1,293 research outputs found

    Probabilistic Programming Interfaces for Random Graphs::Markov Categories, Graphons, and Nominal Sets

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    We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way.We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers 'black-and-white' graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons

    Classical and quantum algorithms for scaling problems

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    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

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Mining Butterflies in Streaming Graphs

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    This thesis introduces two main-memory systems sGrapp and sGradd for performing the fundamental analytic tasks of biclique counting and concept drift detection over a streaming graph. A data-driven heuristic is used to architect the systems. To this end, initially, the growth patterns of bipartite streaming graphs are mined and the emergence principles of streaming motifs are discovered. Next, the discovered principles are (a) explained by a graph generator called sGrow; and (b) utilized to establish the requirements for efficient, effective, explainable, and interpretable management and processing of streams. sGrow is used to benchmark stream analytics, particularly in the case of concept drift detection. sGrow displays robust realization of streaming growth patterns independent of initial conditions, scale and temporal characteristics, and model configurations. Extensive evaluations confirm the simultaneous effectiveness and efficiency of sGrapp and sGradd. sGrapp achieves mean absolute percentage error up to 0.05/0.14 for the cumulative butterfly count in streaming graphs with uniform/non-uniform temporal distribution and a processing throughput of 1.5 million data records per second. The throughput and estimation error of sGrapp are 160x higher and 0.02x lower than baselines. sGradd demonstrates an improving performance over time, achieves zero false detection rates when there is not any drift and when drift is already detected, and detects sequential drifts in zero to a few seconds after their occurrence regardless of drift intervals

    Data-driven deep-learning methods for the accelerated simulation of Eulerian fluid dynamics

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    Deep-learning (DL) methods for the fast inference of the temporal evolution of fluid-dynamics systems, based on the previous recognition of features underlying large sets of fluid-dynamics data, have been studied. Specifically, models based on convolution neural networks (CNNs) and graph neural networks (GNNs) were proposed and discussed. A U-Net, a popular fully-convolutional architecture, was trained to infer wave dynamics on liquid surfaces surrounded by walls, given as input the system state at previous time-points. A term for penalising the error of the spatial derivatives was added to the loss function, which resulted in a suppression of spurious oscillations and a more accurate location and length of the predicted wavefronts. This model proved to accurately generalise to complex wall geometries not seen during training. As opposed to the image data-structures processed by CNNs, graphs offer higher freedom on how data is organised and processed. This motivated the use of graphs to represent the state of fluid-dynamic systems discretised by unstructured sets of nodes, and GNNs to process such graphs. Graphs have enabled more accurate representations of curvilinear geometries and higher resolution placement exclusively in areas where physics is more challenging to resolve. Two novel GNN architectures were designed for fluid-dynamics inference: the MuS-GNN, a multi-scale GNN, and the REMuS-GNN, a rotation-equivariant multi-scale GNN. Both architectures work by repeatedly passing messages from each node to its nearest nodes in the graph. Additionally, lower-resolutions graphs, with a reduced number of nodes, are defined from the original graph, and messages are also passed from finer to coarser graphs and vice-versa. The low-resolution graphs allowed for efficiently capturing physics encompassing a range of lengthscales. Advection and fluid flow, modelled by the incompressible Navier-Stokes equations, were the two types of problems used to assess the proposed GNNs. Whereas a single-scale GNN was sufficient to achieve high generalisation accuracy in advection simulations, flow simulation highly benefited from an increasing number of low-resolution graphs. The generalisation and long-term accuracy of these simulations were further improved by the REMuS-GNN architecture, which processes the system state independently of the orientation of the coordinate system thanks to a rotation-invariant representation and carefully designed components. To the best of the author’s knowledge, the REMuS-GNN architecture was the first rotation-equivariant and multi-scale GNN. The simulations were accelerated between one (in a CPU) and three (in a GPU) orders of magnitude with respect to a CPU-based numerical solver. Additionally, the parallelisation of multi-scale GNNs resulted in a close-to-linear speedup with the number of CPU cores or GPUs.Open Acces

    Hereditary classes of graphs : a parametric approach

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    The world of hereditary classes is rich and diverse and it contains a variety of classes of theoretical and practical importance. Thousands of results in the literature are devoted to individual classes and only a few of them analyse the universe of hereditary classes as a whole. To shift the analysis into a new level, in the present paper we exploit an approach, where we operate by infinite families of classes, rather than individual classes. Each family is associated with a graph parameter and is characterized by classes that are critical with respect to the parameter. In particular, we obtain a complete parametric description of the bottom of the lattice of hereditary classes and discuss a number of open questions related to this approach

