We provide a systematic analysis of nonnegative matrix factorization (NMF) relating to data clustering. We generalize the usual X = FG{sup T} decomposition to the symmetric W = HH{sup T} and W = HSH{sup T} decompositions. We show that (1) W = HH{sup T} is equivalent to Kernel K-means clustering and the Laplacian-based spectral clustering. (2) X = FG{sup T} is equivalent to simultaneous clustering of rows and columns of a bipartite graph. We emphasizes the importance of orthogonality in NMF and soft clustering nature of NMF. These results are verified with experiments on face images and newsgroups.

When designing a FFAG accelerator for protons we found convenient to follow this procedure [1]. (1) We assume a Non-Scaling Lattice (NSL) because the aim is toward a compact layout, though we are aware of the issue of multiple resonance crossing. (2) We take a periodic sequence of FDF triplets as these have been proven to exhibit a very small dispersion function. (3) The reference trajectory is taken to be the injection orbit that corresponds to the lowest value of the acceleration momentum range. (4) Finally, the magnets in the triplet have all a linear field profile. We have indeed found recently [2,3] that the Adjusted Field Profile (AFP) to cancel the horizontal chromaticity is exceedingly non-linear and it causes a too large betatron tune variation with the amplitude of motion. A sequence of FDF triplets is shown in Figure 1. They are made of sector magnets having parallel entrance and exit planes facing each other. Only for the injection orbit the trajectory in the magnets is made of arcs of circle. The magnets have sharp edges and there is no entrance or exit angle only for the reference (injection) orbit [4]. This solution minimizes magnet width, has the most …