In many applications that deal with high dimensional data, it is important to not store the high dimensional object itself, but its representation in a data sparse way. This aims to reduce the storage and computational complexity. There is a general scheme for representing tensors with the help of sums of elementary tensors, where the summation structure is defined by a graph/network. This scheme allows to generalize commonly used approaches in representing a large amount of numerical data (that can be interpreted as a high dimensional object) using sums of elementary tensors. The classification does not only distinguish between elementary tensors and non-elementary tensors, but also describes the number of terms that is needed to represent an object of the tensor space. This classification is referred to as tensor network (format). This work uses the tensor network based approach and describes non-linear block Gauss-Seidel methods (ALS and DMRG) in the context of the general tensor network framework. Another contribution of the thesis is the general conversion of different tensor formats. We are able to efficiently change the underlying graph topology of a given tensor representation while using the similarities (if present) of both the original and the desired structure. This is an important feature in case only minor structural changes are required. In all approximation cases involving iterative methods, it is crucial to find and use a proper initial guess. For linear iteration schemes, a good initial guess helps to decrease the number of iteration steps that are needed to reach a certain accuracy, but it does not change the approximation result. For non-linear iteration schemes, the approximation result may depend on the initial guess. This work introduces a method to successively create an initial guess that improves some approximation results. This algorithm is based on successive rank 1 increments for the r-term format. There are still open questions about how to find the optimal tensor format for a given general problem (e.g. storage, operations, etc.). For instance in the case where a physical background is given, it might be efficient to use this knowledge to create a good network structure. There is however, no guarantee that a better (with respect to the problem) representation structure does not exist.

<p>Knowledge base completion is an important task that allows scientists to reason over knowledge bases and discover new facts. In this thesis, a patient-centric knowledge base</p> <p>is designed and constructed using medical entities and relations extracted from the health forum r/cancer. The knowledge base stores information in binary relation triplets. It is enhanced with an is-a relation that is able to represent the hierarchical relationship between different medical entities. An enhanced Neural Tensor Network that utilizes the frequency of occurrence of relation triplets in the dataset is then developed to infer new facts from</p> <p>the enhanced knowledge base. The results show that when the enhanced inference model uses the enhanced knowledge base, a higher accuracy (73.2 %) and recall@10 (35.4%) are obtained. In addition, this thesis describes a methodology for knowledge base and associated</p> <p>inference model design that can be applied to other chronic diseases.</p>

