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### How Graph Neural Networks (GNN) work: introduction to graph convolutions from scratch

In this tutorial, we will explore graph neural networks and graph convolutions. Graphs are a super general representation of data with intrinsic structure. I will make clear some fuzzy concepts for beginners in this field. The most intuitive transition to graphs is by starting from images. Because images are highly structured data. Their components (pixels) are arranged in a meaningful way. If you change the way pixels are structured the image loses its meaning.

### Post-processing for Individual Fairness

Post-processing in algorithmic fairness is a versatile approach for correcting bias in ML systems that are already used in production. The main appeal of post-processing is that it avoids expensive retraining. In this work, we propose general post-processing algorithms for individual fairness (IF). We consider a setting where the learner only has access to the predictions of the original model and a similarity graph between individuals, guiding the desired fairness constraints. We cast the IF post-processing problem as a graph smoothing problem corresponding to graph Laplacian regularization that preserves the desired "treat similar individuals similarly" interpretation. Our theoretical results demonstrate the connection of the new objective function to a local relaxation of the original individual fairness. Empirically, our post-processing algorithms correct individual biases in large-scale NLP models such as BERT, while preserving accuracy.

### On the Stability of Low Pass Graph Filter With a Large Number of Edge Rewires

Recently, the stability of graph filters has been studied as one of the key theoretical properties driving the highly successful graph convolutional neural networks (GCNs). The stability of a graph filter characterizes the effect of topology perturbation on the output of a graph filter, a fundamental building block for GCNs. Many existing results have focused on the regime of small perturbation with a small number of edge rewires. However, the number of edge rewires can be large in many applications. To study the latter case, this work departs from the previous analysis and proves a bound on the stability of graph filter relying on the filter's frequency response. Assuming the graph filter is low pass, we show that the stability of the filter depends on perturbation to the community structure. As an application, we show that for stochastic block model graphs, the graph filter distance converges to zero when the number of nodes approaches infinity. Numerical simulations validate our findings.

### Deep Unsupervised Feature Selection by Discarding Nuisance and Correlated Features

Modern datasets often contain large subsets of correlated features and nuisance features, which are not or loosely related to the main underlying structures of the data. Nuisance features can be identified using the Laplacian score criterion, which evaluates the importance of a given feature via its consistency with the Graph Laplacians' leading eigenvectors. We demonstrate that in the presence of large numbers of nuisance features, the Laplacian must be computed on the subset of selected features rather than on the complete feature set. To do this, we propose a fully differentiable approach for unsupervised feature selection, utilizing the Laplacian score criterion to avoid the selection of nuisance features. We employ an autoencoder architecture to cope with correlated features, trained to reconstruct the data from the subset of selected features. Building on the recently proposed concrete layer that allows controlling for the number of selected features via architectural design, simplifying the optimization process. Experimenting on several real-world datasets, we demonstrate that our proposed approach outperforms similar approaches designed to avoid only correlated or nuisance features, but not both. Several state-of-the-art clustering results are reported.

### Knowledge Sheaves: A Sheaf-Theoretic Framework for Knowledge Graph Embedding

Knowledge graph embedding involves learning representations of entities -- the vertices of the graph -- and relations -- the edges of the graph -- such that the resulting representations encode the known factual information represented by the knowledge graph are internally consistent and can be used in the inference of new relations. We show that knowledge graph embedding is naturally expressed in the topological and categorical language of \textit{cellular sheaves}: learning a knowledge graph embedding corresponds to learning a \textit{knowledge sheaf} over the graph, subject to certain constraints. In addition to providing a generalized framework for reasoning about knowledge graph embedding models, this sheaf-theoretic perspective admits the expression of a broad class of prior constraints on embeddings and offers novel inferential capabilities. We leverage the recently developed spectral theory of sheaf Laplacians to understand the local and global consistency of embeddings and develop new methods for reasoning over composite relations through harmonic extension with respect to the sheaf Laplacian. We then implement these ideas to highlight the benefits of the extensions inspired by this new perspective.

