Unregularized online learning algorithms with general loss functions

In this paper, we consider unregularized online learning algorithms in a Reproducing Kernel Hilbert Space (RKHS). Firstly, we derive explicit convergence rates of the unregularized online learning algorithms for classification associated with a general α-activating loss (see Definition 1 below). Our results extend and refine the results in [30] for the least square loss and the recent result [3] for the loss function with a Lipschitz-continuous gradient. Moreover, we establish a very general condition on the step sizes which guarantees the convergence of the last iterate of such algorithms. Secondly, we establish, for the first time, the convergence of the unregularized pairwise learning algorithm with a general loss function and derive explicit rates under the assumption of polynomially decaying step sizes. Concrete examples are used to illustrate our main results. The main techniques are tools from convex analysis, refined inequalities of Gaussian averages [5], and an induction approach.     

Efficient domain partitioning algorithms for global optimization of rational and Lipschitz continuous functions

A domain partitioning algorithm for minimizing or maximizing a Lipschitz continuous function is enhanced to yield two new, more efficient algorithms. The use of interval arithmetic in the case of rational functions and the estimates of Lipschitz constants valid in subsets of the domain in the case of others and the addition of local optimization have resulted in an algorithm which, in tests on standard functions, performs well.     

Efficient algorithms for tree reconstruction

In this note, we show thatO(n logn) operations are sufficient to reconstruct an ordered binary tree given its inorder traversal and either its preorder or postorder traversal. An alternative linear representation allows reconstruction usingO(n) operations.     

Nonstationary multisplittings with general weighting matrices for non-hermitian positive definite systems

We discuss the nonstationary multisplittings and two-stage multisplittings to solve the linear systems of algebraic equations Ax = b when the coefficient matrix is a non-Hermitian positive definite matrix, and establish the convergence theories with general weighting matrices. This not only eliminates the restrictive condition that it is usually assumed for scalar weighting matrices, but also generalizes it to a general positive definite matrix.     

Area-efficient algorithms for straight-line tree drawings

We investigate several straight-line drawing problems for bounded-degree trees in the integer grid without edge crossings under various types of drawings: (1) upward drawings whose edges are drawn as vertically monotone chains, a sequence of line segments, from a parent to its children, (2) order-preserving drawings which preserve the left-to-right order of the children of each vertex, and (3) orthogonal straight-line drawings in which each edge is represented as a single vertical or horizontal segment.

Main contribution of this paper is a unified framework to reduce the upper bound on area for the straight-line drawing problems from O(nlogn) (Crescenzi et al., 1992) to O(nloglogn). This is the first solution of an open problem stated by Garg et al. (1993). We also show that any binary tree admits a small area drawing satisfying any given aspect ratio in the orthogonal straight-line drawing type.

Our results are briefly summarized as follows. Let T be a bounded-degree tree with n vertices. Firstly, we show that T admits an upward straight-line drawing with area O(nloglogn). If T is binary, we can obtain an O(nloglogn)-area upward orthogonal drawing in which each edge is drawn as a chain of at most two orthogonal segments and which has O(n/logn) bends in total. Secondly, we present O(nloglogn)-area (respectively, -volume) orthogonal straight-line drawing algorithms for binary trees with arbitrary aspect ratios in 2-dimension (respectively, 3-dimension). Finally, we present some experimental results which shows the area requirements, in practice, for (order-preserving) upward drawing are much smaller than theoretical bounds obtained through analysis.     

Continuous bottleneck tree partitioning problems

We study continuous partitioning problems on tree network spaces whose edges and nodes are points in Euclidean spaces. A continuous partition of this space into p connected components is a collection of p subtrees, such that no pair of them intersect at more than one point, and their union is the tree space. An edge-partition is a continuous partition defined by selecting p−1 cut points along the edges of the underlying tree, which is assumed to have n nodes. These cut points induce a partition into p subtrees (connected components). The objective is to minimize (maximize) the maximum (minimum) “size” of the components (the min–max (max–min) problem). When the size is the length of a subtree, the min–max and the max–min partitioning problems are NP-hard. We present O(n² log(min(p,n))) algorithms for the edge-partitioning versions of the problem. When the size is the diameter, the min–max problems coincide with the continuous p-center problem. We describe O(n log³ n) and O(n log² n) algorithms for the max–min partitioning and edge-partitioning problems, respectively, where the size is the diameter of a component.     

A macroscopic consitutive model on induced anisotropy for polymers with weighting functions

The alignment of polymer chains is a well known microstructural evolution effect due to straining of polymers. This has a drastic influence on the macroscopic properties of the initially isotropic material, such as a pronounced strength in the loading direction of stretched films. Experiments on strain induced anisotropy at room temperature are analyzed by optical measurements. For modeling the effect of strain induced anisotropy a macroscopic constitutive model is presented. As a key idea, weighting functions are introduced to represent a strain-softening/hardening-effect to account for induced anisotropy. These functions represent the ratio between the total strain rate and a structural tensor. In this way, material parameters are used as a sum of weighted direction related quantities. In the finite element examples we simulate the cold-forming of amorphous thermoplastic films below the glass transition temperature subjected to different re-loading directions. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New branch-and-bound algorithms for k-cardinality tree problems

Given an undirected graph, the k-cardinality tree problem (KCTP) is the problem of finding a subtree with exactly k edges whose sum of weights is minimum. In this paper we present a lower bound for KCTP based on the work by Kataoka et al. [Kataoka, S., N. Araki and T. Yamada, Upper and lower bounding procedures for the minimum rooted k-subtree problem, European Journal of Operational Research, 122 (2000), 561–569]. This new bound is the basis for the development of a branch-and-bound algorithm for the problem. Experiments carried out on instances from KCTLib revealed that the new exact algorithm largely outperforms the previous approach.     

