads to more
powerful analyses, however the number of locations in a typical neuroimage can
preclude standard computation with explicitly spatial models. Here we
contribute a Bayesian spatial regression model for group-level neuroimaging
analyses. We induce regularization of spatially varying regression coefficient
functions through Gaussian process priors. When combined with a simple
nonstationary model for the error process, our prior hierarchy can lead to more
data-adaptive smoothing than standard methods. We achieve computational
tractability through Vecchia approximation of our prior which, critically, can
be constructed for a wide class of spatial correlation functions and results in
prior models that retain full spatial rank. We outline several ways to work
with our model in practice and compare performance against standard vertex-wise
analyses. Finally we illustrate our method in an analysis of cortical surface
fMRI task contrast data from a large cohort of children enrolled in the
Adolescent Brain Cognitive Development study
Impact of conditional modelling for universal autoregressive quantum states
We present a generalized framework to adapt universal quantum state
approximators, enabling them to satisfy rigorous normalization and
autoregressive properties. We also introduce filters as analogues to
convolutional layers in neural networks to incorporate translationally
symmetrized correlations in arbitrary quantum states. By applying this
framework to the Gaussian process state, we enforce autoregressive and/or
filter properties, analyzing the impact of the resulting inductive biases on
variational flexibility, symmetries, and conserved quantities. In doing so we
bring together different autoregressive states under a unified framework for
machine learning-inspired ans\"atze. Our results provide insights into how the
autoregressive construction influences the ability of a variational model to
describe correlations in spin and fermionic lattice models, as well as ab
initio electronic structure problems where the choice of representation affects
accuracy. We conclude that, while enabling efficient and direct sampling, thus
avoiding autocorrelation and loss of ergodicity issues in Metropolis sampling,
the autoregressive construction materially constrains the expressivity of the
model in many systems
Introduction to Riemannian Geometry and Geometric Statistics: from basic theory to implementation with Geomstats
International audienceAs data is a predominant resource in applications, Riemannian geometry is a natural framework to model and unify complex nonlinear sources of data.However, the development of computational tools from the basic theory of Riemannian geometry is laborious.The work presented here forms one of the main contributions to the open-source project geomstats, that consists in a Python package providing efficient implementations of the concepts of Riemannian geometry and geometric statistics, both for mathematicians and for applied scientists for whom most of the difficulties are hidden under high-level functions. The goal of this monograph is two-fold. First, we aim at giving a self-contained exposition of the basic concepts of Riemannian geometry, providing illustrations and examples at each step and adopting a computational point of view. The second goal is to demonstrate how these concepts are implemented in Geomstats, explaining the choices that were made and the conventions chosen. The general concepts are exposed and specific examples are detailed along the text.The culmination of this implementation is to be able to perform statistics and machine learning on manifolds, with as few lines of codes as in the wide-spread machine learning tool scikit-learn. We exemplify this with an introduction to geometric statistics
Law of Large Numbers for Bayesian two-layer Neural Network trained with Variational Inference
We provide a rigorous analysis of training by variational inference (VI) of
Bayesian neural networks in the two-layer and infinite-width case. We consider
a regression problem with a regularized evidence lower bound (ELBO) which is
decomposed into the expected log-likelihood of the data and the
Kullback-Leibler (KL) divergence between the a priori distribution and the
variational posterior. With an appropriate weighting of the KL, we prove a law
of large numbers for three different training schemes: (i) the idealized case
with exact estimation of a multiple Gaussian integral from the
reparametrization trick, (ii) a minibatch scheme using Monte Carlo sampling,
commonly known as Bayes by Backprop, and (iii) a new and computationally
cheaper algorithm which we introduce as Minimal VI. An important result is that
all methods converge to the same mean-field limit. Finally, we illustrate our
results numerically and discuss the need for the derivation of a central limit
theorem
Electron Thermal Runaway in Atmospheric Electrified Gases: a microscopic approach
Thesis elaborated from 2018 to 2023 at the Instituto de AstrofĂsica de AndalucĂa under the supervision of Alejandro Luque (Granada, Spain) and Nikolai Lehtinen (Bergen, Norway). This thesis presents a new database of atmospheric electron-molecule collision cross sections which was published separately under the DOI :
With this new database and a new super-electron management algorithm which significantly enhances high-energy electron statistics at previously unresolved ratios, the thesis explores general facets of the electron thermal runaway process relevant to atmospheric discharges under various conditions of the temperature and gas composition as can be encountered in the wake and formation of discharge channels
Microscopy of spin hydrodynamics and cooperative light scattering in atomic Hubbard systems
Wechselwirkungen zwischen quantenmechanischen Teilchen können zu kollektiven Phänomenen führen, deren Eigenschaften sich vom Verhalten einzelner Teilchen stark unterscheiden. Während solche Quanteneffekte im Allgemeinen schwierig zu beobachten sind, haben sich ultrakalte, in optischen Gittern gefangene atomare Gase als vielseitige experimentelle Plattform zur Erforschung der Quantenvielteilchenphysik erwiesen. In dieser Arbeit setzten wir ein Gitterplatz- und Einzelatom-aufgelöstes Quantengasmikroskop für bosonische Rb-87 Atome ein, um Vielteilchensysteme im und außerhalb des Gleichgewichts zu untersuchen.
