Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs

Laplacian mixture models identify overlapping regions of influence in unlabeled graph and network data in a scalable and computationally efficient way, yielding useful low-dimensional representations. By combining Laplacian eigenspace and finite mixture modeling methods, they provide probabilistic or fuzzy dimensionality reductions or domain decompositions for a variety of input data types, including mixture distributions, feature vectors, and graphs or networks. Provable optimal recovery using the algorithm is analytically shown for a nontrivial class of cluster graphs. Heuristic approximations for scalable high-performance implementations are described and empirically tested. Connections to PageRank and community detection in network analysis demonstrate the wide applicability of this approach. The origins of fuzzy spectral methods, beginning with generalized heat or diffusion equations in physics, are reviewed and summarized. Comparisons to other dimensionality reduction and clustering methods for challenging unsupervised machine learning problems are also discussed.Comment: 13 figures, 35 reference

Entropy-based active learning for object recognition

Most methods for learning object categories require large amounts of labeled training data. However, obtaining such data can be a difficult and time-consuming endeavor. We have developed a novel, entropy-based ldquoactive learningrdquo approach which makes significant progress towards this problem. The main idea is to sequentially acquire labeled data by presenting an oracle (the user) with unlabeled images that will be particularly informative when labeled. Active learning adaptively prioritizes the order in which the training examples are acquired, which, as shown by our experiments, can significantly reduce the overall number of training examples required to reach near-optimal performance. At first glance this may seem counter-intuitive: how can the algorithm know whether a group of unlabeled images will be informative, when, by definition, there is no label directly associated with any of the images? Our approach is based on choosing an image to label that maximizes the expected amount of information we gain about the set of unlabeled images. The technique is demonstrated in several contexts, including improving the efficiency of Web image-search queries and open-world visual learning by an autonomous agent. Experiments on a large set of 140 visual object categories taken directly from text-based Web image searches show that our technique can provide large improvements (up to 10 x reduction in the number of training examples needed) over baseline techniques