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We discuss finitely generated infinite groups on which natural random walks are noise sensitive in total variation as well as ones on which natural random walks are noise stable in total variation.

An additive Runge-Kutta method is used for the time stepping, which integrates the linear stiff terms by an explicit singly diagonally implicit Runge-Kutta (ESDIRK) method and the nonlinear terms by an explicit Runge-Kutta (ERK) method. In each time step, the implicit solve is performed by the recently developed Hierarchical Poincar\'e-Steklov (HPS) method. This is a fast direct solver for elliptic equations that decomposes the space domain into a hierarchical tree of subdomains and builds spectral collocation solvers locally on the subdomains. These ideas are naturally combined in the presented method since the singly diagonal coefficient in ESDIRK and a fixed time-step ensures that the coefficient matrix in the implicit solve of HPS remains the same for all time stages. This means that the precomputed inverse can be efficiently reused, leading to a scheme with complexity (in two dimensions) $\mathcal{O}(N^{1.5})$ for the precomputation where the solution operator to the elliptic problems is built, and then $\mathcal{O}(N \log N)$ for the solve in each time step. The stability of the method is proved for first order in time and any order in space, and numerical evidence substantiates a claim of stability for a much broader class of time discretization methods. Numerical experiments supporting the accuracy of efficiency of the method in one and two dimensions are presented.

The notion of Laplacian of a graph can be generalized to simplicial complexes and hypergraphs, and contains information on the topology of these structures. Even for a graph, the consideration of associated simplicial complexes is interesting to understand its shape. Whereas the Laplacian of a graph has a simple probabilistic interpretation as the generator of a continuous time Markov chain on the graph, things are not so direct when considering simplicial complexes. We define here new Markov chains on simplicial complexes. For a given order~$k$, the state space is the set of $k$-cycles that are chains of $k$-simplexes with null boundary. This new framework is a natural generalization of the canonical Markov chains on graphs. We show that the generator of our Markov chain is the upper Laplacian defined in the context of algebraic topology for discrete structure. We establish several key properties of this new process: in particular, when the number of vertices is finite, the Markov chain is positive recurrent. This result is not trivial, since the cycles can loop over themselves an unbounded number of times. We study the diffusive limits when the simplicial complexes under scrutiny are a sequence of ever refining triangulations of the flat torus. Using the analogy between singular and Hodge homologies, we express this limit as valued in the set of currents. The proof of tightness and the identification of the limiting martingale problem make use of the flat norm and carefully controls of the error terms in the convergence of the generator. Uniqueness of the solution to the martingale problem is left open. An application to hole detection is carried.

Robotic affordances, providing information about what actions can be taken in a given situation, can aid robotic manipulation. However, learning about affordances requires expensive large annotated datasets of interactions or demonstrations. In this work, we argue that well-directed interactions with the environment can mitigate this problem and propose an information-based measure to augment the agent's objective and accelerate the affordance discovery process. We provide a theoretical justification of our approach and we empirically validate the approach both in simulation and real-world tasks. Our method, which we dub IDA, enables the efficient discovery of visual affordances for several action primitives, such as grasping, stacking objects, or opening drawers, strongly improving data efficiency in simulation, and it allows us to learn grasping affordances in a small number of interactions, on a real-world setup with a UFACTORY XArm 6 robot arm.

Prediction models are increasingly proposed for guiding treatment decisions, but most fail to address the special role of treatments, leading to inappropriate use. This paper highlights the limitations of using standard prediction models for treatment decision support. We identify `causal blind spots' in three common approaches to handling treatments in prediction modelling: including treatment as a predictor, restricting data based on treatment status and ignoring treatments. When predictions are used to inform treatment decisions, confounders, colliders and mediators, as well as changes in treatment protocols over time may lead to misinformed decision-making. We illustrate potential harmful consequences in several medical applications. We advocate for an extension of guidelines for development, reporting and evaluation of prediction models to ensure that the intended use of the model is matched to an appropriate risk estimand. When prediction models are intended to inform treatment decisions, prediction models should specify upfront the treatment decisions they aim to support and target a prediction estimand in line with that goal. This requires a shift towards developing predictions under the specific treatment options under consideration (`predictions under interventions'). Predictions under interventions need causal reasoning and inference techniques during development and validation. We argue that this will improve the efficacy of prediction models in guiding treatment decisions and prevent potential negative effects on patient outcomes.

