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Obtaining sparse, interpretable representations of observable data is crucial in many machine learning and signal processing tasks. For data representing flows along the edges of a graph, an intuitively interpretable way to obtain such representations is to lift the graph structure to a simplicial complex: The eigenvectors of the associated Hodge-Laplacian, respectively the incidence matrices of the corresponding simplicial complex then induce a Hodge decomposition, which can be used to represent the observed data in terms of gradient, curl, and harmonic flows. In this paper, we generalize this approach to cellular complexes and introduce the cell inference optimization problem, i.e., the problem of augmenting the observed graph by a set of cells, such that the eigenvectors of the associated Hodge Laplacian provide a sparse, interpretable representation of the observed edge flows on the graph. We show that this problem is NP-hard and introduce an efficient approximation algorithm for its solution. Experiments on real-world and synthetic data demonstrate that our algorithm outperforms current state-of-the-art methods while being computationally efficient.

Language models (LMs) have already demonstrated remarkable abilities in understanding and generating both natural and formal language. Despite these advances, their integration with real-world environments such as large-scale knowledge bases (KBs) remains an underdeveloped area, affecting applications such as semantic parsing and indulging in "hallucinated" information. This paper is an experimental investigation aimed at uncovering the robustness challenges that LMs encounter when tasked with knowledge base question answering (KBQA). The investigation covers scenarios with inconsistent data distribution between training and inference, such as generalization to unseen domains, adaptation to various language variations, and transferability across different datasets. Our comprehensive experiments reveal that even when employed with our proposed data augmentation techniques, advanced small and large language models exhibit poor performance in various dimensions. While the LM is a promising technology, the robustness of the current form in dealing with complex environments is fragile and of limited practicality because of the data distribution issue. This calls for future research on data collection and LM learning paradims.

Due to the imbalanced nature of networked observational data, the causal effect predictions for some individuals can severely violate the positivity/overlap assumption, rendering unreliable estimations. Nevertheless, this potential risk of individual-level treatment effect estimation on networked data has been largely under-explored. To create a more trustworthy causal effect estimator, we propose the uncertainty-aware graph deep kernel learning (GraphDKL) framework with Lipschitz constraint to model the prediction uncertainty with Gaussian process and identify unreliable estimations. To the best of our knowledge, GraphDKL is the first framework to tackle the violation of positivity assumption when performing causal effect estimation with graphs. With extensive experiments, we demonstrate the superiority of our proposed method in uncertainty-aware causal effect estimation on networked data.

Stochastic volatility models, where the volatility is a stochastic process, can capture most of the essential stylized facts of implied volatility surfaces and give more realistic dynamics of the volatility smile or skew. However, they come with the significant issue that they take too long to calibrate. Alternative calibration methods based on Deep Learning (DL) techniques have been recently used to build fast and accurate solutions to the calibration problem. Huge and Savine developed a Differential Deep Learning (DDL) approach, where Machine Learning models are trained on samples of not only features and labels but also differentials of labels to features. The present work aims to apply the DDL technique to price vanilla European options (i.e. the calibration instruments), more specifically, puts when the underlying asset follows a Heston model and then calibrate the model on the trained network. DDL allows for fast training and accurate pricing. The trained neural network dramatically reduces Heston calibration's computation time. In this work, we also introduce different regularisation techniques, and we apply them notably in the case of the DDL. We compare their performance in reducing overfitting and improving the generalisation error. The DDL performance is also compared to the classical DL (without differentiation) one in the case of Feed-Forward Neural Networks. We show that the DDL outperforms the DL.

Despite the promising progress in multi-modal tasks, current large multi-modal models (LMM) are prone to hallucinating inconsistent descriptions with respect to the associated image and human instructions. This paper addresses this issue by introducing the first large and diverse visual instruction tuning dataset, named Large-scale Robust Visual (LRV)-Instruction. Our dataset consists of 120k visual instructions generated by GPT4, covering 16 vision-and-language tasks with open-ended instructions and answers. Unlike existing studies that primarily focus on positive instruction samples, we design LRV-Instruction to include both positive and negative instructions for more robust visual instruction tuning. Our negative instructions are designed at two semantic levels: (i) Nonexistent Element Manipulation and (ii) Existent Element Manipulation. To efficiently measure the hallucination generated by LMMs, we propose GPT4-Assisted Visual Instruction Evaluation (GAVIE), a novel approach to evaluate visual instruction tuning without the need for human-annotated groundtruth answers and can adapt to diverse instruction formats. We conduct comprehensive experiments to investigate the hallucination of LMMs. Our results demonstrate that existing LMMs exhibit significant hallucination when presented with our negative instructions, particularly with Existent Element Manipulation instructions. Moreover, by finetuning MiniGPT4 on LRV-Instruction, we successfully mitigate hallucination while improving performance on public datasets using less training data compared to state-of-the-art methods. Additionally, we observed that a balanced ratio of positive and negative instances in the training data leads to a more robust model. Updates of our project are available at //fuxiaoliu.github.io/LRV/.

