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

Recovery of Low-Rank Matrices under Affine Constraints via a Smoothed Rank Function

In this paper, the problem of matrix rank minimization under affine constraints is addressed. The state-of-the-art algorithms can recover matrices with a rank much less than what is sufficient for the uniqueness of the solution of this optimization problem. We propose an algorithm based on a smooth approximation of the rank function, which practically improves recovery limits on the rank of the solution. This approximation leads to a non-convex program; thus, to avoid getting trapped in local solutions, we use the following scheme. Initially, a rough approximation of the rank function subject to the affine constraints is optimized. As the algorithm proceeds, finer approximations of the rank are optimized and the solver is initialized with the solution of the previous approximation until reaching the desired accuracy. On the theoretical side, benefiting from the spherical section property, we will show that the sequence of the solutions of the approximating function converges to the minimum rank solution. On the experimental side, it will be shown that the proposed algorithm, termed SRF standing for Smoothed Rank Function, can recover matrices which are unique solutions of the rank minimization problem and yet not recoverable by nuclear norm minimization. Furthermore, it will be demonstrated that, in completing partially observed matrices, the accuracy of SRF is considerably and consistently better than some famous algorithms when the number of revealed entries is close to the minimum number of parameters that uniquely represent a low-rank matrix.Comment: Accepted in IEEE TSP on December 4th, 201