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Trace finite element methods have become a popular option for solving surface partial differential equations, especially in problems where surface and bulk effects are coupled. In such methods a surface mesh is formed by approximately intersecting the continuous surface on which the PDE is posed with a three-dimensional (bulk) tetrahedral mesh. In classical $H^1$-conforming trace methods, the surface finite element space is obtained by restricting a bulk finite element space to the surface mesh. It is not clear how to carry out a similar procedure in order to obtain other important types of finite element spaces such as $H({\rm div})$-conforming spaces. Following previous work of Olshanskii, Reusken, and Xu on $H^1$-conforming methods, we develop a ``quasi-trace'' mixed method for the Laplace-Beltrami problem. The finite element mesh is taken to be the intersection of the surface with a regular tetrahedral bulk mesh as previously described, resulting in a surface triangulation that is highly unstructured and anisotropic but satisfies a classical maximum angle condition. The mixed method is then employed on this mesh. Optimal error estimates with respect to the bulk mesh size are proved along with superconvergent estimates for the projection of the scalar error and a postprocessed scalar approximation.

We consider a sharp interface formulation for the multi-phase Mullins-Sekerka flow. The flow is characterized by a network of curves evolving such that the total surface energy of the curves is reduced, while the areas of the enclosed phases are conserved. Making use of a variational formulation, we introduce a fully discrete finite element method. Our discretization features a parametric approximation of the moving interfaces that is independent of the discretization used for the equations in the bulk. The scheme can be shown to be unconditionally stable and to satisfy an exact volume conservation property. Moreover, an inherent tangential velocity for the vertices on the discrete curves leads to asymptotically equidistributed vertices, meaning no remeshing is necessary in practice. Several numerical examples, including a convergence experiment for the three-phase Mullins-Sekerka flow, demonstrate the capabilities of the introduced method.

Neural radiance fields (NeRFs) are a powerful tool for implicit scene representations, allowing for differentiable rendering and the ability to make predictions about previously unseen viewpoints. From a robotics perspective, there has been growing interest in object and scene-based localisation using NeRFs, with a number of recent works relying on sampling-based or Monte-Carlo localisation schemes. Unfortunately, these can be extremely computationally expensive, requiring multiple network forward passes to infer camera or object pose. To alleviate this, a variety of sampling strategies have been applied, many relying on keypoint recognition techniques from classical computer vision. This work conducts a systematic empirical comparison of these approaches and shows that in contrast to conventional feature matching approaches for geometry-based localisation, sampling-based localisation using NeRFs benefits significantly from stable features. Results show that rendering stable features can result in a tenfold reduction in the number of forward passes required, a significant speed improvement.

Point source localisation is generally modelled as a Lasso-type problem on measures. However, optimisation methods in non-Hilbert spaces, such as the space of Radon measures, are much less developed than in Hilbert spaces. Most numerical algorithms for point source localisation are based on the Frank-Wolfe conditional gradient method, for which ad hoc convergence theory is developed. We develop extensions of proximal-type methods to spaces of measures. This includes forward-backward splitting, its inertial version, and primal-dual proximal splitting. Their convergence proofs follow standard patterns. We demonstrate their numerical efficacy.

We examine a stochastic formulation for data-driven optimization wherein the decision-maker is not privy to the true distribution, but has knowledge that it lies in some hypothesis set and possesses a historical data set, from which information about it can be gleaned. We define a prescriptive solution as a decision rule mapping such a data set to decisions. As there does not exist prescriptive solutions that are generalizable over the entire hypothesis set, we define out-of-sample optimality as a local average over a neighbourhood of hypotheses, and averaged over the sampling distribution. We prove sufficient conditions for local out-of-sample optimality, which reduces to functions of the sufficient statistic of the hypothesis family. We present an optimization problem that would solve for such an out-of-sample optimal solution, and does so efficiently by a combination of sampling and bisection search algorithms. Finally, we illustrate our model on the newsvendor model, and find strong performance when compared against alternatives in the literature. There are potential implications of our research on end-to-end learning and Bayesian optimization.

