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In a previous paper, a technique was suggested to avoid order reduction with any explicit exponential Runge-Kutta method when integrating initial boundary value nonlinear problems with time-dependent boundary conditions. In this paper, we significantly simplify the full discretization formulas to be applied under conditions which are nearly always satisfied in practice. Not only a simpler linear combination of $\varphi_j$-functions is given for both the stages and the solution, but also the information required on the boundary is so much simplified that, in order to get local order three, it is no longer necessary to resort to numerical differentiation in space. The technique is then shown to be computationally competitive against other widely used methods with high enough stiff order through the standard method of lines.

In reinforcement learning (RL), sparse rewards can present a significant challenge. Fortunately, expert actions can be utilized to overcome this issue. However, acquiring explicit expert actions can be costly, and expert observations are often more readily available. This paper presents a new approach that uses expert observations for learning in robot manipulation tasks with sparse rewards from pixel observations. Specifically, our technique involves using expert observations as intermediate visual goals for a goal-conditioned RL agent, enabling it to complete a task by successively reaching a series of goals. We demonstrate the efficacy of our method in five challenging block construction tasks in simulation and show that when combined with two state-of-the-art agents, our approach can significantly improve their performance while requiring 4-20 times fewer expert actions during training. Moreover, our method is also superior to a hierarchical baseline.

In this paper we derive tight lower bounds resolving the hardness status of several fundamental weighted matroid problems. One notable example is budgeted matroid independent set, for which we show there is no fully polynomial-time approximation scheme (FPTAS), indicating the Efficient PTAS of [Doron-Arad, Kulik and Shachnai, SOSA 2023] is the best possible. Furthermore, we show that there is no pseudo-polynomial time algorithm for exact weight matroid independent set, implying the algorithm of [Camerini, Galbiati and Maffioli, J. Algorithms 1992] for representable matroids cannot be generalized to arbitrary matroids. Similarly, we show there is no Fully PTAS for constrained minimum basis of a matroid and knapsack cover with a matroid, implying the existing Efficient PTAS for the former is optimal. For all of the above problems, we obtain unconditional lower bounds in the oracle model, where the independent sets of the matroid can be accessed only via a membership oracle. We complement these results by showing that the same lower bounds hold under standard complexity assumptions, even if the matroid is encoded as part of the instance. All of our bounds are based on a specifically structured family of paving matroids.

High-resolution wide-angle fisheye images are becoming more and more important for robotics applications such as autonomous driving. However, using ordinary convolutional neural networks or vision transformers on this data is problematic due to projection and distortion losses introduced when projecting to a rectangular grid on the plane. We introduce the HEAL-SWIN transformer, which combines the highly uniform Hierarchical Equal Area iso-Latitude Pixelation (HEALPix) grid used in astrophysics and cosmology with the Hierarchical Shifted-Window (SWIN) transformer to yield an efficient and flexible model capable of training on high-resolution, distortion-free spherical data. In HEAL-SWIN, the nested structure of the HEALPix grid is used to perform the patching and windowing operations of the SWIN transformer, resulting in a one-dimensional representation of the spherical data with minimal computational overhead. We demonstrate the superior performance of our model for semantic segmentation and depth regression tasks on both synthetic and real automotive datasets. Our code is available at //github.com/JanEGerken/HEAL-SWIN.

A regular expression specifies a set of strings formed by single characters combined with concatenation, union, and Kleene star operators. Given a regular expression $R$ and a string $Q$, the regular expression matching problem is to decide if $Q$ matches any of the strings specified by $R$. Regular expressions are a fundamental concept in formal languages and regular expression matching is a basic primitive for searching and processing data. A standard textbook solution [Thompson, CACM 1968] constructs and simulates a nondeterministic finite automaton, leading to an $O(nm)$ time algorithm, where $n$ is the length of $Q$ and $m$ is the length of $R$. Despite considerable research efforts only polylogarithmic improvements of this bound are known. Recently, conditional lower bounds provided evidence for this lack of progress when Backurs and Indyk [FOCS 2016] proved that, assuming the strong exponential time hypothesis (SETH), regular expression matching cannot be solved in $O((nm)^{1-\epsilon})$, for any constant $\epsilon > 0$. Hence, the complexity of regular expression matching is essentially settled in terms of $n$ and $m$. In this paper, we take a new approach and go beyond worst-case analysis in $n$ and $m$. We introduce a \emph{density} parameter, $\Delta$, that captures the amount of nondeterminism in the NFA simulation on $Q$. The density is at most $nm+1$ but can be significantly smaller. Our main result is a new algorithm that solves regular expression matching in $$O\left(\Delta \log \log \frac{nm}{\Delta} +n + m\right)$$ time. This essentially replaces $nm$ with $\Delta$ in the complexity of regular expression matching. We complement our upper bound by a matching conditional lower bound that proves that we cannot solve regular expression matching in time $O(\Delta^{1-\epsilon})$ for any constant $\epsilon > 0$ assuming SETH.

