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The Densest Subgraph Problem requires to find, in a given graph, a subset of vertices whose induced subgraph maximizes a measure of density. The problem has received a great deal of attention in the algorithmic literature since the early 1970s, with many variants proposed and many applications built on top of this basic definition. Recent years have witnessed a revival of research interest in this problem with several important contributions, including some groundbreaking results, published in 2022 and 2023. This survey provides a deep overview of the fundamental results and an exhaustive coverage of the many variants proposed in the literature, with a special attention to the most recent results. The survey also presents a comprehensive overview of applications and discusses some interesting open problems for this evergreen research topic.

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Advancements like Generative Adversarial Networks have attracted the attention of researchers toward face image synthesis to generate ever more realistic images. Thereby, the need for the evaluation criteria to assess the realism of the generated images has become apparent. While FID utilized with InceptionV3 is one of the primary choices for benchmarking, concerns about InceptionV3's limitations for face images have emerged. This study investigates the behavior of diverse feature extractors -- InceptionV3, CLIP, DINOv2, and ArcFace -- considering a variety of metrics -- FID, KID, Precision\&Recall. While the FFHQ dataset is used as the target domain, as the source domains, the CelebA-HQ dataset and the synthetic datasets generated using StyleGAN2 and Projected FastGAN are used. Experiments include deep-down analysis of the features: $L_2$ normalization, model attention during extraction, and domain distributions in the feature space. We aim to give valuable insights into the behavior of feature extractors for evaluating face image synthesis methodologies. The code is publicly available at //github.com/ThEnded32/AnalyzingFeatureExtractors.

The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: the best known decidability results are for LRS with special properties (e.g., low order recurrences). But these problems are easier for "uninitialized" variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided in polynomial time (Tiwari in 2004, Braverman in 2006). In this paper, we consider problems that lie between the initialized and uninitialized variants. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighbourhood is given as part of the input, then robust Skolem and robust positivity are Diophantine hard, i.e., solving either would entail major breakthroughs in Diophantine approximations, as happens for (non-robust) positivity. However, if one asks whether such a neighbourhood exists, then the problems turn out to be decidable with PSPACE complexity. Our techniques also allow us to tackle robustness for ultimate positivity, which asks whether there is a bound on the number of steps after which the LRS remains positive. There are two variants depending on whether we ask for a "uniform" bound on this number of steps. For the non-uniform variant, when the neighbourhood is open, the problem turns out to be tractable, even when the neighbourhood is given as input.

This paper presents a comparative study of sampling methods within the FedHome framework, designed for personalized in-home health monitoring. FedHome leverages federated learning (FL) and generative convolutional autoencoders (GCAE) to train models on decentralized edge devices while prioritizing data privacy. A notable challenge in this domain is the class imbalance in health data, where critical events such as falls are underrepresented, adversely affecting model performance. To address this, the research evaluates six oversampling techniques using Stratified K-fold cross-validation: SMOTE, Borderline-SMOTE, Random OverSampler, SMOTE-Tomek, SVM-SMOTE, and SMOTE-ENN. These methods are tested on FedHome's public implementation over 200 training rounds with and without stratified K-fold cross-validation. The findings indicate that SMOTE-ENN achieves the most consistent test accuracy, with a standard deviation range of 0.0167-0.0176, demonstrating stable performance compared to other samplers. In contrast, SMOTE and SVM-SMOTE exhibit higher variability in performance, as reflected by their wider standard deviation ranges of 0.0157-0.0180 and 0.0155-0.0180, respectively. Similarly, the Random OverSampler method shows a significant deviation range of 0.0155-0.0176. SMOTE-Tomek, with a deviation range of 0.0160-0.0175, also shows greater stability but not as much as SMOTE-ENN. This finding highlights the potential of SMOTE-ENN to enhance the reliability and accuracy of personalized health monitoring systems within the FedHome framework.

Cardinality Estimation over Knowledge Graphs (KG) is crucial for query optimization, yet remains a challenging task due to the semi-structured nature and complex correlations of typical Knowledge Graphs. In this work, we propose GNCE, a novel approach that leverages knowledge graph embeddings and Graph Neural Networks (GNN) to accurately predict the cardinality of conjunctive queries. GNCE first creates semantically meaningful embeddings for all entities in the KG, which are then integrated into the given query, which is processed by a GNN to estimate the cardinality of the query. We evaluate GNCE on several KGs in terms of q-Error and demonstrate that it outperforms state-of-the-art approaches based on sampling, summaries, and (machine) learning in terms of estimation accuracy while also having lower execution time and less parameters. Additionally, we show that GNCE can inductively generalise to unseen entities, making it suitable for use in dynamic query processing scenarios. Our proposed approach has the potential to significantly improve query optimization and related applications that rely on accurate cardinality estimates of conjunctive queries.

