The classical Minkowski problem for convex bodies has deeply influenced the development of differential geometry. During the past several decades, abundant mathematical theories have been developed for studying the solutions of the Minkowski problem, however, the numerical solution of this problem has been largely left behind, with only few methods available to achieve that goal. In this article, focusing on the two-dimensional Minkowski problem with Dirichlet boundary conditions, we introduce two solution methods, both based on operator-splitting. One of these two methods deals directly with the Dirichlet condition, while the other method uses an approximation of this Dirichlet condition. This relaxation of the Dirichlet condition makes this second method better suited than the first one to treat those situations where the Minkowski and the Dirichlet condition are not compatible. Both methods are generalizations of the solution method for the canonical Monge-Amp\`{e}re equation discussed by Glowinski et al. (Journal of Scientific Computing, 79(1), 1-47, 2019); as such they take advantage of a divergence formulation of the Minkowski problem, well-suited to a mixed finite element approximation, and to the the time-discretization via an operator-splitting scheme, of an associated initial value problem. Our methodology can be easily implemented on convex domains of rather general shape (with curved boundaries, possibly). The numerical experiments we performed validate both methods and show that if one uses continuous piecewise affine finite element approximations of the smooth solution of the Minkowski problem and of its three second order derivatives, these two methods provide nearly second order accuracy for the $L^2$ and $L^{\infty}$ error. One can extend easily the methods discussed in this article, to address the solution of three-dimensional Minkowski problem.
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Neural networks have become a prominent approach to solve inverse problems in recent years. While a plethora of such methods was developed to solve inverse problems empirically, we are still lacking clear theoretical guarantees for these methods. On the other hand, many works proved convergence to optimal solutions of neural networks in a more general setting using overparametrization as a way to control the Neural Tangent Kernel. In this work we investigate how to bridge these two worlds and we provide deterministic convergence and recovery guarantees for the class of unsupervised feedforward multilayer neural networks trained to solve inverse problems. We also derive overparametrization bounds under which a two-layers Deep Inverse Prior network with smooth activation function will benefit from our guarantees.
Datasets that pair Knowledge Graphs (KG) and text together (KG-T) can be used to train forward and reverse neural models that generate text from KG and vice versa. However models trained on datasets where KG and text pairs are not equivalent can suffer from more hallucination and poorer recall. In this paper, we verify this empirically by generating datasets with different levels of noise and find that noisier datasets do indeed lead to more hallucination. We argue that the ability of forward and reverse models trained on a dataset to cyclically regenerate source KG or text is a proxy for the equivalence between the KG and the text in the dataset. Using cyclic evaluation we find that manually created WebNLG is much better than automatically created TeKGen and T-REx. Guided by these observations, we construct a new, improved dataset called LAGRANGE using heuristics meant to improve equivalence between KG and text and show the impact of each of the heuristics on cyclic evaluation. We also construct two synthetic datasets using large language models (LLMs), and observe that these are conducive to models that perform significantly well on cyclic generation of text, but less so on cyclic generation of KGs, probably because of a lack of a consistent underlying ontology.
Gender bias in education gained considerable relevance in the literature over the years. However, while the problem of gender bias in education has been widely addressed from a student perspective, it is still not fully analysed from an academic point of view. In this work, we study the problem of gender bias in academic promotions (i.e., from Researcher to Associated Professor and from Associated to Full Professor) in the informatics (INF) and software engineering (SE) Italian communities. In particular, we first conduct a literature review to assess how the problem of gender bias in academia has been addressed so far. Next, we describe a process to collect and preprocess the INF and SE data needed to analyse gender bias in Italian academic promotions. Subsequently, we apply a formal bias metric to these data to assess the amount of bias and look at its variation over time. From the conducted analysis, we observe how the SE community presents a higher bias in promotions to Associate Professors and a smaller bias in promotions to Full Professors compared to the overall INF community.
Following the success of GPT4, there has been a surge in interest in multimodal large language model (MLLM) research. This line of research focuses on developing general-purpose LLMs through fine-tuning pre-trained LLMs and vision models. However, catastrophic forgetting, a notorious phenomenon where the fine-tuned model fails to retain similar performance compared to the pre-trained model, still remains an inherent problem in multimodal LLMs (MLLM). In this paper, we introduce EMT: Evaluating MulTimodality for evaluating the catastrophic forgetting in MLLMs, by treating each MLLM as an image classifier. We first apply EMT to evaluate several open-source fine-tuned MLLMs and we discover that almost all evaluated MLLMs fail to retain the same performance levels as their vision encoders on standard image classification tasks. Moreover, we continue fine-tuning LLaVA, an MLLM and utilize EMT to assess performance throughout the fine-tuning. Interestingly, our results suggest that early-stage fine-tuning on an image dataset improves performance across other image datasets, by enhancing the alignment of text and visual features. However, as fine-tuning proceeds, the MLLMs begin to hallucinate, resulting in a significant loss of generalizability, even when the image encoder remains frozen. Our results suggest that MLLMs have yet to demonstrate performance on par with their vision models on standard image classification tasks and the current MLLM fine-tuning procedure still has room for improvement.
