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Quantum networks crucially rely on the availability of high-quality entangled pairs of qubits, known as entangled links, distributed across distant nodes. Maintaining the quality of these links is a challenging task due to the presence of time-dependent noise, also known as decoherence. Entanglement purification protocols offer a solution by converting multiple low-quality entangled states into a smaller number of higher-quality ones. In this work, we introduce a framework to analyse the performance of entanglement buffering setups that combine entanglement consumption, decoherence, and entanglement purification. We propose two key metrics: the availability, which is the steady-state probability that an entangled link is present, and the average consumed fidelity, which quantifies the steady-state quality of consumed links. We then investigate a two-node system, where each node possesses two quantum memories: one for long-term entanglement storage, and another for entanglement generation. We model this setup as a continuous-time stochastic process and derive analytical expressions for the performance metrics. Our findings unveil a trade-off between the availability and the average consumed fidelity. We also bound these performance metrics for a buffering system that employs the well-known bilocal Clifford purification protocols. Importantly, our analysis demonstrates that, in the presence of noise, consistently purifying the buffered entanglement increases the average consumed fidelity, even when some buffered entanglement is discarded due to purification failures.

The diffusion of AI and big data is reshaping decision-making processes by increasing the amount of information that supports decisions while reducing direct interaction with data and empirical evidence. This paradigm shift introduces new sources of uncertainty, as limited data observability results in ambiguity and a lack of interpretability. The need for the proper analysis of data-driven strategies motivates the search for new models that can describe this type of bounded access to knowledge. This contribution presents a novel theoretical model for uncertainty in knowledge representation and its transfer mediated by agents. We provide a dynamical description of knowledge states by endowing our model with a structure to compare and combine them. Specifically, an update is represented through combinations, and its explainability is based on its consistency in different dimensional representations. We look at inequivalent knowledge representations in terms of multiplicity of inferences, preference relations, and information measures. Furthermore, we define a formal analogy with two scenarios that illustrate non-classical uncertainty in terms of ambiguity (Ellsberg's model) and reasoning about knowledge mediated by other agents observing data (Wigner's friend). Finally, we discuss some implications of the proposed model for data-driven strategies, with special attention to reasoning under uncertainty about business value dimensions and the design of measurement tools for their assessment.

Complex and nonlinear dynamical systems often involve parameters that change with time, accurate tracking of which is essential to tasks such as state estimation, prediction, and control. Existing machine-learning methods require full state observation of the underlying system and tacitly assume adiabatic changes in the parameter. Formulating an inverse problem and exploiting reservoir computing, we develop a model-free and fully data-driven framework to accurately track time-varying parameters from partial state observation in real time. In particular, with training data from a subset of the dynamical variables of the system for a small number of known parameter values, the framework is able to accurately predict the parameter variations in time. Low- and high-dimensional, Markovian and non-Markovian nonlinear dynamical systems are used to demonstrate the power of the machine-learning based parameter-tracking framework. Pertinent issues affecting the tracking performance are addressed.

Unsupervised deep learning approaches have recently become one of the crucial research areas in imaging owing to their ability to learn expressive and powerful reconstruction operators even when paired high-quality training data is scarcely available. In this chapter, we review theoretically principled unsupervised learning schemes for solving imaging inverse problems, with a particular focus on methods rooted in optimal transport and convex analysis. We begin by reviewing the optimal transport-based unsupervised approaches such as the cycle-consistency-based models and learned adversarial regularization methods, which have clear probabilistic interpretations. Subsequently, we give an overview of a recent line of works on provably convergent learned optimization algorithms applied to accelerate the solution of imaging inverse problems, alongside their dedicated unsupervised training schemes. We also survey a number of provably convergent plug-and-play algorithms (based on gradient-step deep denoisers), which are among the most important and widely applied unsupervised approaches for imaging problems. At the end of this survey, we provide an overview of a few related unsupervised learning frameworks that complement our focused schemes. Together with a detailed survey, we provide an overview of the key mathematical results that underlie the methods reviewed in the chapter to keep our discussion self-contained.

A simple way of obtaining robust estimates of the "center" (or the "location") and of the "scatter" of a dataset is to use the maximum likelihood estimate with a class of heavy-tailed distributions, regardless of the "true" distribution generating the data. We observe that the maximum likelihood problem for the Cauchy distributions, which have particularly heavy tails, is geodesically convex and therefore efficiently solvable (Cauchy distributions are parametrized by the upper half plane, i.e. by the hyperbolic plane). Moreover, it has an appealing geometrical meaning: the datapoints, living on the boundary of the hyperbolic plane, are attracting the parameter by unit forces, and we search the point where these forces are in equilibrium. This picture generalizes to several classes of multivariate distributions with heavy tails, including, in particular, the multivariate Cauchy distributions. The hyperbolic plane gets replaced by symmetric spaces of noncompact type. Geodesic convexity gives us an efficient numerical solution of the maximum likelihood problem for these distribution classes. This can then be used for robust estimates of location and spread, thanks to the heavy tails of these distributions.

