Computing Exact Clustering Posteriors with Subset Convolution

An exponential-time exact algorithm is provided for the task of clustering n items of data into k clusters. Instead of seeking one partition, posterior probabilities are computed for summary statistics: the number of clusters, and pairwise co-occurrence. The method is based on subset convolution, and yields the posterior distribution for the number of clusters in O(n * 3^n) operations, or O(n^3 * 2^n) using fast subset convolution. Pairwise co-occurrence probabilities are then obtained in O(n^3 * 2^n) operations. This is considerably faster than exhaustive enumeration of all partitions.Comment: 6 figure

On All-loop Integrands of Scattering Amplitudes in Planar N=4 SYM

We study the relationship between the momentum twistor MHV vertex expansion of planar amplitudes in N=4 super-Yang-Mills and the all-loop generalization of the BCFW recursion relations. We demonstrate explicitly in several examples that the MHV vertex expressions for tree-level amplitudes and loop integrands satisfy the recursion relations. Furthermore, we introduce a rewriting of the MHV expansion in terms of sums over non-crossing partitions and show that this cyclically invariant formula satisfies the recursion relations for all numbers of legs and all loop orders.Comment: 34 pages, 17 figures; v2: Minor improvements to exposition and discussion, updated references, typos fixe