Zunächst betrachteten wir den quantenmechanischen Phasenübergang zwischen dem suprafluiden und dem Mott-isolierenden Zustand im Bose-Hubbard-Modell, das nativ durch kalte Atome in optischen Gittern realisiert wird, und zeigten, dass sich die Brane-Parität eignet, um nichtlokale Ordnung im konventionell als ungeordnet erachteten zweidimensionalen Mott-Isolator zu identifizieren. Mithilfe eines mikroskopischen Ansatzes zur Realisierung einstellbarer Gittergeometrien und programmierbarer Einheitszellen implementierten wir Quadrats-, Dreiecks-, Kagome- und Lieb-Gitter und beobachteten die Skalierung des Phasenübergangspunkts mit der mittleren Koordinationszahl des Gitters.
In einem eindimensionalen Gitter untersuchten wir zudem den Hochtemperatur-Spintransport im Heisenberg-Modell, das durch Superaustausch in der Mott-isolierenden Phase eines zwei-Spezies Bose-Hubbard-Modells realisiert wurde. Durch Betrachten der Relaxationsdynamik eines als Domänenwand präparierten Anfangszustandes fanden wir eine superdiffusive Raum-Zeit-Skalierung mit einem anomalen dynamischen Exponenten von 3/2. Anschließend untersuchten wir die theoretisch vorhergesagten mikroskopischen Voraussetzungen für Superdiffusion, indem wir reguläre Diffusion im nicht-integrablen, zweidimensionalen Heisenberg-Modell und ballistischen Transport für SU(2)-Symmetrie-gebrochene magnetisierte Anfangszustände nachwiesen. Weiterhin maßen wir die Zählstatistik der durch die Domänenwand transportierten Spins; die sich daraus ergebende schiefe Verteilung deutete auf einen nichtlinearen zugrundeliegenden Transportprozess hin, der an die dynamische Kardar-Parisi-Zhang Universalitätsklasse erinnert.
Mittels Mott-Isolatoren im Limit tiefer Gitter konnten wir darüber hinaus die durch Photonen vermittelten Wechselwirkungen in einem Spinsystem untersuchen, das aus zwei über einen geschlossenen optischen Übergang gekoppelten Zuständen besteht. Durch spektroskopische Untersuchung der Reflexion und Transmission konnten wir die direkte Anregung einer subradianten Eigenmode und kohärente Spiegelung beobachten, was auf die Realisierung einer effizienten, im freien Raum operierenden, paraxialen Licht-Materie-Schnittstelle hindeutet.The interplay of quantum particles can give rise to collective phenomena whose characteristics are distinct from the behavior of individual particles. While quantum effects are generally challenging to observe, ultracold atomic gases trapped in optical lattices have emerged as a versatile experimental platform to study quantum many-body physics. In this thesis, we employed a site– and single-atom–resolved quantum gas microscope of bosonic Rb-87 atoms to explore many-body systems in and out of equilibrium.
We first considered the ground-state quantum phase transition between the superfluid and Mott-insulating state in the Bose–Hubbard model, natively realized by cold atoms in optical lattices, for which we found brane parity to be suitable for detecting nonlocal order in the conventionally unordered two-dimensional Mott insulator. Using a microscopic approach to realizing tunable lattice geometries and programmable unit cells, we implemented square, triangular, kagome and Lieb lattices, and observed the mean-field scaling of the phase transition point with average coordination number.
In a one-dimensional lattice, we furthermore studied high-temperature spin transport in the Heisenberg model, realized by superexchange in the Mott-insulating phase of a two-species Bose–Hubbard model. By tracking the relaxation dynamics of an initial domain-wall state, we found superdiffusive space–time scaling with an anomalous dynamical exponent of 3/2. We then probed the predicted microscopic requirements for superdiffusion, verifying regular diffusion for the integrability-broken two-dimensional Heisenberg model and ballistic transport for SU(2)-symmetry–broken net magnetized initial states. Subsequently, we measured the full counting statistics of spins transported across the domain wall; the resulting skewed distribution implied a nonlinear underlying transport process, reminiscent of the Kardar–Parisi–Zhang dynamical universality class.
Moving to Mott insulators in the deep-lattice limit, we could moreover study photon-mediated interactions on a subwavelength-spaced, array-ordered spin system consisting of states coupled by a closed optical transition. By spectroscopically probing the reflectance and transmittance, we demonstrated the direct excitation of a subradiant eigenmode and observed specular reflection, indicating the realization of an efficient free-space paraxial light–matter interface
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