We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology. Numerically we demonstrate the effectiveness of the method on a class of parametric elliptic PDE problems, showing convergence and robustness of the approximation scheme with respect to the size of the discretization, and compare our method with existing algorithms from the literature.

Many data sets contain an inherent multilevel structure, for example, because of repeated measurements of the same observational units. Taking this structure into account is critical for the accuracy and calibration of any statistical analysis performed on such data. However, the large number of possible model configurations hinders the use of multilevel models in practice. In this work, we propose a flexible framework for efficiently assessing differences between the levels of given grouping variables in the data. The assessed group heterogeneity is valuable in choosing the relevant group coefficients to consider in a multilevel model. Our empirical evaluations demonstrate that the framework can reliably identify relevant multilevel components in both simulated and real data sets.

Online multiple kernel learning (OMKL) has provided an attractive performance in nonlinear function learning tasks. Leveraging a random feature approximation, the major drawback of OMKL, known as the curse of dimensionality, has been recently alleviated. In this paper, we introduce a new research problem, termed (stream-based) active multiple kernel learning (AMKL), in which a learner is allowed to label selected data from an oracle according to a selection criterion. This is necessary in many real-world applications as acquiring true labels is costly or time-consuming. We prove that AMKL achieves an optimal sublinear regret, implying that the proposed selection criterion indeed avoids unuseful label-requests. Furthermore, we propose AMKL with an adaptive kernel selection (AMKL-AKS) in which irrelevant kernels can be excluded from a kernel dictionary 'on the fly'. This approach can improve the efficiency of active learning as well as the accuracy of a function approximation. Via numerical tests with various real datasets, it is demonstrated that AMKL-AKS yields a similar or better performance than the best-known OMKL, with a smaller number of labeled data.

We explore the training of generative adversarial networks with differential privacy to anonymize image data sets. On MNIST, we numerically measure the privacy-utility trade-off using parameters from $\epsilon$-$\delta$ differential privacy and the inception score. Our experiments uncover a saturated training regime where an increasing privacy budget adds little to the quality of generated images. We also explain analytically why differentially private Adam optimization is independent of the gradient clipping parameter. Furthermore, we highlight common errors in previous works on differentially private deep learning, which we uncovered in recent literature. Throughout the treatment of the subject, we hope to prevent erroneous estimates of anonymity in the future.

Neural Ordinary Differential Equations (ODEs) are elegant reinterpretations of deep networks where continuous time can replace the discrete notion of depth, ODE solvers perform forward propagation, and the adjoint method enables efficient, constant memory backpropagation. Neural ODEs are universal approximators only when they are non-autonomous, that is, the dynamics depends explicitly on time. We propose a novel family of Neural ODEs with time-varying weights, where time-dependence is non-parametric, and the smoothness of weight trajectories can be explicitly controlled to allow a tradeoff between expressiveness and efficiency. Using this enhanced expressiveness, we outperform previous Neural ODE variants in both speed and representational capacity, ultimately outperforming standard ResNet and CNN models on select image classification and video prediction tasks.

Application and use of deep learning algorithms for different healthcare applications is gaining interest at a steady pace. However, use of such algorithms can prove to be challenging as they require large amounts of training data that capture different possible variations. This makes it difficult to use them in a clinical setting since in most health applications researchers often have to work with limited data. Less data can cause the deep learning model to over-fit. In this paper, we ask how can we use data from a different environment, different use-case, with widely differing data distributions. We exemplify this use case by using single-sensor accelerometer data from healthy subjects performing activities of daily living - ADLs (source dataset), to extract features relevant to multi-sensor accelerometer gait data (target dataset) for Parkinson's disease classification. We train the pre-trained model using the source dataset and use it as a feature extractor. We show that the features extracted for the target dataset can be used to train an effective classification model. Our pre-trained source model consists of a convolutional autoencoder, and the target classification model is a simple multi-layer perceptron model. We explore two different pre-trained source models, trained using different activity groups, and analyze the influence the choice of pre-trained model has over the task of Parkinson's disease classification.

