K-means clustering algorithms: A comprehensive review, variants analysis, and advances in the era of big data

Graph convolution is the core of most Graph Neural Networks (GNNs) and usually approximated by message passing between direct (one-hop) neighbors. In this work, we remove the restriction of using only the direct neighbors by introducing a powerful, yet spatially localized graph convolution: Graph diffusion convolution (GDC). GDC leverages generalized graph diffusion, examples of which are the heat kernel and personalized PageRank. It alleviates the problem of noisy and often arbitrarily defined edges in real graphs. We show that GDC is closely related to spectral-based models and thus combines the strengths of both spatial (message passing) and spectral methods. We demonstrate that replacing message passing with graph diffusion convolution consistently leads to significant performance improvements across a wide range of models on both supervised and unsupervised tasks and a variety of datasets. Furthermore, GDC is not limited to GNNs but can trivially be combined with any graph-based model or algorithm (e.g. spectral clustering) without requiring any changes to the latter or affecting its computational complexity. Our implementation is available online.

Diffusion Improves Graph Learning

This paper is the first of two written by Brad Efron and Ron Thisted studying the frequency distribution of words in the Shakespearean canon. The key idea due to Fisher in the context of sampling of species is simple and elegant. When applied to Shakespeare the idea appears to be preposterous: an author has a personal vocabulary of word species represented by a distribution G, and text is generated by sampling from this distribution. Most results do not require successive words to be sampled independently, which leaves room for individual style and context, but stationarity is needed for prediction and inference. The expected number of words that occur x ≥ 1 times in a large sample of n words is

Estimating the Number of Unseen Species: How Many Words did Shakespeare Know?

Representation learning of graph-structured data is challenging because both graph structure and node features carry important information. Graph Neural Networks (GNNs) provide an expressive way to fuse information from network structure and node features. However, GNNs are prone to adversarial attacks. Here we introduce Graph Information Bottleneck (GIB), an information-theoretic principle that optimally balances expressiveness and robustness of the learned representation of graph-structured data. Inheriting from the general Information Bottleneck (IB), GIB aims to learn the minimal sufficient representation for a given task by maximizing the mutual information between the representation and the target, and simultaneously constraining the mutual information between the representation and the input data. Different from the general IB, GIB regularizes the structural as well as the feature information. We design two sampling algorithms for structural regularization and instantiate the GIB principle with two new models: GIB-Cat and GIB-Bern, and demonstrate the benefits by evaluating the resilience to adversarial attacks. We show that our proposed models are more robust than state-of-the-art graph defense models. GIB-based models empirically achieve up to 31% improvement with adversarial perturbation of the graph structure as well as node features.

Graph Information Bottleneck.

Graph neural networks (GNNs) have achieved tremendous success on multiple graph-based learning tasks by fusing network structure and node features. Modern GNN models are built upon iterative aggregation of neighbor's/proximity features by message passing. Its prediction performance has been shown to be strongly bounded by assortative mixing in the graph, a key property wherein nodes with similar attributes mix/connect with each other. We observe that real world networks exhibit heterogeneous or diverse mixing patterns and the conventional global measurement of assortativity, such as global assortativity coefficient, may not be a representative statistic in quantifying this mixing. We adopt a generalized concept, node-level assortativity, one that is based at the node level to better represent the diverse patterns and accurately quantify the learnability of GNNs. We find that the prediction performance of a wide range of GNN models is highly correlated with the node level assortativity. To break this limit, in this work, we focus on transforming the input graph into a computation graph which contains both proximity and structural information as distinct type of edges. The resulted multi-relational graph has an enhanced level of assortativity and, more importantly, preserves rich information from the original graph. We then propose to run GNNs on this computation graph and show that adaptively choosing between structure and proximity leads to improved performance under diverse mixing. Empirically, we show the benefits of adopting our transformation framework for semi-supervised node classification task on a variety of real world graph learning benchmarks.

Breaking the Limit of Graph Neural Networks by Improving the Assortativity of Graphs with Local Mixing Patterns

We consider the problem of approximate $K$-means clustering with outliers and side information provided by same-cluster queries and possibly noisy answers. Our solution shows that, under some mild assumptions on the smallest cluster size, one can obtain an $(1+\epsilon)$-approximation for the optimal potential with probability at least $1-\delta$, where $\epsilon>0$ and $\delta\in(0,1)$, using an expected number of $O(\frac{K^3}{\epsilon \delta})$ noiseless same-cluster queries and comparison-based clustering of complexity $O(ndK + \frac{K^3}{\epsilon \delta})$; here, $n$ denotes the number of points and $d$ the dimension of space. Compared to a handful of other known approaches that perform importance sampling to account for small cluster sizes, the proposed query technique reduces the number of queries by a factor of roughly $O(\frac{K^6}{\epsilon^3})$, at the cost of possibly missing very small clusters. We extend this settings to the case where some queries to the oracle produce erroneous information, and where certain points, termed outliers, do not belong to any clusters. Our proof techniques differ from previous methods used for $K$-means clustering analysis, as they rely on estimating the sizes of the clusters and the number of points needed for accurate centroid estimation and subsequent nontrivial generalizations of the double Dixie cup problem. We illustrate the performance of the proposed algorithm both on synthetic and real datasets, including MNIST and CIFAR $10$.