Three-dimensional facial stereophotogrammetry provides a detailed representation of craniofacial soft tissue without the use of ionizing radiation. While manual annotation of landmarks serves as the current gold standard for cephalometric analysis, it is a time-consuming process and is prone to human error. The aim in this study was to develop and evaluate an automated cephalometric annotation method using a deep learning-based approach. Ten landmarks were manually annotated on 2897 3D facial photographs by a single observer. The automated landmarking workflow involved two successive DiffusionNet models and additional algorithms for facial segmentation. The dataset was randomly divided into a training and test dataset. The training dataset was used to train the deep learning networks, whereas the test dataset was used to evaluate the performance of the automated workflow. The precision of the workflow was evaluated by calculating the Euclidean distances between the automated and manual landmarks and compared to the intra-observer and inter-observer variability of manual annotation and the semi-automated landmarking method. The workflow was successful in 98.6% of all test cases. The deep learning-based landmarking method achieved precise and consistent landmark annotation. The mean precision of 1.69 (+/-1.15) mm was comparable to the inter-observer variability (1.31 +/-0.91 mm) of manual annotation. The Euclidean distance between the automated and manual landmarks was within 2 mm in 69%. Automated landmark annotation on 3D photographs was achieved with the DiffusionNet-based approach. The proposed method allows quantitative analysis of large datasets and may be used in diagnosis, follow-up, and virtual surgical planning.

Recently, conditional score-based diffusion models have gained significant attention in the field of supervised speech enhancement, yielding state-of-the-art performance. However, these methods may face challenges when generalising to unseen conditions. To address this issue, we introduce an alternative approach that operates in an unsupervised manner, leveraging the generative power of diffusion models. Specifically, in a training phase, a clean speech prior distribution is learnt in the short-time Fourier transform (STFT) domain using score-based diffusion models, allowing it to unconditionally generate clean speech from Gaussian noise. Then, we develop a posterior sampling methodology for speech enhancement by combining the learnt clean speech prior with a noise model for speech signal inference. The noise parameters are simultaneously learnt along with clean speech estimation through an iterative expectationmaximisation (EM) approach. To the best of our knowledge, this is the first work exploring diffusion-based generative models for unsupervised speech enhancement, demonstrating promising results compared to a recent variational auto-encoder (VAE)-based unsupervised approach and a state-of-the-art diffusion-based supervised method. It thus opens a new direction for future research in unsupervised speech enhancement.

Ordinary differential equations (ODEs) are widely used to model complex dynamics that arises in biology, chemistry, engineering, finance, physics, etc. Calibration of a complicated ODE system using noisy data is generally very difficult. In this work, we propose a two-stage nonparametric approach to address this problem. We first extract the de-noised data and their higher order derivatives using boundary kernel method, and then feed them into a sparsely connected deep neural network with ReLU activation function. Our method is able to recover the ODE system without being subject to the curse of dimensionality and complicated ODE structure. When the ODE possesses a general modular structure, with each modular component involving only a few input variables, and the network architecture is properly chosen, our method is proven to be consistent. Theoretical properties are corroborated by an extensive simulation study that demonstrates the validity and effectiveness of the proposed method. Finally, we use our method to simultaneously characterize the growth rate of Covid-19 infection cases from 50 states of the USA.

Hashing has been widely used in approximate nearest search for large-scale database retrieval for its computation and storage efficiency. Deep hashing, which devises convolutional neural network architecture to exploit and extract the semantic information or feature of images, has received increasing attention recently. In this survey, several deep supervised hashing methods for image retrieval are evaluated and I conclude three main different directions for deep supervised hashing methods. Several comments are made at the end. Moreover, to break through the bottleneck of the existing hashing methods, I propose a Shadow Recurrent Hashing(SRH) method as a try. Specifically, I devise a CNN architecture to extract the semantic features of images and design a loss function to encourage similar images projected close. To this end, I propose a concept: shadow of the CNN output. During optimization process, the CNN output and its shadow are guiding each other so as to achieve the optimal solution as much as possible. Several experiments on dataset CIFAR-10 show the satisfying performance of SRH.

Graph representation learning for hypergraphs can be used to extract patterns among higher-order interactions that are critically important in many real world problems. Current approaches designed for hypergraphs, however, are unable to handle different types of hypergraphs and are typically not generic for various learning tasks. Indeed, models that can predict variable-sized heterogeneous hyperedges have not been available. Here we develop a new self-attention based graph neural network called Hyper-SAGNN applicable to homogeneous and heterogeneous hypergraphs with variable hyperedge sizes. We perform extensive evaluations on multiple datasets, including four benchmark network datasets and two single-cell Hi-C datasets in genomics. We demonstrate that Hyper-SAGNN significantly outperforms the state-of-the-art methods on traditional tasks while also achieving great performance on a new task called outsider identification. Hyper-SAGNN will be useful for graph representation learning to uncover complex higher-order interactions in different applications.