### DROP: Deep relocating option policy for optimal ride-hailing vehicle repositioning

In a ride-hailing system, an optimal relocation of vacant vehicles can significantly reduce fleet idling time and balance the supply-demand distribution, enhancing system efficiency and promoting driver satisfaction and retention. Model-free deep reinforcement learning (DRL) has been shown to dynamically learn the relocating policy by actively interacting with the intrinsic dynamics in large-scale ride-hailing systems. However, the issues of sparse reward signals and unbalanced demand and supply distribution place critical barriers in developing effective DRL models. Conventional exploration strategy (e.g., the $\epsilon$-greedy) may barely work under such an environment because of dithering in low-demand regions distant from high-revenue regions. This study proposes the deep relocating option policy (DROP) that supervises vehicle agents to escape from oversupply areas and effectively relocate to potentially underserved areas. We propose to learn the Laplacian embedding of a time-expanded relocation graph, as an approximation representation of the system relocation policy. The embedding generates task-agnostic signals, which in combination with task-dependent signals, constitute the pseudo-reward function for generating DROPs. We present a hierarchical learning framework that trains a high-level relocation policy and a set of low-level DROPs. The effectiveness of our approach is demonstrated using a custom-built high-fidelity simulator with real-world trip record data. We report that DROP significantly improves baseline models with 15.7% more hourly revenue and can effectively resolve the dithering issue in low-demand areas.

### Pointspectrum: Equivariance Meets Laplacian Filtering for Graph Representation Learning

Graph Representation Learning (GRL) has become essential for modern graph data mining and learning tasks. GRL aims to capture the graph's structural information and exploit it in combination with node and edge attributes to compute low-dimensional representations. While Graph Neural Networks (GNNs) have been used in state-of-the-art GRL architectures, they have been shown to suffer from over smoothing when many GNN layers need to be stacked. In a different GRL approach, spectral methods based on graph filtering have emerged addressing over smoothing; however, up to now, they employ traditional neural networks that cannot efficiently exploit the structure of graph data. Motivated by this, we propose PointSpectrum, a spectral method that incorporates a set equivariant network to account for a graph's structure. PointSpectrum enhances the efficiency and expressiveness of spectral methods, while it outperforms or competes with state-of-the-art GRL methods. Overall, PointSpectrum addresses over smoothing by employing a graph filter and captures a graph's structure through set equivariance, lying on the intersection of GNNs and spectral methods. Our findings are promising for the benefits and applicability of this architectural shift for spectral methods and GRL.

### A Gentle Introduction to the Laplacian

The Laplace operator was first applied to the study of celestial mechanics, or the motion of objects in outer space, by Pierre-Simon de Laplace, and as such has been named after him. The Laplace operator has since been used to describe many different phenomena, from electric potentials, to the diffusion equation for heat and fluid flow, and quantum mechanics. It has also been recasted to the discrete space, where it has been used in applications related to image processing and spectral clustering. In this tutorial, you will discover a gentle introduction to the Laplacian. A Gentle Introduction to the Laplacian Photo by Aziz Acharki, some rights reserved.

### The decomposition of the higher-order homology embedding constructed from the $k$-Laplacian

The null space of the $k$-th order Laplacian $\mathbf{\mathcal L}_k$, known as the {\em $k$-th homology vector space}, encodes the non-trivial topology of a manifold or a network. Understanding the structure of the homology embedding can thus disclose geometric or topological information from the data. The study of the null space embedding of the graph Laplacian $\mathbf{\mathcal L}_0$ has spurred new research and applications, such as spectral clustering algorithms with theoretical guarantees and estimators of the Stochastic Block Model. In this work, we investigate the geometry of the $k$-th homology embedding and focus on cases reminiscent of spectral clustering. Namely, we analyze the {\em connected sum} of manifolds as a perturbation to the direct sum of their homology embeddings. We propose an algorithm to factorize the homology embedding into subspaces corresponding to a manifold's simplest topological components. The proposed framework is applied to the {\em shortest homologous loop detection} problem, a problem known to be NP-hard in general. Our spectral loop detection algorithm scales better than existing methods and is effective on diverse data such as point clouds and images.

### Large sample spectral analysis of graph-based multi-manifold clustering

In this work we study statistical properties of graph-based algorithms for multi-manifold clustering (MMC). In MMC the goal is to retrieve the multi-manifold structure underlying a given Euclidean data set when this one is assumed to be obtained by sampling a distribution on a union of manifolds $\mathcal{M} = \mathcal{M}_1 \cup\dots \cup \mathcal{M}_N$ that may intersect with each other and that may have different dimensions. We investigate sufficient conditions that similarity graphs on data sets must satisfy in order for their corresponding graph Laplacians to capture the right geometric information to solve the MMC problem. Precisely, we provide high probability error bounds for the spectral approximation of a tensorized Laplacian on $\mathcal{M}$ with a suitable graph Laplacian built from the observations; the recovered tensorized Laplacian contains all geometric information of all the individual underlying manifolds. We provide an example of a family of similarity graphs, which we call annular proximity graphs with angle constraints, satisfying these sufficient conditions. We contrast our family of graphs with other constructions in the literature based on the alignment of tangent planes. Extensive numerical experiments expand the insights that our theory provides on the MMC problem.