Parametricity is a property of the syntax of type theory implying e.g. that there is only one function having the type of the polymorphic identity function. Parametricity is usually proven externally, and does not hold internally. Internalising it is difficult because once there is a term witnessing parametricity, it also has to be parametric itself and this results in the appearance of higher dimensional cubes. In previous theories with internal parametricity, either an explicit syntax for higher cubes is present or the theory is extended with a new sort for the interval. In this paper we present a type theory with internal parametricity which is a simple extension of Martin-L\"of type thoery: there are a few new type formers, term formers and equations. Geometry is not explicit in this syntax, but emergent: the new operations and equations only refer to objects up to dimension 3. We show that this theory is modelled by presheaves over the BCH cube category. Fibrancy conditions are not needed because we use span-based rather than relational parametricity. We define a gluing model for this theory implying that external parametricity and canonicity hold. The theory can be seen as a special case of a new kind of modal type theory, and it is the simplest setting in which the computational properties of higher observational type theory can be demonstrated.

Graph Neural Networks (GNNs) have proven to be useful for many different practical applications. However, many existing GNN models have implicitly assumed homophily among the nodes connected in the graph, and therefore have largely overlooked the important setting of heterophily, where most connected nodes are from different classes. In this work, we propose a novel framework called CPGNN that generalizes GNNs for graphs with either homophily or heterophily. The proposed framework incorporates an interpretable compatibility matrix for modeling the heterophily or homophily level in the graph, which can be learned in an end-to-end fashion, enabling it to go beyond the assumption of strong homophily. Theoretically, we show that replacing the compatibility matrix in our framework with the identity (which represents pure homophily) reduces to GCN. Our extensive experiments demonstrate the effectiveness of our approach in more realistic and challenging experimental settings with significantly less training data compared to previous works: CPGNN variants achieve state-of-the-art results in heterophily settings with or without contextual node features, while maintaining comparable performance in homophily settings.

To retrieve more relevant, appropriate and useful documents given a query, finding clues about that query through the text is crucial. Recent deep learning models regard the task as a term-level matching problem, which seeks exact or similar query patterns in the document. However, we argue that they are inherently based on local interactions and do not generalise to ubiquitous, non-consecutive contextual relationships.In this work, we propose a novel relevance matching model based on graph neural networks to leverage the document-level word relationships for ad-hoc retrieval. In addition to the local interactions, we explicitly incorporate all contexts of a term through the graph-of-word text format. Matching patterns can be revealed accordingly to provide a more accurate relevance score. Our approach significantly outperforms strong baselines on two ad-hoc benchmarks. We also experimentally compare our model with BERT and show our ad-vantages on long documents.

Exploration-exploitation is a powerful and practical tool in multi-agent learning (MAL), however, its effects are far from understood. To make progress in this direction, we study a smooth analogue of Q-learning. We start by showing that our learning model has strong theoretical justification as an optimal model for studying exploration-exploitation. Specifically, we prove that smooth Q-learning has bounded regret in arbitrary games for a cost model that explicitly captures the balance between game and exploration costs and that it always converges to the set of quantal-response equilibria (QRE), the standard solution concept for games under bounded rationality, in weighted potential games with heterogeneous learning agents. In our main task, we then turn to measure the effect of exploration in collective system performance. We characterize the geometry of the QRE surface in low-dimensional MAL systems and link our findings with catastrophe (bifurcation) theory. In particular, as the exploration hyperparameter evolves over-time, the system undergoes phase transitions where the number and stability of equilibria can change radically given an infinitesimal change to the exploration parameter. Based on this, we provide a formal theoretical treatment of how tuning the exploration parameter can provably lead to equilibrium selection with both positive as well as negative (and potentially unbounded) effects to system performance.