Jianhao Peng

Chao Pan, Eli Chien, Puoya Tabaghi et al.

Many high-dimensional practical data sets have hierarchical structures induced by graphs or time series. Such data sets are hard to process in Euclidean spaces and one often seeks low-dimensional embeddings in other space forms to perform the required learning tasks. For hierarchical data, the space of choice is a hyperbolic space because it guarantees low-distortion embeddings for tree-like structures. The geometry of hyperbolic spaces has properties not encountered in Euclidean spaces that pose challenges when trying to rigorously analyze algorithmic solutions. We propose a unified framework for learning scalable and simple hyperbolic linear classifiers with provable performance guarantees. The gist of our approach is to focus on Poincaré ball models and formulate the classification problems using tangent space formalisms. Our results include a new hyperbolic perceptron algorithm as well as an efficient and highly accurate convex optimization setup for hyperbolic support vector machine classifiers. Furthermore, we adapt our approach to accommodate second-order perceptrons, where data is preprocessed based on second-order information (correlation) to accelerate convergence, and strategic perceptrons, where potentially manipulated data arrives in an online manner and decisions are made sequentially. The excellent performance of the Poincaré second-order and strategic perceptrons shows that the proposed framework can be extended to general machine learning problems in hyperbolic spaces. Our experimental results, pertaining to synthetic, single-cell RNA-seq expression measurements, CIFAR10, Fashion-MNIST and mini-ImageNet, establish that all algorithms provably converge and have complexity comparable to those of their Euclidean counterparts. Accompanying codes can be found at: https://github.com/thupchnsky/PoincareLinearClassification.

Eli Chien, Chao Pan, Jianhao Peng et al.

Hypergraphs are used to model higher-order interactions amongst agents and there exist many practically relevant instances of hypergraph datasets. To enable efficient processing of hypergraph-structured data, several hypergraph neural network platforms have been proposed for learning hypergraph properties and structure, with a special focus on node classification. However, almost all existing methods use heuristic propagation rules and offer suboptimal performance on many datasets. We propose AllSet, a new hypergraph neural network paradigm that represents a highly general framework for (hyper)graph neural networks and for the first time implements hypergraph neural network layers as compositions of two multiset functions that can be efficiently learned for each task and each dataset. Furthermore, AllSet draws on new connections between hypergraph neural networks and recent advances in deep learning of multiset functions. In particular, the proposed architecture utilizes Deep Sets and Set Transformer architectures that allow for significant modeling flexibility and offer high expressive power. To evaluate the performance of AllSet, we conduct the most extensive experiments to date involving ten known benchmarking datasets and three newly curated datasets that represent significant challenges for hypergraph node classification. The results demonstrate that AllSet has the unique ability to consistently either match or outperform all other hypergraph neural networks across the tested datasets.

Puoya Tabaghi, Chao Pan, Eli Chien et al.

Embedding methods for product spaces are powerful techniques for low-distortion and low-dimensional representation of complex data structures. Here, we address the new problem of linear classification in product space forms -- products of Euclidean, spherical, and hyperbolic spaces. First, we describe novel formulations for linear classifiers on a Riemannian manifold using geodesics and Riemannian metrics which generalize straight lines and inner products in vector spaces. Second, we prove that linear classifiers in $d$-dimensional space forms of any curvature have the same expressive power, i.e., they can shatter exactly $d+1$ points. Third, we formalize linear classifiers in product space forms, describe the first known perceptron and support vector machine classifiers for such spaces and establish rigorous convergence results for perceptrons. Moreover, we prove that the Vapnik-Chervonenkis dimension of linear classifiers in a product space form of dimension $d$ is \emph{at least} $d+1$. We support our theoretical findings with simulations on several datasets, including synthetic data, image data, and single-cell RNA sequencing (scRNA-seq) data. The results show that classification in low-dimensional product space forms for scRNA-seq data offers, on average, a performance improvement of $\sim15\%$ when compared to that in Euclidean spaces of the same dimension.

Puoya Tabaghi, Jianhao Peng, Olgica Milenkovic et al.