We present novel, convex relaxations for rotation and pose estimation problems that can a posteriori guarantee global optimality for practical measurement noise levels. Some such relaxations exist in the literature for specific problem setups that assume the matrix von Mises-Fisher distribution (a.k.a., matrix Langevin distribution or chordal distance)for isotropic rotational uncertainty. However, another common way to represent uncertainty for rotations and poses is to define anisotropic noise in the associated Lie algebra. Starting from a noise model based on the Cayley map, we define our estimation problems, convert them to Quadratically Constrained Quadratic Programs (QCQPs), then relax them to Semidefinite Programs (SDPs), which can be solved using standard interior-point optimization methods; global optimality follows from Lagrangian strong duality. We first show how to carry out basic rotation and pose averaging. We then turn to the more complex problem of trajectory estimation, which involves many pose variables with both individual and inter-pose measurements (or motion priors). Our contribution is to formulate SDP relaxations for all these problems based on the Cayley map (including the identification of redundant constraints) and to show them working in practical settings. We hope our results can add to the catalogue of useful estimation problems whose solutions can be a posteriori guaranteed to be globally optimal.

Graph Neural Networks (GNNs) have gained momentum in graph representation learning and boosted the state of the art in a variety of areas, such as data mining (\emph{e.g.,} social network analysis and recommender systems), computer vision (\emph{e.g.,} object detection and point cloud learning), and natural language processing (\emph{e.g.,} relation extraction and sequence learning), to name a few. With the emergence of Transformers in natural language processing and computer vision, graph Transformers embed a graph structure into the Transformer architecture to overcome the limitations of local neighborhood aggregation while avoiding strict structural inductive biases. In this paper, we present a comprehensive review of GNNs and graph Transformers in computer vision from a task-oriented perspective. Specifically, we divide their applications in computer vision into five categories according to the modality of input data, \emph{i.e.,} 2D natural images, videos, 3D data, vision + language, and medical images. In each category, we further divide the applications according to a set of vision tasks. Such a task-oriented taxonomy allows us to examine how each task is tackled by different GNN-based approaches and how well these approaches perform. Based on the necessary preliminaries, we provide the definitions and challenges of the tasks, in-depth coverage of the representative approaches, as well as discussions regarding insights, limitations, and future directions.

The dominating NLP paradigm of training a strong neural predictor to perform one task on a specific dataset has led to state-of-the-art performance in a variety of applications (eg. sentiment classification, span-prediction based question answering or machine translation). However, it builds upon the assumption that the data distribution is stationary, ie. that the data is sampled from a fixed distribution both at training and test time. This way of training is inconsistent with how we as humans are able to learn from and operate within a constantly changing stream of information. Moreover, it is ill-adapted to real-world use cases where the data distribution is expected to shift over the course of a model's lifetime. The first goal of this thesis is to characterize the different forms this shift can take in the context of natural language processing, and propose benchmarks and evaluation metrics to measure its effect on current deep learning architectures. We then proceed to take steps to mitigate the effect of distributional shift on NLP models. To this end, we develop methods based on parametric reformulations of the distributionally robust optimization framework. Empirically, we demonstrate that these approaches yield more robust models as demonstrated on a selection of realistic problems. In the third and final part of this thesis, we explore ways of efficiently adapting existing models to new domains or tasks. Our contribution to this topic takes inspiration from information geometry to derive a new gradient update rule which alleviate catastrophic forgetting issues during adaptation.

Deep neural networks (DNNs) are successful in many computer vision tasks. However, the most accurate DNNs require millions of parameters and operations, making them energy, computation and memory intensive. This impedes the deployment of large DNNs in low-power devices with limited compute resources. Recent research improves DNN models by reducing the memory requirement, energy consumption, and number of operations without significantly decreasing the accuracy. This paper surveys the progress of low-power deep learning and computer vision, specifically in regards to inference, and discusses the methods for compacting and accelerating DNN models. The techniques can be divided into four major categories: (1) parameter quantization and pruning, (2) compressed convolutional filters and matrix factorization, (3) network architecture search, and (4) knowledge distillation. We analyze the accuracy, advantages, disadvantages, and potential solutions to the problems with the techniques in each category. We also discuss new evaluation metrics as a guideline for future research.

Deep Convolutional Neural Networks (CNNs) are a special type of Neural Networks, which have shown state-of-the-art results on various competitive benchmarks. The powerful learning ability of deep CNN is largely achieved with the use of multiple non-linear feature extraction stages that can automatically learn hierarchical representation from the data. Availability of a large amount of data and improvements in the hardware processing units have accelerated the research in CNNs and recently very interesting deep CNN architectures are reported. The recent race in deep CNN architectures for achieving high performance on the challenging benchmarks has shown that the innovative architectural ideas, as well as parameter optimization, can improve the CNN performance on various vision-related tasks. In this regard, different ideas in the CNN design have been explored such as use of different activation and loss functions, parameter optimization, regularization, and restructuring of processing units. However, the major improvement in representational capacity is achieved by the restructuring of the processing units. Especially, the idea of using a block as a structural unit instead of a layer is gaining substantial appreciation. This survey thus focuses on the intrinsic taxonomy present in the recently reported CNN architectures and consequently, classifies the recent innovations in CNN architectures into seven different categories. These seven categories are based on spatial exploitation, depth, multi-path, width, feature map exploitation, channel boosting and attention. Additionally, it covers the elementary understanding of the CNN components and sheds light on the current challenges and applications of CNNs.

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.

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