Many mathematical imaging problems are posed as non-convex optimization problems. When numerically tractable global optimization procedures are not available, one is often interested in testing ex post facto whether or not a locally convergent algorithm has found the globally optimal solution. If the problem has a statistical maximum likelihood formulation, a local test of global optimality can be constructed. In this paper, we develop an improved test, based on a global maximum validation function proposed by Biernacki, under the assumption that the statistical distribution is in the generalized location family, a condition often satisfied in imaging problems. In addition, a new reparameterization and embedding procedure is presented that exploits knowledge about the forward operator to improve the global maximum validation function. Finally, the reparameterized embedding technique is applied to a physically-motivated joint-inverse problem arising in camera blur estimation. The advantages of the proposed global optimum testing techniques are numerically demonstrated in terms of increased detection accuracy and reduced computation.

The ability of overparameterized deep networks to generalize well has been linked to the fact that stochastic gradient descent (SGD) finds solutions that lie in flat, wide minima in the training loss -- minima where the output of the network is resilient to small random noise added to its parameters. So far this observation has been used to provide generalization guarantees only for neural networks whose parameters are either \textit{stochastic} or \textit{compressed}. In this work, we present a general PAC-Bayesian framework that leverages this observation to provide a bound on the original network learned -- a network that is deterministic and uncompressed. What enables us to do this is a key novelty in our approach: our framework allows us to show that if on training data, the interactions between the weight matrices satisfy certain conditions that imply a wide training loss minimum, these conditions themselves {\em generalize} to the interactions between the matrices on test data, thereby implying a wide test loss minimum. We then apply our general framework in a setup where we assume that the pre-activation values of the network are not too small (although we assume this only on the training data). In this setup, we provide a generalization guarantee for the original (deterministic, uncompressed) network, that does not scale with product of the spectral norms of the weight matrices -- a guarantee that would not have been possible with prior approaches.

Given unpaired data from multiple domains, a key challenge is to efficiently exploit these data sources for modeling a target domain. Variants of this problem have been studied in many contexts, such as cross-domain translation and domain adaptation. We propose AlignFlow, a generative modeling framework for learning from multiple domains via normalizing flows. The use of normalizing flows in AlignFlow allows for a) flexibility in specifying learning objectives via adversarial training, maximum likelihood estimation, or a hybrid of the two methods; and b) exact inference of the shared latent factors across domains at test time. We derive theoretical results for the conditions under which AlignFlow guarantees marginal consistency for the different learning objectives. Furthermore, we show that AlignFlow guarantees exact cycle consistency in mapping datapoints from one domain to another. Empirically, AlignFlow can be used for data-efficient density estimation given multiple data sources and shows significant improvements over relevant baselines on unsupervised domain adaptation.

The standard setting of supervised classification in machine learning assumes that we have access to a training set of samples and to their labels; our goal is then to estimate a classifier able to predict the label of new samples. In many real-world situations, however, collecting training sets of labeled examples is not possible, and alternative learning scenarios must be considered. We focus in this paper on a particular setting where one has access to bags of examples, and where for each bag only the proportions of the labels in the bag are available; the task is still to learn a classifier to predict the label of individual samples. This setting, which following Yu et al. [2013] we refer to as learning from label proportions (LLP), is relevant in many situations where labeling of individual samples is time-consuming, difficult, or just not possible, while side-channel information can be used to reconstruct the proportions of label within a given bag. For example, Musicant et al. [2007] explain how LLP is a natural setting to analyze single particle mass spectrometry data, while Quadrianto et al. [2009] discuss applications in e-commerce, politics or spam filtering.

Active learning (AL) concerns itself with learning a model from as few labelled data as possible through actively and iteratively querying an oracle with selected unlabelled samples. In this paper, we focus on a popular type of AL in which the utility of a sample is measured by a specified goal achieved by the retrained model after accounting for the sample's marginal influence. Such AL strategies attract a lot of attention thanks to their intuitive motivations, yet they typically suffer from impractically high computational costs due to their need for many iterations of model retraining. With the help of influence functions, we present an effective approximation that bypasses model retraining altogether, and propose a general efficient implementation that makes such AL strategies applicable in practice, both in the serial and the more challenging batch-mode setting. Additionally, we present theoretical analyses which call into question a common practice widely adopted in the field. Finally, we carry out empirical studies with both synthetic and real-world datasets to validate our discoveries as well as showcase the potentials and issues with such goal-oriented AL strategies.

