Divide-and-Conquer Anchoring for Near-Separable Nonnegative Matrix Factorization and Completion in High Dimensions

Nonnegative matrix factorization (NMF) becomes tractable in polynomial time with unique solution under separability assumption, which postulates all the data points are contained in the conical hull of a few anchor data points. Recently developed linear programming and greedy pursuit methods can pick out the anchors from noisy data and results in a near-separable NMF. But their efficiency could be seriously weakened in high dimensions. In this paper, we show that the anchors can be precisely located from low-dimensional geometry of the data points even when their high dimensional features suffer from serious incompleteness. Our framework, entitled divide-and-conquer anchoring (DCA), divides the high-dimensional anchoring problem into a few cheaper sub-problems seeking anchors of data projections in low-dimensional random spaces, which can be solved in parallel by any near-separable NMF, and combines all the detected low-dimensional anchors via a fast hypothesis testing to identify the original anchors. We further develop two non-iterative anchoring algorithms in 1D and 2D spaces for data in convex hull and conical hull, respectively. These two rapid algorithms in the ultra low dimensions suffice to generate a robust and efficient near-separable NMF for high-dimensional or incomplete data via DCA. Compared to existing methods, two vital advantages of DCA are its scalability for big data, and capability of handling incomplete and high-dimensional noisy data. A rigorous analysis proves that DCA is able to find the correct anchors of a rank-k matrix by solving math cal O(klog k) sub-problems. Finally, we show DCA outperforms state-of-the-art methods on various datasets and tasks.