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Devising deep latent variable models for multi-modal data has been a long-standing theme in machine learning research. Multi-modal Variational Autoencoders (VAEs) have been a popular generative model class that learns latent representations which jointly explain multiple modalities. Various objective functions for such models have been suggested, often motivated as lower bounds on the multi-modal data log-likelihood or from information-theoretic considerations. In order to encode latent variables from different modality subsets, Product-of-Experts (PoE) or Mixture-of-Experts (MoE) aggregation schemes have been routinely used and shown to yield different trade-offs, for instance, regarding their generative quality or consistency across multiple modalities. In this work, we consider a variational bound that can tightly lower bound the data log-likelihood. We develop more flexible aggregation schemes that generalise PoE or MoE approaches by combining encoded features from different modalities based on permutation-invariant neural networks. Our numerical experiments illustrate trade-offs for multi-modal variational bounds and various aggregation schemes. We show that tighter variational bounds and more flexible aggregation models can become beneficial when one wants to approximate the true joint distribution over observed modalities and latent variables in identifiable models.

A significant limitation of one-class classification anomaly detection methods is their reliance on the assumption that unlabeled training data only contains normal instances. To overcome this impractical assumption, we propose two novel classification-based anomaly detection methods. Firstly, we introduce a semi-supervised shallow anomaly detection method based on an unbiased risk estimator. Secondly, we present a semi-supervised deep anomaly detection method utilizing a nonnegative (biased) risk estimator. We establish estimation error bounds and excess risk bounds for both risk minimizers. Additionally, we propose techniques to select appropriate regularization parameters that ensure the nonnegativity of the empirical risk in the shallow model under specific loss functions. Our extensive experiments provide strong evidence of the effectiveness of the risk-based anomaly detection methods.

We introduce new control-volume finite-element discretization schemes suitable for solving the Stokes problem. Within a common framework, we present different approaches for constructing such schemes. The first and most established strategy employs a non-overlapping partitioning into control volumes. The second represents a new idea by splitting into two sets of control volumes, the first set yielding a partition of the domain and the second containing the remaining overlapping control volumes required for stability. The third represents a hybrid approach where finite volumes are combined with finite elements based on a hierarchical splitting of the ansatz space. All approaches are based on typical finite element function spaces but yield locally mass and momentum conservative discretization schemes that can be interpreted as finite volume schemes. We apply all strategies to the inf-sub stable MINI finite-element pair. Various test cases, including convergence tests and the numerical observation of the boundedness of the number of preconditioned Krylov solver iterations, as well as more complex scenarios of flow around obstacles or through a three-dimensional vessel bifurcation, demonstrate the stability and robustness of the schemes.

Hawkes processes are often applied to model dependence and interaction phenomena in multivariate event data sets, such as neuronal spike trains, social interactions, and financial transactions. In the nonparametric setting, learning the temporal dependence structure of Hawkes processes is generally a computationally expensive task, all the more with Bayesian estimation methods. In particular, for generalised nonlinear Hawkes processes, Monte-Carlo Markov Chain methods applied to compute the doubly intractable posterior distribution are not scalable to high-dimensional processes in practice. Recently, efficient algorithms targeting a mean-field variational approximation of the posterior distribution have been proposed. In this work, we first unify existing variational Bayes approaches under a general nonparametric inference framework, and analyse the asymptotic properties of these methods under easily verifiable conditions on the prior, the variational class, and the nonlinear model. Secondly, we propose a novel sparsity-inducing procedure, and derive an adaptive mean-field variational algorithm for the popular sigmoid Hawkes processes. Our algorithm is parallelisable and therefore computationally efficient in high-dimensional setting. Through an extensive set of numerical simulations, we also demonstrate that our procedure is able to adapt to the dimensionality of the parameter of the Hawkes process, and is partially robust to some type of model mis-specification.