Understanding large-scale patterns in student course enrollment is a problem of great interest to university administrators and educational researchers. Yet important decisions are often made without a good quantitative framework of the process underlying student choices. We propose a probabilistic approach to modelling course enrollment decisions, drawing inspiration from multilabel classification and mixture models. We use ten years of anonymized student transcripts from a large university to construct a Gaussian latent variable model that learns the joint distribution over course enrollments. The models allow for a diverse set of inference queries and robustness to data sparsity. We demonstrate the efficacy of this approach in comparison to others, including deep learning architectures, and demonstrate its ability to infer the underlying student interests that guide enrollment decisions.

Recurrent neural networks (RNNs) have been extraordinarily successful for prediction with sequential data. To tackle highly variable and noisy real-world data, we introduce Particle Filter Recurrent Neural Networks (PF-RNNs), a new RNN family that explicitly models uncertainty in its internal structure: while an RNN relies on a long, deterministic latent state vector, a PF-RNN maintains a latent state distribution, approximated as a set of particles. For effective learning, we provide a fully differentiable particle filter algorithm that updates the PF-RNN latent state distribution according to the Bayes rule. Experiments demonstrate that the proposed PF-RNNs outperform the corresponding standard gated RNNs on a synthetic robot localization dataset and 10 real-world sequence prediction datasets for text classification, stock price prediction, etc.

While tasks could come with varying number of instances in realistic settings, the existing meta-learning approaches for few-shot classfication assume even task distributions where the number of instances for each task and class are fixed. Due to such restriction, they learn to equally utilize the meta-knowledge across all the tasks, even when the number of instances per task and class largely varies. Moreover, they do not consider distributional difference in unseen tasks at the meta-test time, on which the meta-knowledge may have varying degree of usefulness depending on the task relatedness. To overcome these limitations, we propose a novel meta-learning model that adaptively balances the effect of the meta-learning and task-specific learning, and also class-specific learning within each task. Through the learning of the balancing variables, we can decide whether to obtain a solution close to the initial parameter or far from it. We formulate this objective into a Bayesian inference framework and solve it using variational inference. Our Bayesian Task-Adaptive Meta-Learning (Bayesian-TAML) significantly outperforms existing meta-learning approaches on benchmark datasets for both few-shot and realistic class- and task-imbalanced datasets, with especially higher gains on the latter.

Deep learning (DL) has solved a problem that as little as five years ago was thought by many to be intractable - the automatic recognition of patterns in data; and it can do so with accuracy that often surpasses human beings. It has solved problems beyond the realm of traditional, hand-crafted machine learning algorithms and captured the imagination of practitioners trying to make sense out of the flood of data that now inundates our society. As public awareness of the efficacy of DL increases so does the desire to make use of it. But even for highly trained professionals it can be daunting to approach the rapidly increasing body of knowledge produced by experts in the field. Where does one start? How does one determine if a particular model is applicable to their problem? How does one train and deploy such a network? A primer on the subject can be a good place to start. With that in mind, we present an overview of some of the key multilayer ANNs that comprise DL. We also discuss some new automatic architecture optimization protocols that use multi-agent approaches. Further, since guaranteeing system uptime is becoming critical to many computer applications, we include a section on using neural networks for fault detection and subsequent mitigation. This is followed by an exploratory survey of several application areas where DL has emerged as a game-changing technology: anomalous behavior detection in financial applications or in financial time-series forecasting, predictive and prescriptive analytics, medical image processing and analysis and power systems research. The thrust of this review is to outline emerging areas of application-oriented research within the DL community as well as to provide a reference to researchers seeking to use it in their work for what it does best: statistical pattern recognition with unparalleled learning capacity with the ability to scale with information.

Determinantal point processes (DPPs) have attracted substantial attention as an elegant probabilistic model that captures the balance between quality and diversity within sets. DPPs are conventionally parameterized by a positive semi-definite kernel matrix, and this symmetric kernel encodes only repulsive interactions between items. These so-called symmetric DPPs have significant expressive power, and have been successfully applied to a variety of machine learning tasks, including recommendation systems, information retrieval, and automatic summarization, among many others. Efficient algorithms for learning symmetric DPPs and sampling from these models have been reasonably well studied. However, relatively little attention has been given to nonsymmetric DPPs, which relax the symmetric constraint on the kernel. Nonsymmetric DPPs allow for both repulsive and attractive item interactions, which can significantly improve modeling power, resulting in a model that may better fit for some applications. We present a method that enables a tractable algorithm, based on maximum likelihood estimation, for learning nonsymmetric DPPs from data composed of observed subsets. Our method imposes a particular decomposition of the nonsymmetric kernel that enables such tractable learning algorithms, which we analyze both theoretically and experimentally. We evaluate our model on synthetic and real-world datasets, demonstrating improved predictive performance compared to symmetric DPPs, which have previously shown strong performance on modeling tasks associated with these datasets.