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Weak supervision searches have in principle the advantages of both being able to train on experimental data and being able to learn distinctive signal properties. However, the practical applicability of such searches is limited by the fact that successfully training a neural network via weak supervision can require a large amount of signal. In this work, we seek to create neural networks that can learn from less experimental signal by using transfer and meta-learning. The general idea is to first train a neural network on simulations, thereby learning concepts that can be reused or becoming a more efficient learner. The neural network would then be trained on experimental data and should require less signal because of its previous training. We find that transfer and meta-learning can substantially improve the performance of weak supervision searches.

This paper presents a method for thematic agreement assessment of geospatial data products of different semantics and spatial granularities, which may be affected by spatial offsets between test and reference data. The proposed method uses a multi-scale framework allowing for a probabilistic evaluation whether thematic disagreement between datasets is induced by spatial offsets due to different nature of the datasets or not. We test our method using real-estate derived settlement locations and remote-sensing derived building footprint data.

The theory of mixed finite element methods for solving different types of elliptic partial differential equations in saddle point formulation is well established since many decades. This topic was mostly studied for variational formulations defined upon the same product spaces of both shape- and test-pairs of primal variable-multiplier. Whenever either these spaces or the two bilinear forms involving the multiplier are distinct, the saddle point problem is asymmetric. The three inf-sup conditions to be satisfied by the product spaces stipulated in work on the subject, in order to guarantee well-posedness, are well known. However, the material encountered in the literature addressing the approximation of this class of problems left room for improvement and clarifications. After making a brief review of the existing contributions to the topic that justifies such an assertion, in this paper we set up finer global error bounds for the pair primal variable-multiplier solving an asymmetric saddle point problem. Besides well-posedness, the three constants in the aforementioned inf-sup conditions are identified as all that is needed for determining the stability constant appearing therein, whose expression is exhibited. As a complement, refined error bounds depending only on these three constants are given for both unknowns separately.

Source conditions are a key tool in regularisation theory that are needed to derive error estimates and convergence rates for ill-posed inverse problems. In this paper, we provide a recipe to practically compute source condition elements as the solution of convex minimisation problems that can be solved with first-order algorithms. We demonstrate the validity of our approach by testing it on two inverse problem case studies in machine learning and image processing: sparse coefficient estimation of a polynomial via LASSO regression and recovering an image from a subset of the coefficients of its discrete Fourier transform. We further demonstrate that the proposed approach can easily be modified to solve the machine learning task of identifying the optimal sampling pattern in the Fourier domain for a given image and variational regularisation method, which has applications in the context of sparsity promoting reconstruction from magnetic resonance imaging data.

Understanding a surgical scene is crucial for computer-assisted surgery systems to provide any intelligent assistance functionality. One way of achieving this scene understanding is via scene segmentation, where every pixel of a frame is classified and therefore identifies the visible structures and tissues. Progress on fully segmenting surgical scenes has been made using machine learning. However, such models require large amounts of annotated training data, containing examples of all relevant object classes. Such fully annotated datasets are hard to create, as every pixel in a frame needs to be annotated by medical experts and, therefore, are rarely available. In this work, we propose a method to combine multiple partially annotated datasets, which provide complementary annotations, into one model, enabling better scene segmentation and the use of multiple readily available datasets. Our method aims to combine available data with complementary labels by leveraging mutual exclusive properties to maximize information. Specifically, we propose to use positive annotations of other classes as negative samples and to exclude background pixels of binary annotations, as we cannot tell if they contain a class not annotated but predicted by the model. We evaluate our method by training a DeepLabV3 on the publicly available Dresden Surgical Anatomy Dataset, which provides multiple subsets of binary segmented anatomical structures. Our approach successfully combines 6 classes into one model, increasing the overall Dice Score by 4.4% compared to an ensemble of models trained on the classes individually. By including information on multiple classes, we were able to reduce confusion between stomach and colon by 24%. Our results demonstrate the feasibility of training a model on multiple datasets. This paves the way for future work further alleviating the need for one large, fully segmented datasets.

We consider the statistical linear inverse problem of making inference on an unknown source function in an elliptic partial differential equation from noisy observations of its solution. We employ nonparametric Bayesian procedures based on Gaussian priors, leading to convenient conjugate formulae for posterior inference. We review recent results providing theoretical guarantees on the quality of the resulting posterior-based estimation and uncertainty quantification, and we discuss the application of the theory to the important classes of Gaussian series priors defined on the Dirichlet-Laplacian eigenbasis and Mat\'ern process priors. We provide an implementation of posterior inference for both classes of priors, and investigate its performance in a numerical simulation study.

We present an asymptotic-preserving (AP) numerical method for solving the three-temperature radiative transfer model, which holds significant importance in inertial confinement fusion. A carefully designedsplitting method is developed that can provide a general framework of extending AP schemes for the gray radiative transport equation to the more complex three-temperature radiative transfer model. The proposed scheme captures two important limiting models: the three-temperature radiation diffusion equation (3TRDE) when opacity approaches infinity and the two-temperature limit when the ion-electron coupling coefficient goes to infinity. We have rigorously demonstrated the AP property and energy conservation characteristics of the proposed scheme and its efficiency has been validated through a series of benchmark tests in the numerical part.

Heuristic tools from statistical physics have been used in the past to locate the phase transitions and compute the optimal learning and generalization errors in the teacher-student scenario in multi-layer neural networks. In this contribution, we provide a rigorous justification of these approaches for a two-layers neural network model called the committee machine. We also introduce a version of the approximate message passing (AMP) algorithm for the committee machine that allows to perform optimal learning in polynomial time for a large set of parameters. We find that there are regimes in which a low generalization error is information-theoretically achievable while the AMP algorithm fails to deliver it, strongly suggesting that no efficient algorithm exists for those cases, and unveiling a large computational gap.

In this short note we formulate a stabilizer formalism in the language of noncommutative graphs. The classes of noncommutative graphs we consider are obtained via unitary representations of compact groups, and suitably chosen operators on finite-dimensional Hilbert spaces. Furthermore, in this framework, we generalize previous results in this area for determining when such noncommutative graphs have anticliques.

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.