The Fastest Mixing Markov Process on a Graph and a Connection to a Maximum Variance Unfolding Problem

We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue lambda_2 of the Laplacian of the weighted graph. In this paper we consider the problem of assigning transition rates to the edges so as to maximize lambda_2 subject to a linear constraint on the rates. This is the problem of finding the fastest mixing Markov process (FMMP) on the graph. We show that the FMMP problem is a convex optimization problem, which can in turn be expressed as a semidefinite program, and therefore effectively solved numerically. We formulate a dual of the FMMP problem and show that it has a natural geometric interpretation as a maximum variance unfolding (MVU) problem, , the problem of choosing a set of points to be as far apart as possible, measured by their variance, while respecting local distance constraints. This MVU problem is closely related to a problem recently proposed by Weinberger and Saul as a method for "unfolding" high-dimensional data that lies on a low-dimensional manifold. The duality between the FMMP and MVU problems sheds light on both problems, and allows us to characterize and, in some cases, find optimal solutions

A Duality View of Spectral Methods for Dimensionality Reduction

We present a unified duality view of several recently emerged spectral methods for nonlinear dimensionality reduction, including Isomap, locally linear embedding, Laplacian eigenmaps, and maximum variance unfolding. We discuss the duality theory for the maximum variance unfolding problem, and show that other methods are directly related to either its primal formulation or its dual formulation, or can be interpreted from the optimality conditions. This duality framework reveals close connections between these seemingly quite different algorithms. In particular, it resolves the myth about these methods in using either the top eigenvectors of a dense matrix, or the bottom eigenvectors of a sparse matrix — these two eigenspaces are exactly aligned at primaldual optimality

Synthesis and characterization of 2-(2-benzhydrylnaphthyliminomethyl)pyridylnickel halides: formation of branched polyethylene

A series of 2-(2-benzhydrylnaphthyliminomethyl)pyridine derivatives (L1–L3) was prepared and used to synthesize the corresponding bis-ligated nickel(II) halide complexes (Ni1–Ni6) in good yield. The molecular structures of representative complexes, namely the bromide Ni3 and the chloride complex Ni6, were confirmed by single crystal X-ray diffraction, and revealed a distorted octahedral geometry at nickel. Upon activation with either methylaluminoxane (MAO) or modified methylaluminoxane (MMAO), all nickel complex pre-catalysts exhibited high activities (up to 2.02 × 10⁷ g(PE) mol⁻¹(Ni) h⁻¹) towards ethylene polymerization, producing branched polyethylene of low molecular weight and narrow polydispersity. The influence of the reaction parameters and the nature of the ligands on the catalytic behavior of the title nickel complexes were investigated

Methyl 2-{[2-(4,4,5,5-tetra­methyl-1,3-dioxyl-4,5-dihydro­imidazol-2-yl)phen­yl]­oxy}acetate

In the title compound, C16H21N2O5, the benzene ring is nearly perpendicular to the imidazole ring, making a torsion angle of 88.6 (8)°·The crystal structure is stabilized by non-classical C—H⋯O and C—H⋯π inter­actions, which build up a three-dimensional network