First order shape optimization methods, in general, require a large number of iterations until they reach a locally optimal design. While higher order methods can significantly reduce the number of iterations, they exhibit only local convergence properties, necessitating a sufficiently close initial guess. In this work, we present an unregularized shape-Newton method and combine shape optimization with homotopy (or continuation) methods in order to allow for the use of higher order methods even if the initial design is far from a solution. The idea of homotopy methods is to continuously connect the problem of interest with a simpler problem and to follow the corresponding solution path by a predictor-corrector scheme. We use a shape-Newton method as a corrector and arbitrary order shape derivatives for the predictor. Moreover, we apply homotopy methods also to the case of multi-objective shape optimization to efficiently obtain well-distributed points on a Pareto front. Finally, our results are substantiated with a set of numerical experiments.

The unique solvability and error analysis of the original Lagrange multiplier approach proposed in [8] for gradient flows is studied in this paper. We identify a necessary and sufficient condition that must be satisfied for the nonlinear algebraic equation arising from the original Lagrange multiplier approach to admit a unique solution in the neighborhood of its exact solution, and propose a modified Lagrange multiplier approach so that the computation can continue even if the aforementioned condition is not satisfied. Using Cahn-Hilliard equation as an example, we prove rigorously the unique solvability and establish optimal error estimates of a second-order Lagrange multiplier scheme assuming this condition and that the time step is sufficient small. We also present numerical results to demonstrate that the modified Lagrange multiplier approach is much more robust and can use much larger time step than the original Lagrange multiplier approach.

When multitudes of features can plausibly be associated with a response, both privacy considerations and model parsimony suggest grouping them to increase the predictive power of a regression model. Specifically, the identification of groups of predictors significantly associated with the response variable eases further downstream analysis and decision-making. This paper proposes a new data analysis methodology that utilizes the high-dimensional predictor space to construct an implicit network with weighted edges %and weights on the edges to identify significant associations between the response and the predictors. Using a population model for groups of predictors defined via network-wide metrics, a new supervised grouping algorithm is proposed to determine the correct group, with probability tending to one as the sample size diverges to infinity. For this reason, we establish several theoretical properties of the estimates of network-wide metrics. A novel model-assisted bootstrap procedure that substantially decreases computational complexity is developed, facilitating the assessment of uncertainty in the estimates of network-wide metrics. The proposed methods account for several challenges that arise in the high-dimensional data setting, including (i) a large number of predictors, (ii) uncertainty regarding the true statistical model, and (iii) model selection variability. The performance of the proposed methods is demonstrated through numerical experiments, data from sports analytics, and breast cancer data.

The emerging behaviors of swarms have fascinated scientists and gathered significant interest in the field of robotics. Traditionally, swarms are viewed as egalitarian, with robots sharing identical roles and capabilities. However, recent findings highlight the importance of hierarchy for deploying robot swarms more effectively in diverse scenarios. Despite nature's preference for hierarchies, the robotics field has clung to the egalitarian model, partly due to a lack of empirical evidence for the conditions favoring hierarchies. Our research demonstrates that while egalitarian swarms excel in environments proportionate to their collective sensing abilities, they struggle in larger or more complex settings. Hierarchical swarms, conversely, extend their sensing reach efficiently, proving successful in larger, more unstructured environments with fewer resources. We validated these concepts through simulations and physical robot experiments, using a complex radiation cleanup task. This study paves the way for developing adaptable, hierarchical swarm systems applicable in areas like planetary exploration and autonomous vehicles. Moreover, these insights could deepen our understanding of hierarchical structures in biological organisms.

We introduce the Birkhoff completion as the smallest distributive lattice in which a given finite lattice can be embedded as semi-lattice. We discuss its relationship to implicational theories, in particular to R. Wille's simply-implicational theories. By an example, we show how the Birkhoff completion can be used as a tool for ordinal data science.