We address the non-convex optimisation problem of finding a sparse matrix on the Stiefel manifold (matrices with mutually orthogonal columns of unit length) that maximises (or minimises) a quadratic objective function. Optimisation problems on the Stiefel manifold occur for example in spectral relaxations of various combinatorial problems, such as graph matching, clustering, or permutation synchronisation. Although sparsity is a desirable property in such settings, it is mostly neglected in spectral formulations since existing solvers, e.g. based on eigenvalue decomposition, are unable to account for sparsity while at the same time maintaining global optimality guarantees. We fill this gap and propose a simple yet effective sparsity-promoting modification of the Orthogonal Iteration algorithm for finding the dominant eigenspace of a matrix. By doing so, we can guarantee that our method finds a Stiefel matrix that is globally optimal with respect to the quadratic objective function, while in addition being sparse. As a motivating application we consider the task of permutation synchronisation, which can be understood as a constrained clustering problem that has particular relevance for matching multiple images or 3D shapes in computer vision, computer graphics, and beyond. We demonstrate that the proposed approach outperforms previous methods in this domain.

Node features and structural information of a graph are both crucial for semi-supervised node classification problems. A variety of graph neural network (GNN) based approaches have been proposed to tackle these problems, which typically determine output labels through feature aggregation. This can be problematic, as it implies conditional independence of output nodes given hidden representations, despite their direct connections in the graph. To learn the direct influence among output nodes in a graph, we propose the Explicit Pairwise Factorized Graph Neural Network (EPFGNN), which models the whole graph as a partially observed Markov Random Field. It contains explicit pairwise factors to model output-output relations and uses a GNN backbone to model input-output relations. To balance model complexity and expressivity, the pairwise factors have a shared component and a separate scaling coefficient for each edge. We apply the EM algorithm to train our model, and utilize a star-shaped piecewise likelihood for the tractable surrogate objective. We conduct experiments on various datasets, which shows that our model can effectively improve the performance for semi-supervised node classification on graphs.

We address the problem of uncertainty calibration and introduce a novel calibration method, Parametrized Temperature Scaling (PTS). Standard deep neural networks typically yield uncalibrated predictions, which can be transformed into calibrated confidence scores using post-hoc calibration methods. In this contribution, we demonstrate that the performance of accuracy-preserving state-of-the-art post-hoc calibrators is limited by their intrinsic expressive power. We generalize temperature scaling by computing prediction-specific temperatures, parameterized by a neural network. We show with extensive experiments that our novel accuracy-preserving approach consistently outperforms existing algorithms across a large number of model architectures, datasets and metrics.

We address the problem of uncertainty calibration. While standard deep neural networks typically yield uncalibrated predictions, calibrated confidence scores that are representative of the true likelihood of a prediction can be achieved using post-hoc calibration methods. However, to date the focus of these approaches has been on in-domain calibration. Our contribution is two-fold. First, we show that existing post-hoc calibration methods yield highly over-confident predictions under domain shift. Second, we introduce a simple strategy where perturbations are applied to samples in the validation set before performing the post-hoc calibration step. In extensive experiments, we demonstrate that this perturbation step results in substantially better calibration under domain shift on a wide range of architectures and modelling tasks.

Can we learn how to explore unknown spaces efficiently? To answer this question, we study the problem of Online Graph Exploration, the online version of the Traveling Salesperson Problem. We reformulate graph exploration as a reinforcement learning problem and apply Direct Future Prediction (Dosovitskiy and Koltun, 2016) to solve it. As the graph is discovered online, the corresponding Markov Decision Process entails a dynamic state space, namely the observable graph and a dynamic action space, namely the nodes forming the graph's frontier. To the best of our knowledge, this is the first attempt to solve online graph exploration in a data-driven way. We conduct experiments on six data sets of procedurally generated graphs and three real city road networks. We demonstrate that our agent can learn strategies superior to many well known graph traversal algorithms, confirming that exploration can be learned.

