FI-GRL: Fast Inductive Graph Representation Learning via Projection-Cost Preservation

Graph representation learning aims at transforming graph data into meaningful low-dimensional vectors to facilitate the employment of machine learning and data mining algorithms designed for general data. Most current graph representation learning approaches are transductive, which means that they require all the nodes in the graph are known when learning graph representations and these approaches cannot naturally generalize to unseen nodes. In this paper, we present a Fast Inductive Graph Representation Learning framework (FI-GRL) to learn nodes' low-dimensional representations. Our approach can obtain accurate representations for seen nodes with provable theoretical guarantees and can easily generalize to unseen nodes. Specifically, in order to explicitly decouple nodes' relations expressed by the graph, we transform nodes into a randomized subspace spanned by a random projection matrix. This stage is guaranteed to preserve the projection-cost of the normalized random walk matrix which is highly related to the normalized cut of the graph. Then feature extraction is achieved by conducting singular value decomposition on the obtained matrix sketch. By leveraging the property of projection-cost preservation on the matrix sketch, the obtained representation result is nearly optimal. To deal with unseen nodes, we utilize folding-in technique to learn their meaningful representations. Empirically, when the amount of seen nodes are larger than that of unseen nodes, FI-GRL always achieves excellent results. Our algorithm is fast, simple to implement and theoretically guaranteed. Extensive experiments on real datasets demonstrate the superiority of our algorithm on both efficacy and efficiency over both macroscopic level (clustering) and microscopic level (structural hole detection) applications.Comment: ICDM 2018, Full Versio

A Framework for Deep Constrained Clustering -- Algorithms and Advances

The area of constrained clustering has been extensively explored by researchers and used by practitioners. Constrained clustering formulations exist for popular algorithms such as k-means, mixture models, and spectral clustering but have several limitations. A fundamental strength of deep learning is its flexibility, and here we explore a deep learning framework for constrained clustering and in particular explore how it can extend the field of constrained clustering. We show that our framework can not only handle standard together/apart constraints (without the well documented negative effects reported earlier) generated from labeled side information but more complex constraints generated from new types of side information such as continuous values and high-level domain knowledge.Comment: Updated for ECML/PKDD 201

Asymptotically efficient estimators for stochastic blockmodels: the naive MLE, the rank-constrained MLE, and the spectral

We establish asymptotic normality results for estimation of the block probability matrix $\mathbf{B}$ in stochastic blockmodel graphs using spectral embedding when the average degrees grows at the rate of $\omega(\sqrt{n})$ in $n$, the number of vertices. As a corollary, we show that when $\mathbf{B}$ is of full-rank, estimates of $\mathbf{B}$ obtained from spectral embedding are asymptotically efficient. When $\mathbf{B}$ is singular the estimates obtained from spectral embedding can have smaller mean square error than those obtained from maximizing the log-likelihood under no rank assumption, and furthermore, can be almost as efficient as the true MLE that assume known $\mathrm{rk}(\mathbf{B})$. Our results indicate, in the context of stochastic blockmodel graphs, that spectral embedding is not just computationally tractable, but that the resulting estimates are also admissible, even when compared to the purportedly optimal but computationally intractable maximum likelihood estimation under no rank assumption.Comment: 34 pages, 2 figure

Robust Unsupervised Flexible Auto-weighted Local-Coordinate Concept Factorization for Image Clustering

We investigate the high-dimensional data clustering problem by proposing a novel and unsupervised representation learning model called Robust Flexible Auto-weighted Local-coordinate Concept Factorization (RFA-LCF). RFA-LCF integrates the robust flexible CF, robust sparse local-coordinate coding and the adaptive reconstruction weighting learning into a unified model. The adaptive weighting is driven by including the joint manifold preserving constraints on the recovered clean data, basis concepts and new representation. Specifically, our RFA-LCF uses a L2,1-norm based flexible residue to encode the mismatch between clean data and its reconstruction, and also applies the robust adaptive sparse local-coordinate coding to represent the data using a few nearby basis concepts, which can make the factorization more accurate and robust to noise. The robust flexible factorization is also performed in the recovered clean data space for enhancing representations. RFA-LCF also considers preserving the local manifold structures of clean data space, basis concept space and the new coordinate space jointly in an adaptive manner way. Extensive comparisons show that RFA-LCF can deliver enhanced clustering results.Comment: Accepted at the 44th IEEE International Conference on Acoustics, Speech, and Signal Processing(ICASSP 2019

