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We develop a new, unsupervised symmetry learning method that starts with raw data, and gives the minimal (discrete) generator of an underlying Lie group of symmetries, together with a symmetry equivariant representation of the data. The method is able to learn the pixel translation operator from a dataset with only an approximate translation symmetry, and can learn quite different types of symmetries which are not apparent to the naked eye, equally well. The method is based on the formulation of an information-theoretic loss function that measures both the degree to which the dataset is symmetric under a given candidate symmetry, and also, the degree of locality of the samples in the dataset with respect to this symmetry. We demonstrate that this coupling between symmetry and locality, together with a special optimization technique developed for entropy estimation, results in a highly stable system that gives reproducible results. The symmetry actions we consider are group representations, however, we believe the approach has the potential to be generalized to more general, nonlinear actions of non-commutative Lie groups.

The aim of this paper is to present three construction methods for quasi-copulas based on recent developments: a representation of multivariate quasi-copulas by means of infima and suprema of copulas, an extension of a classical result on shuffles of min to the setting of quasi-copulas, and a construction method for quasi-copulas obeying a given signed mass pattern on a patch.

To analyze the topological properties of the given discrete data, one needs to consider a continuous transform called filtration. Persistent homology serves as a tool to track changes of homology in the filtration. The outcome of the topological analysis of data varies depending on the choice of filtration, making the selection of filtration crucial. Filtration learning is an attempt to find an optimal filtration that minimizes the loss function. Exact Multi-parameter Persistent Homology (EMPH) has been recently proposed, particularly for topological time-series analysis, that utilizes the exact formula of rank invariant instead of calculating it. In this paper, we propose a framework for filtration learning of EMPH. We formulate an optimization problem and propose an algorithm for solving the problem. We then apply the proposed algorithm to several classification problems. Particularly, we derive the exact formula of the gradient of the loss function with respect to the filtration parameters, which makes it possible to directly update the filtration without using automatic differentiation, significantly enhancing the learning process.

Modeling the complex relationships between multiple categorical response variables as a function of predictors is a fundamental task in the analysis of categorical data. However, existing methods can be difficult to interpret and may lack flexibility. To address these challenges, we introduce a penalized likelihood method for multivariate categorical response regression that relies on a novel subspace decomposition to parameterize interpretable association structures. Our approach models the relationships between categorical responses by identifying mutual, joint, and conditionally independent associations, which yields a linear problem within a tensor product space. We establish theoretical guarantees for our estimator, including error bounds in high-dimensional settings, and demonstrate the method's interpretability and prediction accuracy through comprehensive simulation studies.

We develop confidence sets which provide spatial uncertainty guarantees for the output of a black-box machine learning model designed for image segmentation. To do so we adapt conformal inference to the imaging setting, obtaining thresholds on a calibration dataset based on the distribution of the maximum of the transformed logit scores within and outside of the ground truth masks. We prove that these confidence sets, when applied to new predictions of the model, are guaranteed to contain the true unknown segmented mask with desired probability. We show that learning appropriate score transformations on a learning dataset before performing calibration is crucial for optimizing performance. We illustrate and validate our approach on a polpys tumor dataset. To do so we obtain the logit scores from a deep neural network trained for polpys segmentation and show that using distance transformed scores to obtain outer confidence sets and the original scores for inner confidence sets enables tight bounds on tumor location whilst controlling the false coverage rate.

Driven by the need to generate real-world evidence from multi-site collaborative studies, we introduce an efficient collaborative learning approach to evaluate average treatment effect in a multi-site setting under data sharing constraints. Specifically, the proposed method operates in a federated manner, using individual-level data from a user-defined target population and summary statistics from other source populations, to construct efficient estimator for the average treatment effect on the target population of interest. Our federated approach does not require iterative communications between sites, making it particularly suitable for research consortia with limited resources for developing automated data-sharing infrastructures. Compared to existing work data integration methods in causal inference, it allows distributional shifts in outcomes, treatments and baseline covariates distributions, and achieves semiparametric efficiency bound under appropriate conditions. We illustrate the magnitude of efficiency gains from incorporating extra data sources by examining the effect of insulin vs. non-insulin treatments on heart failure for patients with type II diabetes using electronic health record data collected from the All of Us program.