Philosophical research in AI has hitherto largely focused on the ethics of AI. In this paper we, an ethicist of belief and a machine learning scientist, suggest that we need to pursue a novel area of philosophical research in AI - the epistemology of AI, and in particular an ethics of belief for AI. Here we take the ethics of belief, a field that has been defined in various ways, to refer to a sub-field within epistemology. This subfield is concerned with the study of possible moral, practical, and other non-alethic dimensions of belief. And in this paper, we will primarily be concerned with the normative question within the ethics of belief regarding what agents - both human and artificial - ought to believe, rather than with descriptive questions concerning whether certain beliefs meet various evaluative standards such as being true, being justified or warranted, constituting knowledge, and so on. We suggest four topics in extant work in the ethics of (human) belief that can be applied to an ethics of AI belief: doxastic wronging by AI; morally owed beliefs; pragmatic and moral encroachment on AI beliefs; and moral responsibility for AI beliefs. We also indicate two relatively nascent areas of philosophical research that haven't yet been generally recognized as ethics of AI belief research, but that do fall within this field of research in virtue of investigating various moral and practical dimensions of belief: the epistemic and ethical decolonization of AI; and epistemic injustice in AI.
The Metaverse is a new paradigm that aims to create a virtual environment consisting of numerous worlds, each of which will offer a different set of services. To deal with such a dynamic and complex scenario, considering the stringent quality of service requirements aimed at the 6th generation of communication systems (6G), one potential approach is to adopt self-sustaining strategies, which can be realized by employing Adaptive Artificial Intelligence (Adaptive AI) where models are continually re-trained with new data and conditions. One aspect of self-sustainability is the management of multiple access to the frequency spectrum. Although several innovative methods have been proposed to address this challenge, mostly using Deep Reinforcement Learning (DRL), the problem of adapting agents to a non-stationary environment has not yet been precisely addressed. This paper fills in the gap in the current literature by investigating the problem of multiple access in multi-channel environments to maximize the throughput of the intelligent agent when the number of active User Equipments (UEs) may fluctuate over time. To solve the problem, a Double Deep Q-Learning (DDQL) technique empowered by Continual Learning (CL) is proposed to overcome the non-stationary situation, while the environment is unknown. Numerical simulations demonstrate that, compared to other well-known methods, the CL-DDQL algorithm achieves significantly higher throughputs with a considerably shorter convergence time in highly dynamic scenarios.
For estimating the proportion of false null hypotheses in multiple testing, a family of estimators by Storey (2002) is widely used in the applied and statistical literature, with many methods suggested for selecting the parameter $\lambda$. Inspired by change-point concepts, our new approach to the latter problem first approximates the $p$-value plot with a piecewise linear function with a single change-point and then selects the $p$-value at the change-point location as $\lambda$. Simulations show that our method has among the smallest RMSE across various settings, and we extend it to address the estimation in cases of superuniform $p$-values. We provide asymptotic theory for our estimator, relying on the theory of quantile processes. Additionally, we propose an application in the change-point literature and illustrate it using high-dimensional CNV data.
Incompleteness is a common problem for existing knowledge graphs (KGs), and the completion of KG which aims to predict links between entities is challenging. Most existing KG completion methods only consider the direct relation between nodes and ignore the relation paths which contain useful information for link prediction. Recently, a few methods take relation paths into consideration but pay less attention to the order of relations in paths which is important for reasoning. In addition, these path-based models always ignore nonlinear contributions of path features for link prediction. To solve these problems, we propose a novel KG completion method named OPTransE. Instead of embedding both entities of a relation into the same latent space as in previous methods, we project the head entity and the tail entity of each relation into different spaces to guarantee the order of relations in the path. Meanwhile, we adopt a pooling strategy to extract nonlinear and complex features of different paths to further improve the performance of link prediction. Experimental results on two benchmark datasets show that the proposed model OPTransE performs better than state-of-the-art methods.
Deep Convolutional Neural Networks have pushed the state-of-the art for semantic segmentation provided that a large amount of images together with pixel-wise annotations is available. Data collection is expensive and a solution to alleviate it is to use transfer learning. This reduces the amount of annotated data required for the network training but it does not get rid of this heavy processing step. We propose a method of transfer learning without annotations on the target task for datasets with redundant content and distinct pixel distributions. Our method takes advantage of the approximate content alignment of the images between two datasets when the approximation error prevents the reuse of annotation from one dataset to another. Given the annotations for only one dataset, we train a first network in a supervised manner. This network autonomously learns to generate deep data representations relevant to the semantic segmentation. Then the images in the new dataset, we train a new network to generate a deep data representation that matches the one from the first network on the previous dataset. The training consists in a regression between feature maps and does not require any annotations on the new dataset. We show that this method reaches performances similar to a classic transfer learning on the PASCAL VOC dataset with synthetic transformations.
We investigate the problem of automatically determining what type of shoe left an impression found at a crime scene. This recognition problem is made difficult by the variability in types of crime scene evidence (ranging from traces of dust or oil on hard surfaces to impressions made in soil) and the lack of comprehensive databases of shoe outsole tread patterns. We find that mid-level features extracted by pre-trained convolutional neural nets are surprisingly effective descriptors for this specialized domains. However, the choice of similarity measure for matching exemplars to a query image is essential to good performance. For matching multi-channel deep features, we propose the use of multi-channel normalized cross-correlation and analyze its effectiveness. Our proposed metric significantly improves performance in matching crime scene shoeprints to laboratory test impressions. We also show its effectiveness in other cross-domain image retrieval problems: matching facade images to segmentation labels and aerial photos to map images. Finally, we introduce a discriminatively trained variant and fine-tune our system through our proposed metric, obtaining state-of-the-art performance.
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