Brendan P. W. Ames

Nuclear norm minimization for the planted clique and biclique problems

We consider the problems of finding a maximum clique in a graph and finding a maximum-edge biclique in a bipartite graph. Both problems are NP-hard. We write both problems as matrix-rank minimization and then relax them using the nuclear norm. This technique, which may be regarded as a generalization of compressive sensing, has recently been shown to be an effective way to solve rank optimization problems. In the special case that the input graph has a planted clique or biclique (i.e., a single large clique or biclique plus diversionary edges), our algorithm successfully provides an exact solution to the original instance. For each problem, we provide two analyses of when our algorithm succeeds. In the first analysis, the diversionary edges are placed by an adversary. In the second, they are placed at random. In the case of random edges for the planted clique problem, we obtain the same bound as Alon, Krivelevich and Sudakov as well as Feige and Krauthgamer, but we use different techniques.

Convex optimization for the planted k-disjoint-clique problem

We consider the

Solving ptychography with a convex relaxation

k

A Leapfrog Strategy for Pursuit-Evasion in a Polygonal Environment

k-disjoint-clique problem. The input is an undirected graph

Alternating direction method of multipliers for sparse zero-variance discriminant analysis and principal component analysis

G

Asymptotic expansions of the ordered spectrum of symmetric matrices

G in which the nodes represent data items, and edges indicate a similarity between the corresponding items. The problem is to find within the graph