Recently, pretraining methods for the Graph Neural Networks (GNNs) have been successful at learning effective representations from unlabeled graph data. However, most of these methods rely on pairwise relations in the graph and do not capture the underling higher-order relations between entities. Hypergraphs are versatile and expressive structures that can effectively model higher-order relationships among entities in the data. Despite the efforts to adapt GNNs to hypergraphs (HyperGNN), there are currently no fully self-supervised pretraining methods for HyperGNN on heterogeneous hypergraphs. In this paper, we present SPHH, a novel self-supervised pretraining framework for heterogeneous HyperGNNs. Our method is able to effectively capture higher-order relations among entities in the data in a self-supervised manner. SPHH is consist of two self-supervised pretraining tasks that aim to simultaneously learn both local and global representations of the entities in the hypergraph by using informative representations derived from the hypergraph structure. Overall, our work presents a significant advancement in the field of self-supervised pretraining of HyperGNNs, and has the potential to improve the performance of various graph-based downstream tasks such as node classification and link prediction tasks which are mapped to hypergraph configuration. Our experiments on two real-world benchmarks using four different HyperGNN models show that our proposed SPHH framework consistently outperforms state-of-the-art baselines in various downstream tasks. The results demonstrate that SPHH is able to improve the performance of various HyperGNN models in various downstream tasks, regardless of their architecture or complexity, which highlights the robustness of our framework.

Theoretical analyses for graph learning methods often assume a complete observation of the input graph. Such an assumption might not be useful for handling any-size graphs due to the scalability issues in practice. In this work, we develop a theoretical framework for graph classification problems in the partial observation setting (i.e., subgraph samplings). Equipped with insights from graph limit theory, we propose a new graph classification model that works on a randomly sampled subgraph and a novel topology to characterize the representability of the model. Our theoretical framework contributes a theoretical validation of mini-batch learning on graphs and leads to new learning-theoretic results on generalization bounds as well as size-generalizability without assumptions on the input.

In this paper, we study the graph classification problem from the graph homomorphism perspective. We consider the homomorphisms from $F$ to $G$, where $G$ is a graph of interest (e.g. molecules or social networks) and $F$ belongs to some family of graphs (e.g. paths or non-isomorphic trees). We show that graph homomorphism numbers provide a natural invariant (isomorphism invariant and $\mathcal{F}$-invariant) embedding maps which can be used for graph classification. Viewing the expressive power of a graph classifier by the $\mathcal{F}$-indistinguishable concept, we prove the universality property of graph homomorphism vectors in approximating $\mathcal{F}$-invariant functions. In practice, by choosing $\mathcal{F}$ whose elements have bounded tree-width, we show that the homomorphism method is efficient compared with other methods.

Abduction is a form of inference that seeks the best explanation for the given observation. Because it provides a reasoning process based on background knowledge, it is used in applications that need convincing explanations. In this study, we consider weighted abduction, which is one of the commonly used mathematical models for abduction. The main difficulty associated with applying weighted abduction to real problems is its computational complexity. A state-of-the-art method formulates weighted abduction as an integer linear programming (ILP) problem and solves it using efficient ILP solvers; however, it is still limited to solving problems that include at most 100 rules of background knowledge and observations. In this study, we first formulate the weighted abduction problem as a Max-SAT problem whose hard clauses are mostly Horn clauses. Then, we propose to solve the problem using modern Max-SAT solvers. In our experiments, the proposed method solved the problems much faster than the state-of-the-art ILP-based weighted abduction.

We present a simple proof for the universality of invariant and equivariant tensorized graph neural networks. Our approach considers a restricted intermediate hypothetical model named Graph Homomorphism Model to reach the universality conclusions including an open case for higher-order output. We find that our proposed technique not only leads to simple proofs of the universality properties but also gives a natural explanation for the tensorization of the previously studied models. Finally, we give some remarks on the connection between our model and the continuous representation of graphs.

