Proximity Graph Maintenance for Fast Online Nearest Neighbor Search

Approximate Nearest Neighbor (ANN) search is a fundamental technique for (e.g.,) the deployment of recommender systems. Recent studies bring proximity graph-based methods into practitioners' attention -- proximity graph-based methods outperform other solutions such as quantization, hashing, and tree-based ANN algorithm families. In current recommendation systems, data point insertions, deletions, and queries are streamed into the system in an online fashion as users and items change dynamically. As proximity graphs are constructed incrementally by inserting data points as new vertices into the graph, online insertions and queries are well-supported in proximity graph. However, a data point deletion incurs removing a vertex from the proximity graph index, while no proper graph index updating mechanisms are discussed in previous studies. To tackle the challenge, we propose an incremental proximity graph maintenance (IPGM) algorithm for online ANN. IPGM supports both vertex deletion and insertion on proximity graphs. Given a vertex deletion request, we thoroughly investigate solutions to update the connections of the vertex. The proposed updating scheme eliminates the performance drop in online ANN methods on proximity graphs, making the algorithm suitable for practical systems.

Optimal Bipartite Network Clustering

We consider the problem of bipartite community detection in networks, or more generally the network biclustering problem. We present a fast two-stage procedure based on spectral initialization followed by the application of a pseudo-likelihood classifier twice. Under mild regularity conditions, we establish the weak consistency of the procedure (i.e., the convergence of the misclassification rate to zero) under a general bipartite stochastic block model. We show that the procedure is optimal in the sense that it achieves the optimal convergence rate that is achievable by a biclustering oracle, adaptively over the whole class, up to constants. The optimal rate we obtain sharpens some of the existing results and generalizes others to a wide regime of average degree growth. As a special case, we recover the known exact recovery threshold in the $\log n$ regime of sparsity. To obtain the general consistency result, as part of the provable version of the algorithm, we introduce a sub-block partitioning scheme that is also computationally attractive, allowing for distributed implementation of the algorithm without sacrificing optimality. The provable version of the algorithm is derived from a general blueprint for pseudo-likelihood biclustering algorithms that employ simple EM type updates. We show the effectiveness of this general class by numerical simulations.

Analysis of spectral clustering algorithms for community detection: the general bipartite setting

We consider the analysis of spectral clustering algorithms for community detection under a stochastic block model (SBM). A general spectral clustering algorithm consists of three steps: (1) regularization of an appropriate adjacency or Laplacian matrix (2) a form of spectral truncation and (3) a k-means type algorithm in the reduced spectral domain. By varying each step, one can obtain different spectral algorithms. In light of the recent developments in refining consistency results for the spectral clustering, we identify the necessary bounds at each of these three steps, and then derive and compare consistency results for some existing spectral algorithms as well as a new variant that we propose. The focus of the paper is on providing a better understanding of the analysis of spectral methods for community detection, with an emphasis on the bipartite setting which has received less theoretical consideration. We show how the variations in the spectral truncation step reflects in the consistency results under a general SBM. We also investigate the necessary bounds for the k-means step in some detail, allowing one to replace this step with any algorithm (k-means type or otherwise) that guarantees the necessary bound. We discuss some of the neglected aspects of the bipartite setting, e.g., the role of the mismatch between the communities of the two sides on the performance of spectral methods. Finally, we show how the consistency results can be extended beyond SBMs to the problem of clustering inhomogeneous random graph models that can be approximated by SBMs in a certain sense.