We present a formulation for high-order generalized periodicity conditions in the context of a high-order electromechanical theory including flexoelectricity, strain gradient elasticity and gradient dielectricity, with the goal of studying periodic architected metamaterials. Such theory results in fourth-order governing partial differential equations, and the periodicity conditions involve continuity across the periodic boundary of primal fields (displacement and electric potential) and their normal derivatives, continuity of the corresponding dual generalized forces (tractions, double tractions, surface charge density and double surface charge density). Rather than imposing these conditions numerically as explicit constraints, we develop an approximation space which fulfils generalized periodicity by construction. Our method naturally allows us to impose general macroscopic fields (strains/stresses and electric fields/electric displacements) along arbitrary directions, enabling the characterization of the material anisotropy. We apply the proposed method to study periodic architected metamaterials with apparent piezoelectricity. We first verify the method by directly comparing the results with a large periodic structure, then apply it to evaluate the anisotropic apparently piezoelectricity of a geometrically polarized 2D lattice, and finally demonstrate the application of the method in a 3D architected metamaterial.

Feedforward neural networks (FNNs) are typically viewed as pure prediction algorithms, and their strong predictive performance has led to their use in many machine-learning applications. However, their flexibility comes with an interpretability trade-off; thus, FNNs have been historically less popular among statisticians. Nevertheless, classical statistical theory, such as significance testing and uncertainty quantification, is still relevant. Supplementing FNNs with methods of statistical inference, and covariate-effect visualisations, can shift the focus away from black-box prediction and make FNNs more akin to traditional statistical models. This can allow for more inferential analysis, and, hence, make FNNs more accessible within the statistical-modelling context.

Quantum supervised learning, utilizing variational circuits, stands out as a promising technology for NISQ devices due to its efficiency in hardware resource utilization during the creation of quantum feature maps and the implementation of hardware-efficient ansatz with trainable parameters. Despite these advantages, the training of quantum models encounters challenges, notably the barren plateau phenomenon, leading to stagnation in learning during optimization iterations. This study proposes an innovative approach: an evolutionary-enhanced ansatz-free supervised learning model. In contrast to parametrized circuits, our model employs circuits with variable topology that evolves through an elitist method, mitigating the barren plateau issue. Additionally, we introduce a novel concept, the superposition of multi-hot encodings, facilitating the treatment of multi-classification problems. Our framework successfully avoids barren plateaus, resulting in enhanced model accuracy. Comparative analysis with variational quantum classifiers from the technology's state-of-the-art reveal a substantial improvement in training efficiency and precision. Furthermore, we conduct tests on a challenging dataset class, traditionally problematic for conventional kernel machines, demonstrating a potential alternative path for achieving quantum advantage in supervised learning for NISQ era.

Knowledge graphs (KGs) of real-world facts about entities and their relationships are useful resources for a variety of natural language processing tasks. However, because knowledge graphs are typically incomplete, it is useful to perform knowledge graph completion or link prediction, i.e. predict whether a relationship not in the knowledge graph is likely to be true. This paper serves as a comprehensive survey of embedding models of entities and relationships for knowledge graph completion, summarizing up-to-date experimental results on standard benchmark datasets and pointing out potential future research directions.

Recent advances in 3D fully convolutional networks (FCN) have made it feasible to produce dense voxel-wise predictions of volumetric images. In this work, we show that a multi-class 3D FCN trained on manually labeled CT scans of several anatomical structures (ranging from the large organs to thin vessels) can achieve competitive segmentation results, while avoiding the need for handcrafting features or training class-specific models. To this end, we propose a two-stage, coarse-to-fine approach that will first use a 3D FCN to roughly define a candidate region, which will then be used as input to a second 3D FCN. This reduces the number of voxels the second FCN has to classify to ~10% and allows it to focus on more detailed segmentation of the organs and vessels. We utilize training and validation sets consisting of 331 clinical CT images and test our models on a completely unseen data collection acquired at a different hospital that includes 150 CT scans, targeting three anatomical organs (liver, spleen, and pancreas). In challenging organs such as the pancreas, our cascaded approach improves the mean Dice score from 68.5 to 82.2%, achieving the highest reported average score on this dataset. We compare with a 2D FCN method on a separate dataset of 240 CT scans with 18 classes and achieve a significantly higher performance in small organs and vessels. Furthermore, we explore fine-tuning our models to different datasets. Our experiments illustrate the promise and robustness of current 3D FCN based semantic segmentation of medical images, achieving state-of-the-art results. Our code and trained models are available for download: //github.com/holgerroth/3Dunet_abdomen_cascade.