The last decade has witnessed a boom of deep learning research and applications achieving state-of-the-art results in various domains. However, most advances have been established empirically, and their theoretical analysis remains lacking. One major issue is that our current interpretation of neural networks (NNs) as function approximators is too generic to support in-depth analysis. In this paper, we remedy this by proposing an alternative interpretation that identifies NNs as chain graphs (CGs) and feed-forward as an approximate inference procedure. The CG interpretation specifies the nature of each NN component within the rich theoretical framework of probabilistic graphical models, while at the same time remains general enough to cover real-world NNs with arbitrary depth, multi-branching and varied activations, as well as common structures including convolution / recurrent layers, residual block and dropout. We demonstrate with concrete examples that the CG interpretation can provide novel theoretical support and insights for various NN techniques, as well as derive new deep learning approaches such as the concept of partially collapsed feed-forward inference. It is thus a promising framework that deepens our understanding of neural networks and provides a coherent theoretical formulation for future deep learning research.

Computer-aided drug discovery is an essential component of modern drug development. Therein, deep learning has become an important tool for rapid screening of billions of molecules in silico for potential hits containing desired chemical features. Despite its importance, substantial challenges persist in training these models, such as severe class imbalance, high decision thresholds, and lack of ground truth labels in some datasets. In this work we argue in favor of directly optimizing the receiver operating characteristic (ROC) in such cases, due to its robustness to class imbalance, its ability to compromise over different decision thresholds, certain freedom to influence the relative weights in this compromise, fidelity to typical benchmarking measures, and equivalence to positive/unlabeled learning. We also propose new training schemes (coherent mini-batch arrangement, and usage of out-of-batch samples) for cost functions based on the ROC, as well as a cost function based on the logAUC metric that facilitates early enrichment (i.e. improves performance at high decision thresholds, as often desired when synthesizing predicted hit compounds). We demonstrate that these approaches outperform standard deep learning approaches on a series of PubChem high-throughput screening datasets that represent realistic and diverse drug discovery campaigns on major drug target families.

In active learning, sampling bias could pose a serious inconsistency problem and hinder the algorithm from finding the optimal hypothesis. However, many methods for neural networks are hypothesis space agnostic and do not address this problem. We examine active learning with convolutional neural networks through the principled lens of version space reduction. We identify the connection between two approaches---prior mass reduction and diameter reduction---and propose a new diameter-based querying method---the minimum Gibbs-vote disagreement. By estimating version space diameter and bias, we illustrate how version space of neural networks evolves and examine the realizability assumption. With experiments on MNIST, Fashion-MNIST, SVHN and STL-10 datasets, we demonstrate that diameter reduction methods reduce the version space more effectively and perform better than prior mass reduction and other baselines, and that the Gibbs vote disagreement is on par with the best query method.

The ability of deep learning models to generalize well across different scenarios depends primarily on the quality and quantity of annotated data. Labeling large amounts of data for all possible scenarios that a model may encounter would not be feasible; if even possible. We propose a framework to deal with limited labeled training data and demonstrate it on the application of vision-based vehicle control. We show how limited steering angle data available for only one condition can be transferred to multiple different weather scenarios. This is done by leveraging unlabeled images in a teacher-student learning paradigm complemented with an image-to-image translation network. The translation network transfers the images to a new domain, whereas the teacher provides soft supervised targets to train the student on this domain. Furthermore, we demonstrate how utilization of auxiliary networks can reduce the size of a model at inference time, without affecting the accuracy. The experiments show that our approach generalizes well across multiple different weather conditions using only ground truth labels from one domain.

We take the novel perspective to view data not as a probability distribution but rather as a current. Primarily studied in the field of geometric measure theory, $k$-currents are continuous linear functionals acting on compactly supported smooth differential forms and can be understood as a generalized notion of oriented $k$-dimensional manifold. By moving from distributions (which are $0$-currents) to $k$-currents, we can explicitly orient the data by attaching a $k$-dimensional tangent plane to each sample point. Based on the flat metric which is a fundamental distance between currents, we derive FlatGAN, a formulation in the spirit of generative adversarial networks but generalized to $k$-currents. In our theoretical contribution we prove that the flat metric between a parametrized current and a reference current is Lipschitz continuous in the parameters. In experiments, we show that the proposed shift to $k>0$ leads to interpretable and disentangled latent representations which behave equivariantly to the specified oriented tangent planes.