/pdf/query-k-means-clustering-and-the-double-dixie-cup-problem-3h5nfpclgs.pdf

Query K-means Clustering and the Double Dixie Cup Problem

We consider the problem of approximate $K$-means clustering with outliers and side information provided by same-cluster queries and possibly noisy answers. Our solution shows that, under some mild assumptions on the smallest cluster size, one can obtain an $(1+\epsilon)$-approximation for the optimal potential with probability at least $1-\delta$, where $\epsilon>0$ and $\delta\in(0,1)$, using an expected number of $O(\frac{K^3}{\epsilon \delta})$ noiseless same-cluster queries and comparison-based clustering of complexity $O(ndK + \frac{K^3}{\epsilon \delta})$, here, $n$ denotes the number of points and $d$ the dimension of space. Compared to a handful of other known approaches that perform importance sampling to account for small cluster sizes, the proposed query technique reduces the number of queries by a factor of roughly $O(\frac{K^6}{\epsilon^3})$, at the cost of possibly missing very small clusters. We extend this settings to the case where some queries to the oracle produce erroneous information, and where certain points, termed outliers, do not belong to any clusters. Our proof techniques differ from previous methods used for $K$-means clustering analysis, as they rely on estimating the sizes of the clusters and the number of points needed for accurate centroid estimation and subsequent nontrivial generalizations of the double Dixie cup problem. We illustrate the performance of the proposed algorithm both on synthetic and real datasets, including MNIST and CIFAR $10$.

We introduce a new framework for estimating the support size of an unknown distribution which improves upon known approximation-based techniques. Our main contributions include describing a rigorous new weighted Chebyshev polynomial approximation method and introducing regularization terms into the problem formulation that provably improve the performance of stateof-the-art approximation-based approaches. In particular, we present both theoretical and computer simulation results that illustrate the utility and performance improvements of our method. The theoretical analysis relies on jointly optimizing the bias and variance components of the risk, and combining new weighted minmax polynomial approximation techniques with discretized semi-infinite programming solvers. Such a setting allows for casting the estimation problem as a linear program (LP) with a small number of variables and constraints that may be solved as efficiently as the original Chebyshev approximation-based problem. The described approach also applies to the support coverage and entropy estimation problems. Our newly developed technique is tested on synthetic data and used to estimate the number of bacterial species in the human gut. On synthetic datasets, we observed up to five-fold improvements in the value of the worst-case risk. For the bioinformatics application, metagenomic data from the NIH Human Gut and the American Gut Microbiome was combined and processed to obtain lists of bacterial taxonomies. These were subsequently used to compute the bacterial species histograms and estimate the number of bacterial species in the human gut to roughly 2350, with the species being represented by trillions of cells. Equal contribution Department of ECE, UIUC. Correspondence to: I (Eli) Chien , Olgica Milenkovic .

Support Estimation via Regularized and Weighted Chebyshev Approximations.

Landing probabilities (LP) of random walks (RW) over graphs encode rich information regarding graph topology. Generalized PageRanks (GPR), which represent weighted sums of LPs of RWs, utilize the discriminative power of LP features to enable many graph-based learning studies. Previous work in the area has mostly focused on evaluating suitable weights for GPRs, and only a few studies so far have attempted to derive the optimal weights of GPRs for a given application. We take a fundamental step forward in this direction by using random graph models to better our understanding of the behavior of GPRs. In this context, we provide a rigorous non-asymptotic analysis for the convergence of LPs and GPRs to their mean-field values on edge-independent random graphs. Although our theoretical results apply to many problem settings, we focus on the task of seed-expansion community detection over stochastic block models. There, we find that the predictive power of LPs decreases significantly slower than previously reported based on asymptotic findings. Given this result, we propose a new GPR, termed Inverse PR (IPR), with LP weights that increase for the initial few steps of the walks. Extensive experiments on both synthetic and real, large-scale networks illustrate the superiority of IPR compared to other GPRs for seeded community detection.

/pdf/optimizing-generalized-pagerank-methods-for-seed-expansion-2dto2eusa7.pdf

Optimizing Generalized PageRank Methods for Seed-Expansion Community Detection

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Papers

Query K-means Clustering and the Double Dixie Cup Problem

Query K-means Clustering and the Double Dixie Cup Problem

Support Estimation via Regularized and Weighted Chebyshev Approximations.

Optimizing Generalized PageRank Methods for Seed-Expansion Community Detection