The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics

The singular value matrix decomposition plays a ubiquitous role throughout statistics and related fields. Myriad applications including clustering, classification, and dimensionality reduction involve studying and exploiting the geometric structure of singular values and singular vectors. This paper provides a novel collection of technical and theoretical tools for studying the geometry of singular subspaces using the two-to-infinity norm. Motivated by preliminary deterministic Procrustes analysis, we consider a general matrix perturbation setting in which we derive a new Procrustean matrix decomposition. Together with flexible machinery developed for the two-to-infinity norm, this allows us to conduct a refined analysis of the induced perturbation geometry with respect to the underlying singular vectors even in the presence of singular value multiplicity. Our analysis yields singular vector entrywise perturbation bounds for a range of popular matrix noise models, each of which has a meaningful associated statistical inference task. In addition, we demonstrate how the two-to-infinity norm is the preferred norm in certain statistical settings. Specific applications discussed in this paper include covariance estimation, singular subspace recovery, and multiple graph inference. Both our Procrustean matrix decomposition and the technical machinery developed for the two-to-infinity norm may be of independent interest.Comment: 36 page

Multi-View Spectral Clustering via Structured Low-Rank Matrix Factorization

Multi-view data clustering attracts more attention than their single view counterparts due to the fact that leveraging multiple independent and complementary information from multi-view feature spaces outperforms the single one. Multi-view Spectral Clustering aims at yielding the data partition agreement over their local manifold structures by seeking eigenvalue-eigenvector decompositions. However, as we observed, such classical paradigm still suffers from (1) overlooking the flexible local manifold structure, caused by (2) enforcing the low-rank data correlation agreement among all views; worse still, (3) LRR is not intuitively flexible to capture the latent data clustering structures. In this paper, we present the structured LRR by factorizing into the latent low-dimensional data-cluster representations, which characterize the data clustering structure for each view. Upon such representation, (b) the laplacian regularizer is imposed to be capable of preserving the flexible local manifold structure for each view. (c) We present an iterative multi-view agreement strategy by minimizing the divergence objective among all factorized latent data-cluster representations during each iteration of optimization process, where such latent representation from each view serves to regulate those from other views, such intuitive process iteratively coordinates all views to be agreeable. (d) We remark that such data-cluster representation can flexibly encode the data clustering structure from any view with adaptive input cluster number. To this end, (e) a novel non-convex objective function is proposed via the efficient alternating minimization strategy. The complexity analysis are also presented. The extensive experiments conducted against the real-world multi-view datasets demonstrate the superiority over state-of-the-arts.Comment: Accepted to appear at IEEE Trans on Neural Networks and Learning System

Multi-Level Network Embedding with Boosted Low-Rank Matrix Approximation

As opposed to manual feature engineering which is tedious and difficult to scale, network representation learning has attracted a surge of research interests as it automates the process of feature learning on graphs. The learned low-dimensional node vector representation is generalizable and eases the knowledge discovery process on graphs by enabling various off-the-shelf machine learning tools to be directly applied. Recent research has shown that the past decade of network embedding approaches either explicitly factorize a carefully designed matrix to obtain the low-dimensional node vector representation or are closely related to implicit matrix factorization, with the fundamental assumption that the factorized node connectivity matrix is low-rank. Nonetheless, the global low-rank assumption does not necessarily hold especially when the factorized matrix encodes complex node interactions, and the resultant single low-rank embedding matrix is insufficient to capture all the observed connectivity patterns. In this regard, we propose a novel multi-level network embedding framework BoostNE, which can learn multiple network embedding representations of different granularity from coarse to fine without imposing the prevalent global low-rank assumption. The proposed BoostNE method is also in line with the successful gradient boosting method in ensemble learning as multiple weak embeddings lead to a stronger and more effective one. We assess the effectiveness of the proposed BoostNE framework by comparing it with existing state-of-the-art network embedding methods on various datasets, and the experimental results corroborate the superiority of the proposed BoostNE network embedding framework

