Now,since|N(v)|=β,itholds (Px)[v]= a+b 2, thus verifying the first claim of the lemma as the choice ofv was arbitrary. This construction essentially generalizes the graph demonstrated in Figure 1 of the main paper (see Sec. 7). The following lemma shows that onsuch graphs, the filter responses ofgθ for aconstant signal will encode some geometric information, butwill not distinguish between the cycles inthe graph. These responses with appropriate color coding give the illustration in Figure 1 in the main paper. Validation & testing procedure: All tests were done using train-validation-test splits of the datasets, where validation accuracy is used for tuning hyperparameters and test accuracy is reportedinthecomparisontable.

For any node v V, according to Eq. 8, we can write (Px)[v ] = This construction essentially generalizes the graph demonstrated in Figure 1 of the main paper (see Sec. 7). Therefore, computing these boils down to a simple proof by cases: 1. Let us now revisit the example given in the main paper (see Figure 1 or a copy in Figure 1). These responses with appropriate color coding give the illustration in Figure 1 in the main paper. As many applied fields such as Bioinformatics and Neuroscience heavily rely on the analysis of graph-structured data, the study of reliable classification methods has received much attention lately.

Graph neural networks (GNNs) have achieved great success for a variety of tasks such as node classification, graph classification, and link prediction. However, the use of GNNs (and machine learning more generally) to solve combinatorial optimization (CO) problems is much less explored. Here, we introduce a novel GNN architecture which leverages a complex filter bank and localized attention mechanisms designed to solve CO problems on graphs. We show how our method differentiates itself from prior GNN-based CO solvers and how it can be effectively applied to the maximum clique, minimum dominating set, and maximum cut problems in a self-supervised learning setting. In addition to demonstrating competitive overall performance across all tasks, we establish state-of-the-art results for the max cut problem.

The recent works proposing transformer-based models for graphs have proven the inadequacy of Vanilla Transformer for graph representation learning. To understand this inadequacy, there is a need to investigate if spectral analysis of the transformer will reveal insights into its expressive power. Similar studies already established that spectral analysis of Graph neural networks (GNNs) provides extra perspectives on their expressiveness. In this work, we systematically study and establish the link between the spatial and spectral domain in the realm of the transformer. We further provide a theoretical analysis and prove that the spatial attention mechanism in the transformer cannot effectively capture the desired frequency response, thus, inherently limiting its expressiveness in spectral space. Therefore, we propose FeTA, a framework that aims to perform attention over the entire graph spectrum (i.e., actual frequency components of the graphs) analogous to the attention in spatial space. Empirical results suggest that FeTA provides homogeneous performance gain against vanilla transformer across all tasks on standard benchmarks and can easily be extended to GNN-based models with low-pass characteristics (e.g., GAT).

Painting has played a great part in human life. Since thousand years human represents their culture, art of living, and various more through painting. There is a saying "a picture is worth a thousand words", means many ideas can be shared through a painting. Let's how we can generate an artistic image automatically within a short period. After the rise of machine intelligence, we have taken it to another level whereby the help of Neural Network which designed in such a way that mimics the human brain can synthesize images and gives a new flavor to it in an artistic manner.

Although deep learning models have been successfully applied to a variety of tasks, due to the millions of parameters, they are becoming increasingly opaque and complex. In order to establish trust for their widespread commercial use, it is important to formalize a principled framework to reason over these models. In this work, we use ideas from causal inference to describe a general framework to reason over CNN models. Specifically, we build a Structural Causal Model (SCM) as an abstraction over a specific aspect of the CNN. We also formulate a method to quantitatively rank the filters of a convolution layer according to their counterfactual importance. We illustrate our approach with popular CNN architectures such as LeNet5, VGG19, and ResNet32.

Global average pooling (GAP) has been used previously to generate class activation maps. The motivation behind AdGAP comes from the fact that the convolutional filters possess position information of the essential features and hence, combination of the feature maps could help us locate the class instances in an image. Our novel architecture generates promising results and unlike previous methods, the architecture is not sensitive to the size of the input image, thus promising wider application.

In an effort to understand the meaning of the intermediate representations captured by deep networks, recent papers have tried to associate specific semantic concepts to individual neural network filter responses, where interesting correlations are often found, largely by focusing on extremal filter responses. In this paper, we show that this approach can favor easy-to-interpret cases that are not necessarily representative of the average behavior of a representation. A more realistic but harder-to-study hypothesis is that semantic representations are distributed, and thus filters must be studied in conjunction. In order to investigate this idea while enabling systematic visualization and quantification of multiple filter responses, we introduce the Net2Vec framework, in which semantic concepts are mapped to vectorial embeddings based on corresponding filter responses. By studying such embeddings, we are able to show that 1., in most cases, multiple filters are required to code for a concept, that 2., often filters are not concept specific and help encode multiple concepts, and that 3., compared to single filter activations, filter embeddings are able to better characterize the meaning of a representation and its relationship to other concepts.

Convolutional Neural Networks have been a subject of great importance over the past decade and great strides have been made in their utility for producing state of the art performance in many computer vision problems. However, the behavior of deep networks is yet to be fully understood and is still an active area of research. In this work, we present an intriguing behavior: pre-trained CNNs can be made to improve their predictions by structurally perturbing the input. We observe that these perturbations - referred as Guided Perturbations - enable a trained network to improve its prediction performance without any learning or change in network weights. We perform various ablative experiments to understand how these perturbations affect the local context and feature representations. Furthermore, we demonstrate that this idea can improve performance of several existing approaches on semantic segmentation and scene labeling tasks on the PASCAL VOC dataset and supervised classification tasks on MNIST and CIFAR10 datasets.

We consider the task of learning to extract motion from videos. To this end, we show that the detection of spatial transformations can be viewed as the detection of synchrony between the image sequence and a sequence of features undergoing the motion we wish to detect. We show that learning about synchrony is possible using very fast, local learning rules, by introducing multiplicative "gating" interactions between hidden units across frames. This makes it possible to achieve competitive performance in a wide variety of motion estimation tasks, using a small fraction of the time required to learn features, and to outperform hand-crafted spatio-temporal features by a large margin. We also show how learning about synchrony can be viewed as performing greedy parameter estimation in the well-known motion energy model.