Splitting Methods for Convex Clustering

Clustering is a fundamental problem in many scientific applications. Standard methods such as k-means, Gaussian mixture models, and hierarchical clustering, however, are beset by local minima, which are sometimes drastically suboptimal. Recently introduced convex relaxations of k-means and hierarchical clustering shrink cluster centroids toward one another and ensure a unique global minimizer. In this work, we present two splitting methods for solving the convex clustering problem. The first is an instance of the alternating direction method of multipliers (ADMM); the second is an instance of the alternating minimization algorithm (AMA). In contrast to previously considered algorithms, our ADMM and AMA formulations provide simple and unified frameworks for solving the convex clustering problem under the previously studied norms and open the door to potentially novel norms. We demonstrate the performance of our algorithm on both simulated and real data examples. While the differences between the two algorithms appear to be minor on the surface, complexity analysis and numerical experiments show AMA to be significantly more efficient. This article has supplementary materials available online.     

Proof of the Razumov-Stroganov conjecture

The Razumov-Stroganov conjecture relates the ground-state coefficients in the periodic even-length dense O(1) loop model to the enumeration of fully-packed loop configurations on the square, with alternating boundary conditions, refined according to the link pattern for the boundary points.Here we prove this conjecture, by means of purely combinatorial methods. The main ingredient is a generalization of the Wieland proof technique for the dihedral symmetry of these classes, based on the ‘gyration’ operation, whose full strength we will investigate in a companion paper.     

A numerical scheme for particle-laden thin film flow in two dimensions

The physics of particle-laden thin film flow is not fully understood, and recent experiments have raised questions with current theory. There is a need for fully two-dimensional simulations to compare with experimental data. To this end, a numerical scheme is presented for a lubrication model derived for particle-laden thin film flow in two dimensions with surface tension. The scheme relies on an ADI process to handle the higher-order terms, and an iterative procedure to improve the solution at each timestep. This is the first paper to simulate the two-dimensional particle-laden thin film lubrication model. Several aspects of the scheme are examined for a test problem, such as the timestep, runtime, and number of iterations. The results from the simulation are compared to experimental data. The simulation shows good qualitative agreement. It also suggests further lines of inquiry for the physical model.     

N-苯基马来酰亚胺与环己烯共聚物的制备和表征

Ｎ苯基马来酰亚胺（ＰＨＭＩ）是一种很有应用价值的共聚单体,它能赋予共聚物很好的耐热性．有关ＰＨＭＩ与甲基丙烯酸甲酯、苯乙烯及乙烯基醚的二元共聚合反应已有文献报道［１～３］．而有关ＰＨＭＩ与环烯烃特别是环己烯的共聚合研究除我们所做的工作外［４］,报道甚少．作者研究了ＰＨＭＩ与环己烯在溶液中的二元共聚合反应,并对所得的共聚物进行了表征．１　原料ＰＨＭＩ,自制,ｍ．ｐ．８８－７℃;元素分析Ｃ,６９－３３;Ｈ,４－０７;Ｎ,８－１０;理论值Ｃ,６９－３６;Ｈ,４－０７;Ｎ,８－０９,１ＨＮＭＲ（δ）…     

Alternating mapping functions

We consider functions f from $[n]:=\{1,2,\dots ,n\}$ into itself satisfying that the labels along the iteration orbit of each $i\in [n]$ are forming an alternating sequence, i.e., $i<f(i)>f^2(i)<f^3(i)>?$ or $i>f(i)<f^2(i)>f^3(i)<?$ . We are able to solve the enumeration problem by stating exact and asymptotic formulæ for the number of such so-called alternating n-mapping functions. Furthermore we study the expected component structure of a randomly chosen alternating n-mapping by determining the probability that the underlying mapping graph is connected as well as the limiting distribution of the number of components. Moreover, the corresponding enumeration problem for weakly alternating n-mapping functions has also been solved.     

Unconstrained Optimization Techniques for the Acceleration of Alternating Projection Methods

Alternating projection methods have been extensively used to find the closest point, to a given point, in the intersection of several given sets that belong to a Hilbert space. One of the characteristics of these schemes is the slow convergence that can be observed in practical applications. To overcome this difficulty, several techniques, based on different ideas, have been developed to accelerate their convergence. Recently, a successful acceleration scheme was developed specially for Cimmino's method when applied to the solution of large-scale saddle point problems. This specialized acceleration scheme is based on the use of the well-known conjugate gradient method for minimizing a related convex quadratic map. In this work, we extend and further analyze this optimization approach for several alternating projection methods on different scenarios. In particular, we include a specialized analysis and treatment for the acceleration of von Neumann-Halperin's method and Cimmino's method on subspaces, and Kaczmarz method on linear varieties. For some specific applications we illustrate the advantages of our acceleration schemes with encouraging numerical experiments.     

Algorithms for Fitting the Constrained Lasso

We compare alternative computing strategies for solving the constrained lasso problem. As its name suggests, the constrained lasso extends the widely used lasso to handle linear constraints, which allow the user to incorporate prior information into the model. In addition to quadratic programming, we employ the alternating direction method of multipliers (ADMM) and also derive an efficient solution path algorithm. Through both simulations and benchmark data examples, we compare the different algorithms and provide practical recommendations in terms of efficiency and accuracy for various sizes of data. We also show that, for an arbitrary penalty matrix, the generalized lasso can be transformed to a constrained lasso, while the converse is not true. Thus, our methods can also be used for estimating a generalized lasso, which has wide-ranging applications. Code for implementing the algorithms is freely available in both the Matlab toolbox SparseReg and the Julia package ConstrainedLasso. Supplementary materials for this article are available online.     

The equivariant topology of stable Kneser graphs

The stable Kneser graph SG_n,k, n?1, k?0, introduced by Schrijver (1978) [19], is a vertex critical graph with chromatic number k+2, its vertices are certain subsets of a set of cardinality m=2n+k. Björner and de Longueville (2003) [5] have shown that its box complex is homotopy equivalent to a sphere, Hom(K₂,SG_n,k)?Sk. The dihedral group D2m acts canonically on SG_n,k, the group C₂ with 2 elements acts on K₂. We almost determine the (C₂×D2m)-homotopy type of Hom(K₂,SG_n,k) and use this to prove the following results.The graphs SG_2s,4 are homotopy test graphs, i.e. for every graph H and r?0 such that Hom(SG_2s,4,H) is (r−1)-connected, the chromatic number χ(H) is at least r+6.If k∉{0,1,2,4,8} and n?N(k) then SG_n,k is not a homotopy test graph, i.e. there are a graph G and an r?1 such that Hom(SG_n,k,G) is (r−1)-connected and χ(G)<r+k+2.     

Electric impedance study of elastic alternating propylene-carbon monoxide copolymer (PCO-200)

The electrical properties of elastic alternating propylene-carbon monoxide copolymer (PCO-200) were investigated using the impedance spectroscopy technique. The results revealed a phase transition at about 70 °C where the material transforms from its insulating phase of conductivity in the order of 6×10⁻⁹ to about 9×10⁻⁵ (Ω m)⁻¹, The second phase is characterized by temperature dependent electrical relaxation phenomena. The plot of the complex electric modulus and the complex impedance yields semicircles in the temperature range 70 up to 110 °C and a decreasing radius with increasing temperature. The activation energy was found to be in the order of 0.8 eV.