Face Clustering: Representation and Pairwise Constraints

Clustering face images according to their identity has two important applications: (i) grouping a collection of face images when no external labels are associated with images, and (ii) indexing for efficient large scale face retrieval. The clustering problem is composed of two key parts: face representation and choice of similarity for grouping faces. We first propose a representation based on ResNet, which has been shown to perform very well in image classification problems. Given this representation, we design a clustering algorithm, Conditional Pairwise Clustering (ConPaC), which directly estimates the adjacency matrix only based on the similarity between face images. This allows a dynamic selection of number of clusters and retains pairwise similarity between faces. ConPaC formulates the clustering problem as a Conditional Random Field (CRF) model and uses Loopy Belief Propagation to find an approximate solution for maximizing the posterior probability of the adjacency matrix. Experimental results on two benchmark face datasets (LFW and IJB-B) show that ConPaC outperforms well known clustering algorithms such as k-means, spectral clustering and approximate rank-order. Additionally, our algorithm can naturally incorporate pairwise constraints to obtain a semi-supervised version that leads to improved clustering performance. We also propose an k-NN variant of ConPaC, which has a linear time complexity given a k-NN graph, suitable for large datasets.Comment: This second version is the same as TIFS version. Some experiment results are different from v1 because we correct the protocol

Deep Clustering via Joint Convolutional Autoencoder Embedding and Relative Entropy Minimization

Image clustering is one of the most important computer vision applications, which has been extensively studied in literature. However, current clustering methods mostly suffer from lack of efficiency and scalability when dealing with large-scale and high-dimensional data. In this paper, we propose a new clustering model, called DEeP Embedded RegularIzed ClusTering (DEPICT), which efficiently maps data into a discriminative embedding subspace and precisely predicts cluster assignments. DEPICT generally consists of a multinomial logistic regression function stacked on top of a multi-layer convolutional autoencoder. We define a clustering objective function using relative entropy (KL divergence) minimization, regularized by a prior for the frequency of cluster assignments. An alternating strategy is then derived to optimize the objective by updating parameters and estimating cluster assignments. Furthermore, we employ the reconstruction loss functions in our autoencoder, as a data-dependent regularization term, to prevent the deep embedding function from overfitting. In order to benefit from end-to-end optimization and eliminate the necessity for layer-wise pretraining, we introduce a joint learning framework to minimize the unified clustering and reconstruction loss functions together and train all network layers simultaneously. Experimental results indicate the superiority and faster running time of DEPICT in real-world clustering tasks, where no labeled data is available for hyper-parameter tuning

Bayesian Distance Clustering

Model-based clustering is widely-used in a variety of application areas. However, fundamental concerns remain about robustness. In particular, results can be sensitive to the choice of kernel representing the within-cluster data density. Leveraging on properties of pairwise differences between data points, we propose a class of Bayesian distance clustering methods, which rely on modeling the likelihood of the pairwise distances in place of the original data. Although some information in the data is discarded, we gain substantial robustness to modeling assumptions. The proposed approach represents an appealing middle ground between distance- and model-based clustering, drawing advantages from each of these canonical approaches. We illustrate dramatic gains in the ability to infer clusters that are not well represented by the usual choices of kernel. A simulation study is included to assess performance relative to competitors, and we apply the approach to clustering of brain genome expression data. Keywords: Distance-based clustering; Mixture model; Model-based clustering; Model misspecification; Pairwise distance matrix; Partial likelihood; Robustness