    Chernoff Information in Community Detection

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    In network inference applications, it is desirable to detect community structure, i.e., cluster vertices into potential blocks. Beyond adjacency matrices, many real-world networks also involve vertex covariates that may carry information about underlying block structure. Since accurate inference on random networks depends on exploiting all available signal, we need scalable algorithms that can incorporate both network connectivity data and additional insight from vertex covariates. In addition, it can be prohibitively expensive to observe the entire graph in many real applications, especially for large graphs. Thus it becomes essential to identify vertices that have the most impact on block structure and only check whether there are edges between them given a limited budget. To assess the effects of vertex covariates on block recovery, we consider two model-based spectral algorithms. The first algorithm uses only the adjacency matrix, and directly estimates the block assignments. The second algorithm incorporates both the adjacency matrix and the vertex covariates into the estimation of block assignments. We employ Chernoff information to analytically compare the algorithms’ performance and derive the information-theoretic Chernoff ratio for certain models of interest. Analytic results and simulations suggest that the second algorithm is often preferred: one can better estimate the induced block assignments by first estimating the effect of vertex covariates. In addition, real data experiments also indicate that the second algorithm has the advantage of revealing underlying block structure while considering observed vertex heterogeneity in real applications. Moreover, we propose a dynamic network sampling scheme to optimize block recovery for stochastic blockmodel in the case where it is prohibitively expensive to observe the entire graph. Theoretically, we provide justification of our proposed Chernoff-optimal dynamic sampling scheme via the Chernoff information. Practically, we evaluate the performance of our method on several real datasets from different domains. Both theoretically and practically results suggest that our method can identify vertices that have the most impact on block structure so that one can only check whether there are edges between them to save significant resources but still recover the block structure

    Exotic Ground States and Dynamics in Constrained Systems

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    The overarching theme of this thesis is the question of how constraints influence collective behavior. Constraints are crucial in shaping both static and dynamic properties of systems across diverse areas within condensed matter physics and beyond. For example, the simple geometric constraint that hard particles cannot overlap at high density leads to slow dynamics and jamming in glass formers. Constraints also arise effectively at low temperature as a consequence of strong competing interactions in magnetic materials, where they give rise to emergent gauge theories and unconventional magnetic order. Enforcing constraints artificially in turn can be used to protect otherwise fragile quantum information from external noise. This thesis in particular contains progress on the realization of different unconventional phases of matter in constrained systems. The presentation of individual results is organized by the stage of realization of the respective phase. Novel physical phenomena after conceptualization are often exemplified in simple, heuristic models bearing little resemblance of actual matter, but which are interesting enough to motivate efforts with the final goal of realizing them in some way in the lab. One form of progress is then to devise refined models, which retain a degree of simplification while still realizing the same physics and improving the degree of realism in some direction. Finally, direct efforts in realizing either the original models or some refined version in experiment today are mostly two-fold. One route, having grown in importance rapidly during the last two decades, is via the engineering of artificial systems realizing suitable models. The other, more conventional way is to search for realizations of novel phases in materials. The thesis is divided into three parts, where Part I is devoted to the study of two simple models, while artificial systems and real materials are the subject of Part II and Part III respectively. Below, the content of each part is summarized in more detail. After a general introduction to entropic ordering and slow dynamics we present a family of models devised as a lattice analog of hard spheres. These are often studied to explore whether low-dimensional analogues of mean-field glass- and jamming transitions exist, but also serve as the canonical model systems for slow dynamics in granular materials more generally. Arguably the models in this family do not offer a close resemblance of actual granular materials. However, by studying their behavior far from equilibrium, we observe the onset of slow dynamics and a kinetic arrest for which, importantly, we obtain an essentially complete analytical and numerical understanding. Particularly interesting is the fact that this understanding hinges on the (in-)ability to anneal topological defects in the presence of a hardcore constraints, which resonates with some previous proposals for an understanding of the glass transition. As another example of anomalous dynamics arising in a magnetic system, we also present a detailed study of a two-dimensional fracton spin liquid. The model is an Ising system with an energy function designed to give rise to an emergent higher-rank gauge theory at low energy. We show explicitly that the number of zero-energy states in the model scales exponentially with the system size, establishing a finite residual entropy. A purpose-built cluster Monte-Carlo algorithm makes it possible to study the behavior of the model as a function of temperature. We show evidence for a first order transition from a high-temperature paramagnet to a low-temperature phase where correlations match predictions of a higher-rank coulomb phase. Turning away from heuristic models, the second part of the thesis begins with an introduction to quantum error correction, a scheme where constraints are artificially imposed in a quantum system through measurement and feedback. This is done in order to preserve quantum information in the presence of external noise, and is widely believed to be necessary in order to one day harness the full power of quantum computers. Given a certain error-correcting code as well as a noise model, a particularly interesting quantity is the threshold of the code, that is the critical amount of external noise below which quantum error correction becomes possible. For the toric code under independent bit- and phase-flip noise for example, the threshold is well known to map to the paramagnet to ferromagnet transition of the two-dimensional random-bond Ising model along the Nishimori line. Here, we present the first generalization of this mapping to a family of codes with finite rate, that is a family where the number of encoded logical qubits grows linearly with the number of physical qubits. In particular, we show that the threshold of hyperbolic surface codes maps to a paramagnet to ferromagnet transition in what we call the 'dual'' random-bond Ising model on regular tessellations of compact hyperbolic manifolds. This model is related to the usual random-bond Ising model by the Kramers-Wannier duality but distinct from it even on self-dual tessellations. As a corollary, we clarify long-standing issues regarding self-duality of the Ising model in hyperbolic space. The final part of the thesis is devoted to the study of material candidates of quantum spin ice, a three-dimensional quantum spin liquid. The work presented here was done in close collaboration with experiment and focuses on a particular family of materials called dipolar-octupolar pyrochlores. This family of materials is particularly interesting because they might realize novel exotic quantum states such as octupolar spin liquids, while at the same time being described by a relatively simple model Hamiltonian. This thesis contains a detailed study of ground state selection in dipolar-octupolar pyrochlore magnets and its signatures as observable in neutron scattering. First, we present evidence that the two compounds Ce2Zr2O7 and Ce2Sn2O7 despite their similar chemical composition realize an exotic quantum spin liquid state and an ordered state respectively. Then, we also study the ground-state selection in dipolar-octupolar pyrochlores in a magnetic field. Most importantly, we show that the well-known effective one-dimensional physics -- arising when the field is applied along a certain crystallographic axis -- is expected to be stable at experimentally relevant temperatures. Finally, we make predictions for neutron scattering in the large-field phase and compare these to measurements on Ce2Zr2O7