Many data analysis problems can be cast as distance geometry problems in \emph{space forms} -- Euclidean, spherical, or hyperbolic spaces. Often, absolute distance measurements are often unreliable or simply unavailable and only proxies to absolute distances in the form of similarities are available. Hence we ask the following: Given only \emph{comparisons} of similarities amongst a set of entities, what can be said about the geometry of the underlying space form? To study this question, we introduce the notions of the \textit{ordinal capacity} of a target space form and \emph{ordinal spread} of the similarity measurements. The latter is an indicator of complex patterns in the measurements, while the former quantifies the capacity of a space form to accommodate a set of measurements with a specific ordinal spread profile. We prove that the ordinal capacity of a space form is related to its dimension and the sign of its curvature. This leads to a lower bound on the Euclidean and spherical embedding dimension of what we term similarity graphs. More importantly, we show that the statistical behavior of the ordinal spread random variables defined on a similarity graph can be used to identify its underlying space form. We support our theoretical claims with experiments on weighted trees, single-cell RNA expression data and spherical cartographic measurements.

Eli Chien, Jianhao Peng, Pan Li et al.

In many important graph data processing applications the acquired information includes both node features and observations of the graph topology. Graph neural networks (GNNs) are designed to exploit both sources of evidence but they do not optimally trade-off their utility and integrate them in a manner that is also universal. Here, universality refers to independence on homophily or heterophily graph assumptions. We address these issues by introducing a new Generalized PageRank (GPR) GNN architecture that adaptively learns the GPR weights so as to jointly optimize node feature and topological information extraction, regardless of the extent to which the node labels are homophilic or heterophilic. Learned GPR weights automatically adjust to the node label pattern, irrelevant on the type of initialization, and thereby guarantee excellent learning performance for label patterns that are usually hard to handle. Furthermore, they allow one to avoid feature over-smoothing, a process which renders feature information nondiscriminative, without requiring the network to be shallow. Our accompanying theoretical analysis of the GPR-GNN method is facilitated by novel synthetic benchmark datasets generated by the so-called contextual stochastic block model. We also compare the performance of our GNN architecture with that of several state-of-the-art GNNs on the problem of node-classification, using well-known benchmark homophilic and heterophilic datasets. The results demonstrate that GPR-GNN offers significant performance improvement compared to existing techniques on both synthetic and benchmark data.

Anuththari Gamage, Eli Chien, Jianhao Peng et al.

Generative graph models create instances of graphs that mimic the properties of real-world networks. Generative models are successful at retaining pairwise associations in the underlying networks but often fail to capture higher-order connectivity patterns known as network motifs. Different types of graphs contain different network motifs, an example of which are triangles that often arise in social and biological networks. It is hence vital to capture these higher-order structures to simulate real-world networks accurately. We propose Multi-MotifGAN (MMGAN), a motif-targeted Generative Adversarial Network (GAN) that generalizes the benchmark NetGAN approach. The generalization consists of combining multiple biased random walks, each of which captures a different motif structure. MMGAN outperforms NetGAN at creating new graphs that accurately reflect the network motif statistics of input graphs such as Citeseer, Cora and Facebook.

Abhishek Agarwal, Jianhao Peng, Olgica Milenkovic

Matrix factorization (MF) is a versatile learning method that has found wide applications in various data-driven disciplines. Still, many MF algorithms do not adequately scale with the size of available datasets and/or lack interpretability. To improve the computational efficiency of the method, an online (streaming) MF algorithm was proposed in Mairal et al. [2010]. To enable data interpretability, a constrained version of MF, termed convex MF, was introduced in Ding et al. [2010]. In the latter work, the basis vectors are required to lie in the convex hull of the data samples, thereby ensuring that every basis can be interpreted as a weighted combination of data samples. No current algorithmic solutions for online convex MF are known as it is challenging to find adequate convex bases without having access to the complete dataset. We address both problems by proposing the first online convex MF algorithm that maintains a collection of constant-size sets of representative data samples needed for interpreting each of the basis (Ding et al. [2010]) and has the same almost sure convergence guarantees as the online learning algorithm of Mairal et al. [2010]. Our proof techniques combine random coordinate descent algorithms with specialized quasi-martingale convergence analysis. Experiments on synthetic and real world datasets show significant computational savings of the proposed online convex MF method compared to classical convex MF. Since the proposed method maintains small representative sets of data samples needed for convex interpretations, it is related to a body of work in theoretical computer science, pertaining to generating point sets (Blum et al. [2016]), and in computer vision, pertaining to archetypal analysis (Mei et al. [2018]). Nevertheless, it differs from these lines of work both in terms of the objective and algorithmic implementations.