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The prevailing statistical approach to analyzing persistence diagrams is concerned with filtering out topological noise. In this paper, we adopt a different viewpoint and aim at estimating the actual distribution of a random persistence diagram, which captures both topological signal and noise. To that effect, Chazel and Divol (2019) proved that, under general conditions, the expected value of a random persistence diagram is a measure admitting a Lebesgue density, called the persistence intensity function. In this paper, we are concerned with estimating the persistence intensity function and a novel, normalized version of it -- called the persistence density function. We present a class of kernel-based estimators based on an i.i.d. sample of persistence diagrams and derive estimation rates in the supremum norm. As a direct corollary, we obtain uniform consistency rates for estimating linear representations of persistence diagrams, including Betti numbers and persistence surfaces. Interestingly, the persistence density function delivers stronger statistical guarantees.

Friction drag from a turbulent fluid moving past or inside an object plays a crucial role in domains as diverse as transportation, public utility infrastructure, energy technology, and human health. As a direct measure of the shear-induced friction forces, an accurate prediction of the wall-shear stress can contribute to sustainability, conservation of resources, and carbon neutrality in civil aviation as well as enhanced medical treatment of vascular diseases and cancer. Despite such importance for our modern society, we still lack adequate experimental methods to capture the instantaneous wall-shear stress dynamics. In this contribution, we present a holistic approach that derives velocity and wall-shear stress fields with impressive spatial and temporal resolution from flow measurements using a deep optical flow estimator with physical knowledge. The validity and physical correctness of the derived flow quantities is demonstrated with synthetic and real-world experimental data covering a range of relevant fluid flows.

Accurate short-term predictions of phase-resolved water wave conditions are crucial for decision-making in ocean engineering. However, the initialization of remote-sensing-based wave prediction models first requires a reconstruction of wave surfaces from sparse measurements like radar. Existing reconstruction methods either rely on computationally intensive optimization procedures or simplistic modelling assumptions that compromise the real-time capability or accuracy of the subsequent prediction process. We therefore address these issues by proposing a novel approach for phase-resolved wave surface reconstruction using neural networks based on the U-Net and Fourier neural operator (FNO) architectures. Our approach utilizes synthetic yet highly realistic training data on uniform one-dimensional grids, that is generated by the high-order spectral method for wave simulation and a geometric radar modelling approach. The investigation reveals that both models deliver accurate wave reconstruction results and show good generalization for different sea states when trained with spatio-temporal radar data containing multiple historic radar snapshots in each input. Notably, the FNO demonstrates superior performance in handling the data structure imposed by wave physics due to its global approach to learn the mapping between input and output in Fourier space.

The effects of continuous treatments are often characterized through the average dose response function, which is challenging to estimate from observational data due to confounding and positivity violations. Modified treatment policies (MTPs) are an alternative approach that aim to assess the effect of a modification to observed treatment values and work under relaxed assumptions. Estimators for MTPs generally focus on estimating the conditional density of treatment given covariates and using it to construct weights. However, weighting using conditional density models has well-documented challenges. Further, MTPs with larger treatment modifications have stronger confounding and no tools exist to help choose an appropriate modification magnitude. This paper investigates the role of weights for MTPs showing that to control confounding, weights should balance the weighted data to an unobserved hypothetical target population, that can be characterized with observed data. Leveraging this insight, we present a versatile set of tools to enhance estimation for MTPs. We introduce a distance that measures imbalance of covariate distributions under the MTP and use it to develop new weighting methods and tools to aid in the estimation of MTPs. We illustrate our methods through an example studying the effect of mechanical power of ventilation on in-hospital mortality.

Fairness holds a pivotal role in the realm of machine learning, particularly when it comes to addressing groups categorised by sensitive attributes, e.g., gender, race. Prevailing algorithms in fair learning predominantly hinge on accessibility or estimations of these sensitive attributes, at least in the training process. We design a single group-blind projection map that aligns the feature distributions of both groups in the source data, achieving (demographic) group parity, without requiring values of the protected attribute for individual samples in the computation of the map, as well as its use. Instead, our approach utilises the feature distributions of the privileged and unprivileged groups in a boarder population and the essential assumption that the source data are unbiased representation of the population. We present numerical results on synthetic data and real data.

We study signals that are sparse in graph spectral domain and develop explicit algorithms to reconstruct the support set as well as partial components from samples on few vertices of the graph. The number of required samples is independent of the total size of the graph and takes only local properties of the graph into account. Our results rely on an operator based framework for subspace methods and become effective when the spectral eigenfunctions are zero-free or linear independent on small sets of the vertices. The latter has recently been adressed using algebraic methods by the first author.

Robot-assisted navigation is a perfect example of a class of applications requiring flexible control approaches. When the human is reliable, the robot should concede space to their initiative. When the human makes inappropriate choices the robot controller should kick-in guiding them towards safer paths. Shared authority control is a way to achieve this behaviour by deciding online how much of the authority should be given to the human and how much should be retained by the robot. An open problem is how to evaluate the appropriateness of the human's choices. One possible way is to consider the deviation from an ideal path computed by the robot. This choice is certainly safe and efficient, but it emphasises the importance of the robot's decision and relegates the human to a secondary role. In this paper, we propose a different paradigm: a human's behaviour is correct if, at every time, it bears a close resemblance to what other humans do in similar situations. This idea is implemented through the combination of machine learning and adaptive control. The map of the environment is decomposed into a grid. In each cell, we classify the possible motions that the human executes. We use a neural network classifier to classify the current motion, and the probability score is used as a hyperparameter in the control to vary the amount of intervention. The experiments collected for the paper show the feasibility of the idea. A qualitative evaluation, done by surveying the users after they have tested the robot, shows that the participants preferred our control method over a state-of-the-art visco-elastic control.

In indoor scenes, reverberation is a crucial factor in degrading the perceived quality and intelligibility of speech. In this work, we propose a generative dereverberation method. Our approach is based on a probabilistic model utilizing a recurrent variational auto-encoder (RVAE) network and the convolutive transfer function (CTF) approximation. Different from most previous approaches, the output of our RVAE serves as the prior of the clean speech. And our target is the maximum a posteriori (MAP) estimation of clean speech, which is achieved iteratively through the expectation maximization (EM) algorithm. The proposed method integrates the capabilities of network-based speech prior modelling and CTF-based observation modelling. Experiments on single-channel speech dereverberation show that the proposed generative method noticeably outperforms the advanced discriminative networks.

Finding the optimal design of experiments in the Bayesian setting typically requires estimation and optimization of the expected information gain functional. This functional consists of one outer and one inner integral, separated by the logarithm function applied to the inner integral. When the mathematical model of the experiment contains uncertainty about the parameters of interest and nuisance uncertainty, (i.e., uncertainty about parameters that affect the model but are not themselves of interest to the experimenter), two inner integrals must be estimated. Thus, the already considerable computational effort required to determine good approximations of the expected information gain is increased further. The Laplace approximation has been applied successfully in the context of experimental design in various ways, and we propose two novel estimators featuring the Laplace approximation to alleviate the computational burden of both inner integrals considerably. The first estimator applies Laplace's method followed by a Laplace approximation, introducing a bias. The second estimator uses two Laplace approximations as importance sampling measures for Monte Carlo approximations of the inner integrals. Both estimators use Monte Carlo approximation for the remaining outer integral estimation. We provide three numerical examples demonstrating the applicability and effectiveness of our proposed estimators.

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.