Ridge regression (RR) is a regularization technique that penalizes the L2-norm of the coefficients in linear regression. One of the challenges of using RR is the need to set a hyperparameter ($\alpha$) that controls the amount of regularization. Cross-validation is typically used to select the best $\alpha$ from a set of candidates. However, efficient and appropriate selection of $\alpha$ can be challenging, particularly where large amounts of data are analyzed. Because the selected $\alpha$ depends on the scale of the data and predictors, it is not straightforwardly interpretable. Here, we propose to reparameterize RR in terms of the ratio $\gamma$ between the L2-norms of the regularized and unregularized coefficients. This approach, called fractional RR (FRR), has several benefits: the solutions obtained for different $\gamma$ are guaranteed to vary, guarding against wasted calculations, and automatically span the relevant range of regularization, avoiding the need for arduous manual exploration. We provide an algorithm to solve FRR, as well as open-source software implementations in Python and MATLAB (https://github.com/nrdg/fracridge). We show that the proposed method is fast and scalable for large-scale data problems, and delivers results that are straightforward to interpret and compare across models and datasets.

The COVID-19 can cause severe pneumonia and is estimated to have a high impact on the healthcare system. The standard image diagnosis tests for pneumonia are chest X-ray (CXR) and computed tomography (CT) scan. CXR are useful in because it is cheaper, faster and more widespread than CT. This study aims to identify pneumonia caused by COVID-19 from other types and also healthy lungs using only CXR images. In order to achieve the objectives, we have proposed a classification schema considering the multi-class and hierarchical perspectives, since pneumonia can be structured as a hierarchy. Given the natural data imbalance in this domain, we also proposed the use of resampling algorithms in order to re-balance the classes distribution. Our classification schema extract features using some well-known texture descriptors and also using a pre-trained CNN model. We also explored early and late fusion techniques in order to leverage the strength of multiple texture descriptors and base classifiers at once. To evaluate the approach, we composed a database, named RYDLS-20, containing CXR images of pneumonia caused by different pathogens as well as CXR images of healthy lungs. The classes distribution follows a real-world scenario in which some pathogens are more common than others. The proposed approach achieved a macro-avg F1-Score of 0.65 using a multi-class approach and a F1-Score of 0.89 for the COVID-19 identification in the hierarchical classification scenario. As far as we know, we achieved the best nominal rate obtained for COVID-19 identification in an unbalanced environment with more than three classes. We must also highlight the novel proposed hierarchical classification approach for this task, which considers the types of pneumonia caused by the different pathogens and lead us to the best COVID-19 recognition rate obtained here.

In this paper, we show how sparse or isoperimetric cuts of a probability density function relate to Cheeger cuts of its principal eigenfunction, for appropriate definitions of `sparse cut' and `principal eigenfunction'. We construct these appropriate definitions of sparse cut and principal eigenfunction in the probability density setting. Then, we prove Cheeger and Buser type inequalities similar to those for the normalized graph Laplacian of Alon-Milman. We demonstrate that no such inequalities hold for most prior definitions of sparse cut and principal eigenfunction. We apply this result to generate novel algorithms for cutting probability densities and clustering data, including a principled variant of spectral clustering.

Crowdsourcing is a popular approach to collect annotations for unlabeled data instances. It involves collecting a large number of annotations from several, often naive untrained annotators for each data instance which are then combined to estimate the ground truth. Further, annotations for constructs such as affect are often multi-dimensional with annotators rating multiple dimensions, such as valence and arousal, for each instance. Most annotation fusion schemes however ignore this aspect and model each dimension separately. In this work we address this by proposing a generative model for multi-dimensional annotation fusion, which models the dimensions jointly leading to more accurate ground truth estimates. The model we propose is applicable to both global and time series annotation fusion problems and treats the ground truth as a latent variable distorted by the annotators. The model parameters are estimated using the Expectation-Maximization algorithm and we evaluate its performance using synthetic data and real emotion corpora as well as on an artificial task with human annotations