    BDS GNSS for Earth Observation

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    For millennia, human communities have wondered about the possibility of observing phenomena in their surroundings, and in particular those affecting the Earth on which they live. More generally, it can be conceptually defined as Earth observation (EO) and is the collection of information about the biological, chemical and physical systems of planet Earth. It can be undertaken through sensors in direct contact with the ground or airborne platforms (such as weather balloons and stations) or remote-sensing technologies. However, the definition of EO has only become significant in the last 50 years, since it has been possible to send artificial satellites out of Earth’s orbit. Referring strictly to civil applications, satellites of this type were initially designed to provide satellite images; later, their purpose expanded to include the study of information on land characteristics, growing vegetation, crops, and environmental pollution. The data collected are used for several purposes, including the identification of natural resources and the production of accurate cartography. Satellite observations can cover the land, the atmosphere, and the oceans. Remote-sensing satellites may be equipped with passive instrumentation such as infrared or cameras for imaging the visible or active instrumentation such as radar. Generally, such satellites are non-geostationary satellites, i.e., they move at a certain speed along orbits inclined with respect to the Earth’s equatorial plane, often in polar orbit, at low or medium altitude, Low Earth Orbit (LEO) and Medium Earth Orbit (MEO), thus covering the entire Earth’s surface in a certain scan time (properly called ’temporal resolution’), i.e., in a certain number of orbits around the Earth. The first remote-sensing satellites were the American NASA/USGS Landsat Program; subsequently, the European: ENVISAT (ENVironmental SATellite), ERS (European Remote-Sensing satellite), RapidEye, the French SPOT (Satellite Pour l’Observation de laTerre), and the Canadian RADARSAT satellites were launched. The IKONOS, QuickBird, and GeoEye-1 satellites were dedicated to cartography. The WorldView-1 and WorldView-2 satellites and the COSMO-SkyMed system are more recent. The latest generation are the low payloads called Small Satellites, e.g., the Chinese BuFeng-1 and Fengyun-3 series. Also, Global Navigation Satellite Systems (GNSSs) have captured the attention of researchers worldwide for a multitude of Earth monitoring and exploration applications. On the other hand, over the past 40 years, GNSSs have become an essential part of many human activities. As is widely noted, there are currently four fully operational GNSSs; two of these were developed for military purposes (American NAVstar GPS and Russian GLONASS), whilst two others were developed for civil purposes such as the Chinese BeiDou satellite navigation system (BDS) and the European Galileo. In addition, many other regional GNSSs, such as the South Korean Regional Positioning System (KPS), the Japanese quasi-zenital satellite system (QZSS), and the Indian Regional Navigation Satellite System (IRNSS/NavIC), will become available in the next few years, which will have enormous potential for scientific applications and geomatics professionals. In addition to their traditional role of providing global positioning, navigation, and timing (PNT) information, GNSS navigation signals are now being used in new and innovative ways. Across the globe, new fields of scientific study are opening up to examine how signals can provide information about the characteristics of the atmosphere and even the surfaces from which they are reflected before being collected by a receiver. EO researchers monitor global environmental systems using in situ and remote monitoring tools. Their findings provide tools to support decision makers in various areas of interest, from security to the natural environment. GNSS signals are considered an important new source of information because they are a free, real-time, and globally available resource for the EO community
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