For multivariate data with noise variables, tandem clustering is a well-known technique that aims to improve cluster identification by first reducing the dimension. However, the usual approach using principal component analysis (PCA) has been criticized for focusing only on inertia so that the first components do not necessarily retain the structure of interest for clustering. To overcome this drawback, a new tandem clustering approach based on invariant coordinate selection (ICS) is proposed. By jointly diagonalizing two scatter matrices, ICS is designed to find structure in the data while returning affine invariant components. Some theoretical results have already been derived and guarantee that under some elliptical mixture models, the group structure can be highlighted on a subset of the first and/or last components. Nevertheless, ICS has received little attention in a clustering context. Two challenges are the choice of the pair of scatter matrices and the selection of the components to retain. For clustering purposes, it is demonstrated that the best scatter pairs consist of one scatter matrix that captures the within-cluster structure and another that captures the global structure. For the former, local shape or pairwise scatters are of great interest, as is the minimum covariance determinant (MCD) estimator based on a carefully selected subset size that is smaller than usual. The performance of ICS as a dimension reduction method is evaluated in terms of preserving the cluster structure present in data. In an extensive simulation study and in empirical applications with benchmark data sets, different combinations of scatter matrices as well as component selection criteria are compared in situations with and without outliers. Overall, the new approach of tandem clustering with ICS shows promising results and clearly outperforms the approach with PCA.

Blind quantum computation (BQC) protocols enable quantum algorithms to be executed on third-party quantum agents while keeping the data and algorithm confidential. The previous proposals for measurement-based BQC require preparing a highly entangled cluster state. In this paper, we show that such a requirement is not necessary. Our protocol only requires pre-shared bell pairs between delegated quantum agents, and there is no requirement for any classical or quantum information exchange between agents during the execution. Our proposal requires fewer quantum resources than previous proposals by eliminating the need for a universal cluster state.

We present a framework for approximate Bayesian inference when only a limited number of noisy log-likelihood evaluations can be obtained due to computational constraints, which is becoming increasingly common for applications of complex models. We model the log-likelihood function using a Gaussian process (GP) and the main methodological innovation is to apply this model to emulate the progression that an exact Metropolis-Hastings (MH) sampler would take if it was applicable. Informative log-likelihood evaluation locations are selected using a sequential experimental design strategy until the MH accept/reject decision is done accurately enough according to the GP model. The resulting approximate sampler is conceptually simple and sample-efficient. It is also more robust to violations of GP modelling assumptions compared with earlier, related "Bayesian optimisation-like" methods tailored for Bayesian inference. We discuss some theoretical aspects and various interpretations of the resulting approximate MH sampler, and demonstrate its benefits in the context of Bayesian and generalised Bayesian likelihood-free inference for simulator-based statistical models.

We consider dynamical low-rank approximations to parabolic problems on higher-order tensor manifolds in Hilbert spaces. In addition to existence of solutions and their stability with respect to perturbations to the problem data, we show convergence of spatial discretizations. Our framework accommodates various standard low-rank tensor formats for multivariate functions, including tensor train and hierarchical tensors.

Explaining artificial intelligence (AI) predictions is increasingly important and even imperative in many high-stakes applications where humans are the ultimate decision-makers. In this work, we propose two novel architectures of self-interpretable image classifiers that first explain, and then predict (as opposed to post-hoc explanations) by harnessing the visual correspondences between a query image and exemplars. Our models consistently improve (by 1 to 4 points) on out-of-distribution (OOD) datasets while performing marginally worse (by 1 to 2 points) on in-distribution tests than ResNet-50 and a $k$-nearest neighbor classifier (kNN). Via a large-scale, human study on ImageNet and CUB, our correspondence-based explanations are found to be more useful to users than kNN explanations. Our explanations help users more accurately reject AI's wrong decisions than all other tested methods. Interestingly, for the first time, we show that it is possible to achieve complementary human-AI team accuracy (i.e., that is higher than either AI-alone or human-alone), in ImageNet and CUB image classification tasks.

In large-scale systems there are fundamental challenges when centralised techniques are used for task allocation. The number of interactions is limited by resource constraints such as on computation, storage, and network communication. We can increase scalability by implementing the system as a distributed task-allocation system, sharing tasks across many agents. However, this also increases the resource cost of communications and synchronisation, and is difficult to scale. In this paper we present four algorithms to solve these problems. The combination of these algorithms enable each agent to improve their task allocation strategy through reinforcement learning, while changing how much they explore the system in response to how optimal they believe their current strategy is, given their past experience. We focus on distributed agent systems where the agents' behaviours are constrained by resource usage limits, limiting agents to local rather than system-wide knowledge. We evaluate these algorithms in a simulated environment where agents are given a task composed of multiple subtasks that must be allocated to other agents with differing capabilities, to then carry out those tasks. We also simulate real-life system effects such as networking instability. Our solution is shown to solve the task allocation problem to 6.7% of the theoretical optimal within the system configurations considered. It provides 5x better performance recovery over no-knowledge retention approaches when system connectivity is impacted, and is tested against systems up to 100 agents with less than a 9% impact on the algorithms' performance.