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In this paper, we study the outerplanarity of planar graphs, i.e., the number of times that we must (in a planar embedding that we can initially freely choose) remove the outerface vertices until the graph is empty. It is well-known that there are $n$-vertex graphs with outerplanarity $\tfrac{n}{6}+\Theta(1)$, and not difficult to show that the outerplanarity can never be bigger. We give here improved bounds of the form $\tfrac{n}{2g}+2g+O(1)$, where $g$ is the fence-girth, i.e., the length of the shortest cycle with vertices on both sides. This parameter $g$ is at least the connectivity of the graph, and often bigger; for example, our results imply that planar bipartite graphs have outerplanarity $\tfrac{n}{8}+O(1)$. We also show that the outerplanarity of a planar graph $G$ is at most $\tfrac{1}{2}$diam$(G)+O(\sqrt{n})$, where diam$(G)$ is the diameter of the graph. All our bounds are tight up to smaller-order terms, and a planar embedding that achieves the outerplanarity bound can be found in linear time.

We study estimation and inference on causal parameters under finely stratified rerandomization designs, which use baseline covariates to match units into groups (e.g. matched pairs), then rerandomize within-group treatment assignments until a balance criterion is satisfied. We show that finely stratified rerandomization does partially linear regression adjustment by design, providing nonparametric control over the covariates used for stratification, and linear control over the rerandomization covariates. We also introduce novel rerandomization criteria, allowing for nonlinear imbalance metrics and proposing a minimax scheme that optimizes the balance criterion using pilot data or prior information provided by the researcher. While the asymptotic distribution of generalized method of moments (GMM) estimators under stratified rerandomization is generically non-Gaussian, we show how to restore asymptotic normality using optimal ex-post linear adjustment. This allows us to provide simple asymptotically exact inference methods for superpopulation parameters, as well as efficient conservative inference methods for finite population parameters.

In this work, we deal with the problem of re compression based image forgery detection, where some regions of an image are modified illegitimately, hence giving rise to presence of dual compression characteristics within a single image. There have been some significant researches in this direction, in the last decade. However, almost all existing techniques fail to detect this form of forgery, when the first compression factor is greater than the second. We address this problem in re compression based forgery detection, here Recently, Machine Learning techniques have started gaining a lot of importance in the domain of digital image forensics. In this work, we propose a Convolution Neural Network based deep learning architecture, which is capable of detecting the presence of re compression based forgery in JPEG images. The proposed architecture works equally efficiently, even in cases where the first compression ratio is greater than the second. In this work, we also aim to localize the regions of image manipulation based on re compression features, using the trained neural network. Our experimental results prove that the proposed method outperforms the state of the art, with respect to forgery detection and localization accuracy.

In this paper, we present an information-theoretic method for clustering mixed-type data, that is, data consisting of both continuous and categorical variables. The method is a variant of the Deterministic Information Bottleneck algorithm which optimally compresses the data while retaining relevant information about the underlying structure. We compare the performance of the proposed method to that of three well-established clustering methods (KAMILA, K-Prototypes, and Partitioning Around Medoids with Gower's dissimilarity) on simulated and real-world datasets. The results demonstrate that the proposed approach represents a competitive alternative to conventional clustering techniques under specific conditions.

This book is the result of a seminar in which we reviewed multimodal approaches and attempted to create a solid overview of the field, starting with the current state-of-the-art approaches in the two subfields of Deep Learning individually. Further, modeling frameworks are discussed where one modality is transformed into the other, as well as models in which one modality is utilized to enhance representation learning for the other. To conclude the second part, architectures with a focus on handling both modalities simultaneously are introduced. Finally, we also cover other modalities as well as general-purpose multi-modal models, which are able to handle different tasks on different modalities within one unified architecture. One interesting application (Generative Art) eventually caps off this booklet.

In order to overcome the expressive limitations of graph neural networks (GNNs), we propose the first method that exploits vector flows over graphs to develop globally consistent directional and asymmetric aggregation functions. We show that our directional graph networks (DGNs) generalize convolutional neural networks (CNNs) when applied on a grid. Whereas recent theoretical works focus on understanding local neighbourhoods, local structures and local isomorphism with no global information flow, our novel theoretical framework allows directional convolutional kernels in any graph. First, by defining a vector field in the graph, we develop a method of applying directional derivatives and smoothing by projecting node-specific messages into the field. Then we propose the use of the Laplacian eigenvectors as such vector field, and we show that the method generalizes CNNs on an n-dimensional grid, and is provably more discriminative than standard GNNs regarding the Weisfeiler-Lehman 1-WL test. Finally, we bring the power of CNN data augmentation to graphs by providing a means of doing reflection, rotation and distortion on the underlying directional field. We evaluate our method on different standard benchmarks and see a relative error reduction of 8\% on the CIFAR10 graph dataset and 11% to 32% on the molecular ZINC dataset. An important outcome of this work is that it enables to translate any physical or biological problems with intrinsic directional axes into a graph network formalism with an embedded directional field.

In this paper, we propose Latent Relation Language Models (LRLMs), a class of language models that parameterizes the joint distribution over the words in a document and the entities that occur therein via knowledge graph relations. This model has a number of attractive properties: it not only improves language modeling performance, but is also able to annotate the posterior probability of entity spans for a given text through relations. Experiments demonstrate empirical improvements over both a word-based baseline language model and a previous approach that incorporates knowledge graph information. Qualitative analysis further demonstrates the proposed model's ability to learn to predict appropriate relations in context.

This paper is an attempt to explain all the matrix calculus you need in order to understand the training of deep neural networks. We assume no math knowledge beyond what you learned in calculus 1, and provide links to help you refresh the necessary math where needed. Note that you do not need to understand this material before you start learning to train and use deep learning in practice; rather, this material is for those who are already familiar with the basics of neural networks, and wish to deepen their understanding of the underlying math. Don't worry if you get stuck at some point along the way---just go back and reread the previous section, and try writing down and working through some examples. And if you're still stuck, we're happy to answer your questions in the Theory category at forums.fast.ai. Note: There is a reference section at the end of the paper summarizing all the key matrix calculus rules and terminology discussed here. See related articles at //explained.ai

In this paper, we introduce the Reinforced Mnemonic Reader for machine reading comprehension tasks, which enhances previous attentive readers in two aspects. First, a reattention mechanism is proposed to refine current attentions by directly accessing to past attentions that are temporally memorized in a multi-round alignment architecture, so as to avoid the problems of attention redundancy and attention deficiency. Second, a new optimization approach, called dynamic-critical reinforcement learning, is introduced to extend the standard supervised method. It always encourages to predict a more acceptable answer so as to address the convergence suppression problem occurred in traditional reinforcement learning algorithms. Extensive experiments on the Stanford Question Answering Dataset (SQuAD) show that our model achieves state-of-the-art results. Meanwhile, our model outperforms previous systems by over 6% in terms of both Exact Match and F1 metrics on two adversarial SQuAD datasets.

This paper proposes a method to modify traditional convolutional neural networks (CNNs) into interpretable CNNs, in order to clarify knowledge representations in high conv-layers of CNNs. In an interpretable CNN, each filter in a high conv-layer represents a certain object part. We do not need any annotations of object parts or textures to supervise the learning process. Instead, the interpretable CNN automatically assigns each filter in a high conv-layer with an object part during the learning process. Our method can be applied to different types of CNNs with different structures. The clear knowledge representation in an interpretable CNN can help people understand the logics inside a CNN, i.e., based on which patterns the CNN makes the decision. Experiments showed that filters in an interpretable CNN were more semantically meaningful than those in traditional CNNs.