Phase transitions are fundamental phenomena in (quantum) many-body systems. They are associated with changes in the macroscopic physical properties of the system in response to the alteration in the conditions controlled by one or more parameters, like temperature or coupling constants. Quantum phase transitions are particularly intriguing as they reveal new insights into the fundamental nature of matter and the laws of physics. The study of phase transitions in such systems is crucial in aiding our understanding of how materials behave in extreme conditions, which are difficult to replicate in laboratory, and also the behavior of exotic states of matter with unique and potentially useful properties like superconductors and superfluids. Moreover, this understanding has other practical applications and can lead to the development of new materials with specific properties or more efficient technologies, such as quantum computers. Hence, detecting the transition point from one phase of matter to another and constructing the corresponding phase diagram is of great importance for examining many-body systems and predicting their response to external perturbations. Traditionally, phase transitions have been identified either through analytical methods like mean field theory or numerical simulations. The pinpointing of the critical value normally involves the measure of specific quantities such as local observables, correlation functions, energy gaps, etc. reflecting the changes in the physics through the transition. However, the latter approach requires prior knowledge of the system to calculate the order parameter of the transition, which is uniquely associated to its universality class. Recently, another method has gained more and more attention in the physics community. By using raw and very general representative data of the system, one can resort to machine learning techniques to distinguish among patterns within the data belonging to different phases. The relevance of these techniques is rooted in the ability of a properly trained machine to efficiently process complex data for the sake of pursuing classification tasks, pattern recognition, generating brand new data and even developing decision processes. The aim of this thesis is to explore phase transitions from this new and promising data-centric perspective. On the one hand, our work is focused on the developement of new machine learning architectures using state-of-the-art and interpretable models. On the other hand, we are interested in the study of the various possible data which can be fed to the artificial intelligence model for the mapping of a quantum many-body system phase diagram. Our analysis is supported by numerical examples obtained via matrix-product-states (MPS) simulations for several one-dimensional zero-temperature systems on a lattice such as the XXZ model, the Extended Bose-Hubbard model (EBH) and the two-species Bose Hubbard model (BH2S). In Part I, we provide a general introduction to the background concepts for the understanding of the physics and the numerical methods used for the simulations and the analysis with deep learning. In Part II, we first present the models of the quantum many-body systems that we study. Then, we discuss the machine learning protocol to identify phase transitions, namely anomaly detection technique, that involves the training of a model on a dataset of normal behavior and use it to recognize deviations from this behavior on test data. The latter can be applied for our purpose by training in a known phase so that, at test-time, all the other phases of the system are marked as anomalies. Our method is based on Generative Adversarial Networks (GANs) and improves the networks adopted by the previous works in the literature for the anomaly detection scheme taking advantage of the adversarial training procedure. Specifically, we train the GAN on a dataset composed of bipartite entanglement spectra (ES) obtained from Tensor Network simulations for the three aforementioned quantum systems. We focus our study on the detection of the elusive Berezinskii-Kosterlitz-Thouless (BKT) transition that have been object of intense theoretical and experimental studies since its first prediction for the classical two-dimensional XY model. The absence of an explicit symmetry breaking and its gappless-to-gapped nature which characterize such a transition make the latter very subtle to be detected, hence providing a challenging testing ground for the machine-driven method. We train the GAN architecture on the ES data in the gapless side of BKT transition and we show that the GAN is able to automatically distinguish between data from the same phase and beyond the BKT. The protocol that we develop is not supposed to become a substitute to the traditional methods for the phase transitions detection but allows to obtain a qualitative map of a phase diagram with almost no prior knowledge about the nature and the arrangement of the phases -- in this sense we refer to it as agnostic -- in an automatic fashion. Furthermore, it is very general and it can be applied in principle to all kind of representative data of the system coming both from experiments and numerics, as long as they have different patterns (even hidden to the eye) in different phases. Since the kind of data is crucially linked with the success of the detection, together with the ES we investigate another candidate: the probability density function (PDF) of a globally U(1) conserved charge in an extensive sub-portion of the system. The full PDF is one of the possible reductions of the ES which is known to exhibit relations and degeneracies reflecting very peculiar aspects of the physics and the symmetries of the system. Its patterns are often used to tell different kinds of phases apart and embed information about non-local quantum correlations. However, the PDF is measurable, e.g. in quantum gas microscopes experiments, and it is quite general so that it can be considered not only in the cases of the study but also in other systems with different symmetries and dimensionalities. Both the ES and the PDF can be extracted from the simulation of the ground state by dividing the one-dimensional chain into two complementary subportions. For the EBH we calculate the PDF of the bosonic occupation number in a wide range of values of the couplings and we are able to reproduce the very rich phase diagram containing several phases (superfluid, Mott insulator, charge density wave, phase separation of supersolid and superfluid and the topological Haldane insulator) just with an educated gaussian fit of the PDF. Even without resorting to machine learning, this analysis is instrumental to show the importance of the experimentally accessible PDF for the task. Moreover, we highlight some of its properties according to the gapless and gapped nature of the ground state which require a further investigation and extension beyond zero-temperature regimes and one-dimensional systems. The last chapter of the results contains the description of another architecture, namely the Concrete Autoencoder (CAE) which can be used for detecting phase transitions with the anomaly detection scheme while being able to automatically learn what the most relevant components of the input data are. We show that the CAE can recognize the important eigenvalues out of the entire ES for the EBH model in order to characterize the gapless phase. Therefore the latter architecture can be used to provide not only a more compact version of the input data (dimensionality reduction) -- which can improve the training -- but also some meaningful insights in the spirit of machine learning interpretability. In conclusion, in this thesis we describe two advances in the solution to the problem of phase recognition in quantum many-body systems. On one side, we improve the literature standard anomaly detection protocol for an automatic and agnostic identification of the phases by employing the GAN network. Moreover, we implement and test an explainable model which can make the interpretation of the results easier. On the other side we put the focus on the PDF as a new candidate quantity for the scope of discerning phases of matter. We show that it contains a lot of information about the many-body state being very general and experimentally accessible.

We study non-equilibrium and strongly-correlated dynamics in two contexts. We begin by analysing quantum many-body systems out of equilibrium through the lens of cold atomic impurities in Bose gases. Such highly-imbalanced mixtures provide a controlled arena for the study of interactions, dissipation, decoherence and transport in a many-body quantum environment. Specifically we investigate the oscillatory dynamics of a trapped and initially highly-localised impurity interacting with a weakly-interacting trapped quasi low-dimensional Bose gas. This relates to and goes beyond a recent experiment by the Inguscio group in Florence. We witness a delicate interplay between the self-trapping of the impurity and the inhomogeneity of the Bose gas, and describe the dissipation of the energy of the impurity through phononic excitations of the Bose gas. We then study the transport of a driven, periodically-trapped impurity through a quasi one-dimensional Bose gas. We show that placing the weakly-interacting Bose gas in a separate periodic potential leads to a phononic excitation spectrum that closely mimics those in solid state systems. As a result we show that the impurity-Bose gas system exhibits phonon-induced resonances in the impurity current that were predicted to occur in solids decades ago but never clearly observed. Following this, allowing the bosons to interact strongly, we predict the effect of different strongly-correlated phases of the Bose gas on the motion of the impurity. We show that, by observing the impurity, properties of the excitation spectrum of the Bose gas, e.g., gap and bandwidth, may be inferred along with the filling of the bosonic lattice. In other words the impurity acts as a probe of its environment. To describe the dynamics of such a strongly-correlated system we use the powerful and near-exact time-evolving block decimation (TEBD) method, which we describe in detail. The second part of this thesis then analyses, for the first time, the performance of this method when applied to simulate non-equilibrium classical stochastic processes. We study its efficacy for a well-understood model of transport, the totally-asymmetric exclusion process, and find it to be accurate. Next, motivated by the inefficiency of sampling-based numerical methods for high variance observables we adapt and apply TEBD to simulate a path-dependent observable whose variance increases exponentially with system size. Specifically we calculate the expected value of the exponential of the work done by a varying magnetic field on a one-dimensional Ising model undergoing Glauber dynamics. We confirm using Jarzynski's equality that the TEBD method remains accurate and efficient. Therefore TEBD and related methods complement and challenge the usual Monte Carlo-based simulators of non-equilibrium stochastic processes.