We propose an upwind finite volume method for a system of two kinetic equations in one dimension that are coupled through nonlocal interaction terms. These cross-interaction systems were recently obtained as the mean-field limit of a second-order system of ordinary differential equations for two interacting species. Models of this kind are encountered in a myriad of contexts, for instance, to describe large systems of indistinguishable agents such as cell colonies, flocks of birds, schools of fish, herds of sheep. The finite volume method we propose is constructed to conserve mass and preserve positivity. Moreover, convex functionals of the discrete solution are controlled, which we use to show the convergence of the scheme. Finally, we investigate the scheme numerically.

To plan the trajectories of a large and heterogeneous swarm, sequential or synchronous distributed methods usually become intractable, due to the lack of global connectivity and clock synchronization, Moreover, the existing asynchronously distributed schemes usually require recheck-like mechanisms instead of inherently considering the other' moving tendency. To this end, we propose a novel asynchronous protocol to allocate the agents' derivable space in a distributed way, by which each agent can replan trajectory depending on its own timetable. Properties such as collision avoidance and recursive feasibility are theoretically shown and a lower bound of protocol updating is provided. Comprehensive simulations and comparisons with five state-of-the-art methods validate the effectiveness of our method and illustrate the improvement in both the completion time and the moving distance. Finally, hardware experiments are carried out, where 8 heterogeneous unmanned ground vehicles with onboard computation navigate in cluttered scenarios at a high agility.

We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. Motivated by the universal approximation property of this type of networks, both methods extend the extreme learning machine (ELM) approach from low to high dimensions. With the first method the unknown solution field in $d$ dimensions is represented by a randomized feed-forward neural network, in which the hidden-layer parameters are randomly assigned and fixed while the output-layer parameters are trained. The PDE and the boundary/initial conditions, as well as the continuity conditions (for the local variant of the method), are enforced on a set of random interior/boundary collocation points. The resultant linear or nonlinear algebraic system, through its least squares solution, provides the trained values for the network parameters. With the second method the high-dimensional PDE problem is reformulated through a constrained expression based on an Approximate variant of the Theory of Functional Connections (A-TFC), which avoids the exponential growth in the number of terms of TFC as the dimension increases. The free field function in the A-TFC constrained expression is represented by a randomized neural network and is trained by a procedure analogous to the first method. We present ample numerical simulations for a number of high-dimensional linear/nonlinear stationary/dynamic PDEs to demonstrate their performance. These methods can produce accurate solutions to high-dimensional PDEs, in particular with their errors reaching levels not far from the machine accuracy for relatively lower dimensions. Compared with the physics-informed neural network (PINN) method, the current method is both cost-effective and more accurate for high-dimensional PDEs.

Generalized metric spaces are obtained by weakening the requirements (e.g., symmetry) on the distance function and by allowing it to take values in structures (e.g., quantales) that are more general than the set of non-negative real numbers. Quantale-valued metric spaces have gained prominence due to their use in quantitative reasoning on programs/systems, and for defining various notions of behavioral metrics. We investigate imprecision and robustness in the framework of quantale-valued metric spaces, when the quantale is continuous. In particular, we study the relation between the robust topology, which captures robustness of analyses, and the Hausdorff-Smyth hemi-metric. To this end, we define a preorder-enriched monad $\mathsf{P}_S$, called the Hausdorff-Smyth monad, and when $Q$ is a continuous quantale and $X$ is a $Q$-metric space, we relate the topology induced by the metric on $\mathsf{P}_S(X)$ with the robust topology on the powerset $\mathsf{P}(X)$ defined in terms of the metric on $X$.

Embedding entities and relations into a continuous multi-dimensional vector space have become the dominant method for knowledge graph embedding in representation learning. However, most existing models ignore to represent hierarchical knowledge, such as the similarities and dissimilarities of entities in one domain. We proposed to learn a Domain Representations over existing knowledge graph embedding models, such that entities that have similar attributes are organized into the same domain. Such hierarchical knowledge of domains can give further evidence in link prediction. Experimental results show that domain embeddings give a significant improvement over the most recent state-of-art baseline knowledge graph embedding models.