Stochastic algorithms with armijo stepsizes for minimization of functions

Stochastic algorithms for optimization problems, where function evaluations are done by Monte Carlo simulations, are presented. At each iteratex _i, they draw a predetermined numbern(i) of sample points from an underlying probability space; based on these sample points, they compute a feasible-descent direction, an Armijo stepsize, and the next iteratex _i+1. For an appropriate optimality function , corresponding to an optimality condition, it is shown that, ifn(i) , then (x _i) 0, whereJ is a set of integers whose upper density is zero. First, convergence is shown for a general algorithm prototype: then, a steepest-descent algorithm for unconstrained problems and a feasible-direction algorithm for problems with inequality constraints are developed. A numerical example is supplied.     

The power of adaptive algorithms for functions with singularities

This is an overview of recent results on complexity and optimality of adaptive algorithms for integrating and approximating scalar piecewise r-smooth functions with unknown singular points. We provide adaptive algorithms that use at most n function samples and have the worst case errors proportional to n^−r for functions with at most one unknown singularity. This is a tremendous improvement over nonadaptive algorithms whose worst case errors are at best proportional to n⁻¹ for integration and n^−1/p for the L^p approximation problem. For functions with multiple singular points the adaptive algorithms cease to dominate the nonadaptive ones in the worst case setting. Fortunately, they regain their superiority in the asymptotic setting. Indeed, they yield convergence of order n^−r for piecewise r-smooth functions with an arbitrary (unknown but finite) number of singularities. None of these results hold for the L^∞ approximation. However, they hold for the Skorohodmetric, which we argue to be more appropriate than L^∞ for dealing with discontinuous functions. Numerical test results and possible extensions are also discussed.     

VLSN search algorithms for partitioning problems using matching neighbourhoods

In this paper, we propose a general paradigm to design very large-scale neighbourhood search algorithms for generic partitioning-type problems. We identify neighbourhoods of exponential size, called matching neighbourhoods, comprised of the union of a class of exponential neighbourhoods. It is shown that these individual components of the matching neighbourhood can be searched in polynomial time, whereas searching the matching neighbourhood is NP-hard. Matching neighbourhood subsumes a well-known class of exponential neighbourhoods called cyclic-exchange neighbourhoods. Our VLSN algorithm is implemented for two special cases of the partitioning problem; the covering assignment problem and the single source transportation problem. Encouraging experimental results are also reported.     

Quotient tree partitioning of undirected graphs

The partitioning of the vertices of an undirected graph, in a way that makes its quotient graph a tree, mirrors a way of permuting a square symmetric matrix to allow its factoring with little fill-in. We analyze the complexity of finding the best partitioning and show that it is NP-complete. We also give a new and simpler implementation of an algorithm that finds a maximal quotient tree.     

Universal algorithms for evaluating boolean functions

Theorems relating the most important concepts of switching theory to a new kind of difference operator are stated. This difference operator is then used as a unifying concept to solve, by means of a new type of algorithm, several problems arising in logical design.     

Parameterized approximation of fuzzy number with minimum variance weighting functions

In the parameterized fuzzy number expectation and fuzzy number interval approximation with the weighting function method, we once considered the weighting function form with maximum entropy, which has some interesting properties. In the present paper, we will propose another kind of parameterized weighting function with minimum variance, and apply it to the fuzzy number expectation and fuzzy number interval approximation problems. It shows that the minimum variance weighting function also has some similar interesting properties, and can be used to express the decision maker’s preference information in the fuzzy number defuzzification process.     

Polynomially bounded algorithms for locatingp-centers on a tree

We discuss several forms of thep-center location problems on an undirected tree network. Our approach is based on utilizing results for rigid circuit graphs to obtain polynomial algorithms for solving the model. Duality theory on perfect graphs is used to define and solve the dual location model.     

Decomposition algorithms for the tree edit distance problem

We study the behavior of dynamic programming methods for the tree edit distance problem, such as [P. Klein, Computing the edit-distance between unrooted ordered trees, in: Proceedings of 6th European Symposium on Algorithms, 1998, p. 91–102; K. Zhang, D. Shasha, SIAM J. Comput. 18 (6) (1989) 1245–1262]. We show that those two algorithms may be described as decomposition strategies. We introduce the general framework of cover strategies, and we provide an exact characterization of the complexity of cover strategies. This analysis allows us to define a new tree edit distance algorithm, that is optimal for cover strategies.     

Approximation algorithms for constrained generalized tree alignment problem

In generalized tree alignment problem, we are given a set S of k biologically related sequences and we are interested in a minimum cost evolutionary tree for S. In many instances of this problem partial phylogenetic tree for S is known. In such instances, we would like to make use of this knowledge to restrict the tree topologies that we consider and construct a biologically relevant minimum cost evolutionary tree. So, we propose the following natural generalization of the generalized tree alignment problem, a problem known to be MAX-SNP Hard, stated as follows:

Constrained Generalized Tree Alignment Problem [S. Divakaran, Algorithms and heuristics for constrained generalized alignment problem, DIMACS Technical Report 2007-21, 2007]: Given a set S of k related sequences and a phylogenetic forest comprising of node-disjoint phylogenetic trees that specify the topological constraints that an evolutionary tree of S needs to satisfy, construct a minimum cost evolutionary tree for S.

In this paper, we present constant approximation algorithms for the constrained generalized tree alignment problem. For the generalized tree alignment problem, a special case of this problem, our algorithms provide a guaranteed error bound of 2−2/k.