Selecting appropriate regularization coefficients is critical to performance with respect to regularized empirical risk minimization problems. Existing theoretical approaches attempt to determine the coefficients in order for regularized empirical objectives to be upper-bounds of true objectives, uniformly over a hypothesis space. Such an approach is, however, known to be over-conservative, especially in high-dimensional settings with large hypothesis space. In fact, an existing generalization error bound in variance-based regularization is $O(\sqrt{d \log n/n})$, where $d$ is the dimension of hypothesis space, and thus the number of samples required for convergence linearly increases with respect to $d$. This paper proposes an algorithm that calculates regularization coefficient, one which results in faster convergence of generalization error $O(\sqrt{\log n/n})$ and whose leading term is independent of the dimension $d$. This faster convergence without dependence on the size of the hypothesis space is achieved by means of empirical hypothesis space reduction, which, with high probability, successfully reduces a hypothesis space without losing the true optimum solution. Calculation of uniform upper bounds over reduced spaces, then, enables acceleration of the convergence of generalization error.

Data cleansing is a typical approach used to improve the accuracy of machine learning models, which, however, requires extensive domain knowledge to identify the influential instances that affect the models. In this paper, we propose an algorithm that can suggest influential instances without using any domain knowledge. With the proposed method, users only need to inspect the instances suggested by the algorithm, implying that users do not need extensive knowledge for this procedure, which enables even non-experts to conduct data cleansing and improve the model. The existing methods require the loss function to be convex and an optimal model to be obtained, which is not always the case in modern machine learning. To overcome these limitations, we propose a novel approach specifically designed for the models trained with stochastic gradient descent (SGD). The proposed method infers the influential instances by retracing the steps of the SGD while incorporating intermediate models computed in each step. Through experiments, we demonstrate that the proposed method can accurately infer the influential instances. Moreover, we used MNIST and CIFAR10 to show that the models can be effectively improved by removing the influential instances suggested by the proposed method.

Graph neural networks have become one of the most important techniques to solve machine learning problems on graph-structured data. Recent work on vertex classification proposed deep and distributed learning models to achieve high performance and scalability. However, we find that the feature vectors of benchmark datasets are already quite informative for the classification task, and the graph structure only provides a means to denoise the data. In this paper, we develop a theoretical framework based on graph signal processing for analyzing graph neural networks. Our results indicate that graph neural networks only perform low-pass filtering on feature vectors and do not have the non-linear manifold learning property. We further investigate their resilience to feature noise and propose some insights on GCN-based graph neural network design.

Fairness by decision-makers is believed to be auditable by third parties. In this study, we show that this is not always true. We consider the following scenario. Imagine a decision-maker who discloses a subset of his dataset with decisions to make his decisions auditable. If he is corrupt, and he deliberately selects a subset that looks fair even though the overall decision is unfair, can we identify this decision-maker's fraud? We answer this question negatively. We first propose a sampling method that produces a subset whose distribution is biased from the original (to pretend to be fair); however, its differentiation from uniform sampling is difficult. We call such a sampling method as stealthily biased sampling, which is formulated as a Wasserstein distance minimization problem, and is solved through a minimum-cost flow computation. We proved that the stealthily biased sampling minimizes an upper-bound of the indistinguishability. We conducted experiments to see that the stealthily biased sampling is, in fact, difficult to detect.

In an ordinary feature selection procedure, a set of important features is obtained by solving an optimization problem such as the Lasso regression problem, and we expect that the obtained features explain the data well. In this study, instead of the single optimal solution, we consider finding a set of diverse yet nearly optimal solutions. To this end, we formulate the problem as finding a small number of solutions such that the convex hull of these solutions approximates the set of nearly optimal solutions. The proposed algorithm consists of two steps: First, we randomly sample the extreme points of the set of nearly optimal solutions. Then, we select a small number of points using a greedy algorithm. The experimental results indicate that the proposed algorithm can approximate the solution set well. The results also indicate that we can obtain Lasso solutions with a large diversity.