Scalable Laplacian K-modes
We advocate Laplacian K-modes for joint clustering and density mode finding, and propose a concave-convex relaxation of the problem, which yields a parallel algorithm that scales up to large datasets and high dimensions. We optimize a tight bound (auxiliary function) of our relaxation, which, at each iteration, amounts to computing an independent update for each cluster-assignment variable, with guaranteed convergence. Therefore, our bound optimizer can be trivially distributed for large-scale data sets. Furthermore, we show that the density modes can be obtained as byproducts of the assignment variables via simple maximum-value operations whose additional computational cost is linear in the number of data points. Our formulation does not need storing a full affinity matrix and computing its eigenvalue decomposition, neither does it perform expensive projection steps and Lagrangian-dual inner iterates for the simplex constraints of each point. Furthermore, unlike mean-shift, our density-mode estimation does not require inner-loop gradient-ascent iterates. It has a complexity independent of featurespace dimension, yields modes that are valid data points in the input set and is applicable to discrete domains as well as arbitrary kernels. We report comprehensive experiments over various data sets, which show that our algorithm yields very competitive performances in term of optimization quality (i.e., the value of the discrete-variable objective at convergence) and clustering accuracy.
The paper proposed a noval and scalable approach to solve the Laplacian K-modes problem. The primary strategy is to adopt another way to relax the original discrete problem and further minimize a upper bound of the objective. The method is guaranteed to converge and evaluated to show its superiority over some benchmark clustering algorithms in both clustering performance and algorithmic efficiency. The statements are clear and content is easy to follow. The technical part is also correct.    My main criticism is about the significance.  -First, I am not sure if Laplacian K-modes energy is a widely used metric for clustering. Laplacian K-modes energy follows a form that simultaneously maximizes weights within each cluster (mode part) and minimizes weights across clusters (Lap part). There are several objectives falling in this category like correlation clustering. However, in experiments, the authors only compare LK objective with some very preliminary objectives NCUT, k-means, k-modes that are in other categories of clustering algorithms (NCUT: balanced min-cut; k-means, k-modes: max weights within clusters). - Second, although the proposed approach has good scalability, the derivation is kind of ad-hoc and all tricks are for scalability. In the first-step relaxation, an entropy term is subjectively added to handle the constrains. With this term, the objective can hardly be a tight relaxation of Laplacian K-modes energy. The proposed approach only holds convergence guarantee while do not has performance guarantee and the trick to obtain such convergence is not difficult to think of. Also, the efficient method to estimate mode is also ad-hoc. I think the proposed method may be useful in some practical scenarios but it is not significant enough for a strong acceptance of NIPS. Hence, I give a borderline overall score.