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Register a new userScalable Spectral Clustering Using Random Binning Features
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Spectral clustering is one of the most effective clustering approaches that capture hidden cluster structures in the data. However, it does not scale well to large-scale problems due to its quadratic complexity in constructing similarity graphs and computing subsequent eigendecomposition. Although a number of methods have been proposed to accelerate spectral clustering, most of them compromise considerable information loss in the original data for reducing computational bottlenecks. In this paper, we present a novel scalable spectral clustering method using Random Binning features (RB) to simultaneously accelerate both similarity graph construction and the eigendecomposition. Specifically, we implicitly approximate the graph similarity (kernel) matrix by the inner product of a large sparse feature matrix generated by RB. Then we introduce a state-of-the-art SVD solver to effectively compute eigenvectors of this large matrix for spectral clustering. Using these two building blocks, we reduce the computational cost from quadratic to linear in the number of data points while achieving similar accuracy. Our theoretical analysis shows that spectral clustering via RB converges faster to the exact spectral clustering than the standard Random Feature approximation. Extensive experiments on 8 benchmarks show that the proposed method either outperforms or matches the state-of-the-art methods in both accuracy and runtime. Moreover, our method exhibits linear scalability in both the number of data samples and the number of RB features.
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Graph kernels are widely used for measuring the similarity between graphs. Many existing graph kernels, which focus on local patterns within graphs rather than their global properties, suffer from significant structure information loss when representing graphs. Some recent global graph kernels, which utilizes the alignment of geometric node embeddings of graphs, yield state-of-the-art performance. However, these graph kernels are not necessarily positive-definite. More importantly, computing the graph kernel matrix will have at least quadratic {time} complexity in terms of the number and the size of the graphs. In this paper, we propose a new family of global alignment graph kernels, which take into account the global properties of graphs by using geometric node embeddings and an associated node transportation based on earth movers distance. Compared to existing global kernels, the proposed kernel is positive-definite. Our graph kernel is obtained by defining a distribution over emph{random graphs}, which can naturally yield random feature approximations. The random feature approximations lead to our graph embeddings, which is named as random graph embeddings (RGE). In particular, RGE is shown to achieve emph{(quasi-)linear scalability} with respect to the number and the size of the graphs. The experimental results on nine benchmark datasets demonstrate that RGE outperforms or matches twelve state-of-the-art graph classification algorithms.
Kernel method has been developed as one of the standard approaches for nonlinear learning, which however, does not scale to large data set due to its quadratic complexity in the number of samples. A number of kernel approximation methods have thus been proposed in the recent years, among which the random features method gains much popularity due to its simplicity and direct reduction of nonlinear problem to a linear one. The Random Binning (RB) feature, proposed in the first random-feature paper cite{rahimi2007random}, has drawn much less attention than the Random Fourier (RF) feature. In this work, we observe that the RB features, with right choice of optimization solver, could be orders-of-magnitude more efficient than other random features and kernel approximation methods under the same requirement of accuracy. We thus propose the first analysis of RB from the perspective of optimization, which by interpreting RB as a Randomized Block Coordinate Descent in the infinite-dimensional space, gives a faster convergence rate compared to that of other random features. In particular, we show that by drawing $R$ random grids with at least $kappa$ number of non-empty bins per grid in expectation, RB method achieves a convergence rate of $O(1/(kappa R))$, which not only sharpens its $O(1/sqrt{R})$ rate from Monte Carlo analysis, but also shows a $kappa$ times speedup over other random features under the same analysis framework. In addition, we demonstrate another advantage of RB in the L1-regularized setting, where unlike other random features, a RB-based Coordinate Descent solver can be parallelized with guaranteed speedup proportional to $kappa$. Our extensive experiments demonstrate the superior performance of the RB features over other random features and kernel approximation methods. Our code and data is available at { url{https://github.com/teddylfwu/RB_GEN}}.
Deep clustering (DC) has become the state-of-the-art for unsupervised clustering. In principle, DC represents a variety of unsupervised methods that jointly learn the underlying clusters and the latent representation directly from unstructured datasets. However, DC methods are generally poorly applied due to high operational costs, low scalability, and unstable results. In this paper, we first evaluate several popular DC variants in the context of industrial applicability using eight empirical criteria. We then choose to focus on variational deep clustering (VDC) methods, since they mostly meet those criteria except for simplicity, scalability, and stability. To address these three unmet criteria, we introduce four generic algorithmic improvements: initial $gamma$-training, periodic $beta$-annealing, mini-batch GMM (Gaussian mixture model) initialization, and inverse min-max transform. We also propose a novel clustering algorithm S3VDC (simple, scalable, and stable VDC) that incorporates all those improvements. Our experiments show that S3VDC outperforms the state-of-the-art on both benchmark tasks and a large unstructured industrial dataset without any ground truth label. In addition, we analytically evaluate the usability and interpretability of S3VDC.
The computational cost of training with softmax cross entropy loss grows linearly with the number of classes. For the settings where a large number of classes are involved, a common method to speed up training is to sample a subset of classes and utilize an estimate of the loss gradient based on these classes, known as the sampled softmax method. However, the sampled softmax provides a biased estimate of the gradient unless the samples are drawn from the exact softmax distribution, which is again expensive to compute. Therefore, a widely employed practical approach involves sampling from a simpler distribution in the hope of approximating the exact softmax distribution. In this paper, we develop the first theoretical understanding of the role that different sampling distributions play in determining the quality of sampled softmax. Motivated by our analysis and the work on kernel-based sampling, we propose the Random Fourier Softmax (RF-softmax) method that utilizes the powerful Random Fourier Features to enable more efficient and accurate sampling from an approximate softmax distribution. We show that RF-softmax leads to low bias in estimation in terms of both the full softmax distribution and the full softmax gradient. Furthermore, the cost of RF-softmax scales only logarithmically with the number of classes.
Clustering is fundamental for gaining insights from complex networks, and spectral clustering (SC) is a popular approach. Conventional SC focuses on second-order structures (e.g., edges connecting two nodes) without direct consideration of higher-order structures (e.g., triangles and cliques). This has motivated SC extensions that directly consider higher-order structures. However, both approaches are limited to considering a single order. This paper proposes a new Mixed-Order Spectral Clustering (MOSC) approach to model both second-order and third-order structures simultaneously, with two MOSC methods developed based on Graph Laplacian (GL) and Random Walks (RW). MOSC-GL combines edge and triangle adjacency matrices, with theoretical performance guarantee. MOSC-RW combines first-order and second-order random walks for a probabilistic interpretation. We automatically determine the mixing parameter based on cut criteria or triangle density, and construct new structure-aware error metrics for performance evaluation. Experiments on real-world networks show 1) the superior performance of two MOSC methods over existing SC methods, 2) the effectiveness of the mixing parameter determination strategy, and 3) insights offered by the structure-aware error metrics.
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