We propose a new step-wise approach to proving observational equivalence, and in particular reasoning about fragility of observational equivalence. Our approach is based on what we call local reasoning. The local reasoning exploits the graphical concept of neighbourhood, and it extracts a new, formal, concept of robustness as a key sufficient condition of observational equivalence. Moreover, our proof methodology is capable of proving a generalised notion of observational equivalence. The generalised notion can be quantified over syntactically restricted contexts instead of all contexts, and also quantitatively constrained in terms of the number of reduction steps. The operational machinery we use is given by a hypergraph-rewriting abstract machine inspired by Girard's Geometry of Interaction. The behaviour of language features, including function abstraction and application, is provided by hypergraph-rewriting rules. We demonstrate our proof methodology using the call-by-value lambda-calculus equipped with (higher-order) state.

This dissertation studies a fundamental open challenge in deep learning theory: why do deep networks generalize well even while being overparameterized, unregularized and fitting the training data to zero error? In the first part of the thesis, we will empirically study how training deep networks via stochastic gradient descent implicitly controls the networks' capacity. Subsequently, to show how this leads to better generalization, we will derive {\em data-dependent} {\em uniform-convergence-based} generalization bounds with improved dependencies on the parameter count. Uniform convergence has in fact been the most widely used tool in deep learning literature, thanks to its simplicity and generality. Given its popularity, in this thesis, we will also take a step back to identify the fundamental limits of uniform convergence as a tool to explain generalization. In particular, we will show that in some example overparameterized settings, {\em any} uniform convergence bound will provide only a vacuous generalization bound. With this realization in mind, in the last part of the thesis, we will change course and introduce an {\em empirical} technique to estimate generalization using unlabeled data. Our technique does not rely on any notion of uniform-convergece-based complexity and is remarkably precise. We will theoretically show why our technique enjoys such precision. We will conclude by discussing how future work could explore novel ways to incorporate distributional assumptions in generalization bounds (such as in the form of unlabeled data) and explore other tools to derive bounds, perhaps by modifying uniform convergence or by developing completely new tools altogether.

Artificial neural networks thrive in solving the classification problem for a particular rigid task, acquiring knowledge through generalized learning behaviour from a distinct training phase. The resulting network resembles a static entity of knowledge, with endeavours to extend this knowledge without targeting the original task resulting in a catastrophic forgetting. Continual learning shifts this paradigm towards networks that can continually accumulate knowledge over different tasks without the need to retrain from scratch. We focus on task incremental classification, where tasks arrive sequentially and are delineated by clear boundaries. Our main contributions concern 1) a taxonomy and extensive overview of the state-of-the-art, 2) a novel framework to continually determine the stability-plasticity trade-off of the continual learner, 3) a comprehensive experimental comparison of 11 state-of-the-art continual learning methods and 4 baselines. We empirically scrutinize method strengths and weaknesses on three benchmarks, considering Tiny Imagenet and large-scale unbalanced iNaturalist and a sequence of recognition datasets. We study the influence of model capacity, weight decay and dropout regularization, and the order in which the tasks are presented, and qualitatively compare methods in terms of required memory, computation time, and storage.

Machine-learning models have demonstrated great success in learning complex patterns that enable them to make predictions about unobserved data. In addition to using models for prediction, the ability to interpret what a model has learned is receiving an increasing amount of attention. However, this increased focus has led to considerable confusion about the notion of interpretability. In particular, it is unclear how the wide array of proposed interpretation methods are related, and what common concepts can be used to evaluate them. We aim to address these concerns by defining interpretability in the context of machine learning and introducing the Predictive, Descriptive, Relevant (PDR) framework for discussing interpretations. The PDR framework provides three overarching desiderata for evaluation: predictive accuracy, descriptive accuracy and relevancy, with relevancy judged relative to a human audience. Moreover, to help manage the deluge of interpretation methods, we introduce a categorization of existing techniques into model-based and post-hoc categories, with sub-groups including sparsity, modularity and simulatability. To demonstrate how practitioners can use the PDR framework to evaluate and understand interpretations, we provide numerous real-world examples. These examples highlight the often under-appreciated role played by human audiences in discussions of interpretability. Finally, based on our framework, we discuss limitations of existing methods and directions for future work. We hope that this work will provide a common vocabulary that will make it easier for both practitioners and researchers to discuss and choose from the full range of interpretation methods.