Non-decreasing Order Research Articles

High-Dimensional Multivariate Linear Regression with Weighted Nuclear Norm Regularization

We consider a low-rank matrix estimation problem when the data is assumed to be generated from the multivariate linear regression model. To induce the low-rank coefficient matrix, we employ the weighted nuclear norm (WNN) penalty defined as the weighted sum of the singular values of the matrix. The weights are set in a nondecreasing order, which yields the non-convexity of the WNN objective function in the parameter space. Although the objective function has been widely applied, studies on the estimation properties of its resulting estimator are limited. We propose an efficient algorithm under the framework of the alternative directional method of multipliers (ADMM) to estimate the coefficient matrix. The estimator from the suggested algorithm converges to a stationary point of an augmented Lagrangian function. Under the orthogonal design setting, the effects of the weights for estimating the singular values of the ground-truth coefficient matrix are derived. Under the Gaussian design setting, a minimax convergence rate on the estimation error is derived. We also propose a generalized cross-validation (GCV) criterion for selecting the tuning parameter and an iterative algorithm for updating the weights. Simulations and a real data analysis demonstrate the competitive performance of our new method. Supplementary materials for this article are available online.

The Asymptotic Number of Score Sequences

A tournament on a graph is an orientation of its edges. The score sequence lists the in-degrees in non-decreasing order. Works by Winston and Kleitman (J Comb Theory Ser A 35(2):208–230, 1983) and Kim and Pittel (J Comb Theory Ser A 92(2):197–206, 2000) showed that the number S_n of score sequences on the complete graph K_n satisfies S_n=Theta (4^n/n^{5/2}). By combining a recent recurrence relation for S_n in terms of the Erdős–Ginzburg–Ziv numbers N_n with the limit theory for discrete infinitely divisible distributions, we observe that n^{5/2}S_n/4^nrightarrow e^lambda /2sqrt{pi }, where lambda =sum _{k=1}^infty N_k/k4^k. This limit agrees numerically with the asymptotics of S_n conjectured by Takács (J Stat Plan Inference 14(1):123–142, 1986). We also identify the asymptotic number of strong score sequences, and show that the number of irreducible subscores in a random score sequence converges in distribution to a shifted negative binomial with parameters r=2 and p=e^{-lambda }.

Uniform distribution of saddle connection lengths in all textsf{SL}(2,pmb {mathbb {R}}) orbits

For every flat surface, almost every flat surface in its textsf{SL}(2,mathbb {R}) orbit has the following property: the sequence of its saddle connection lengths in non-decreasing order is uniformly distributed in the unit interval.

Counting tournament score sequences

The score sequence of a tournament is the sequence of the out-degrees of its vertices arranged in nondecreasing order. The problem of counting score sequences of a tournament with n n vertices is more than 100 years old [Quart. J. Math. 49 (1920), pp. 1–36]. In 2013 Hanna conjectured a surprising and elegant recursion for these numbers. We settle this conjecture in the affirmative by showing that it is a corollary to our main theorem, which is a factorization of the generating function for score sequences with a distinguished index. We also derive a closed formula and a quadratic time algorithm for counting score sequences.

Asymptotics for the eigenvalues of Toeplitz matrices with a symbol having a power singularity

AbstractThe present work is devoted to the construction of an asymptotic expansion for the eigenvalues of a Toeplitz matrix as goes to infinity, with a continuous and real‐valued symbol having a power singularity of degree with , at one point. The resulting matrix is dense and its entries decrease slowly to zero when moving away from the main diagonal, we apply the so called simple‐loop (SL) method for constructing and justifying a uniform asymptotic expansion for all the eigenvalues. Note however, that the considered symbol does not fully satisfy the conditions imposed in previous works, but only in a small neighborhood of the singularity point. In the present work: (i) We construct and justify the asymptotic formulas of the SL method for the eigenvalues with , where the eigenvalues are arranged in nondecreasing order and is a sufficiently small fixed number. (ii) We show, with the help of numerical calculations, that the obtained formulas give good approximations in the case . (iii) We numerically show that the main term of the asymptotics for eigenvalues with , formally obtained from the formulas of the SL method, coincides with the main term of the asymptotics constructed and justified in the classical works of Widom and Parter.

Pinning Control of Dynamical Networks With Optimal Convergence Rate

Dynamical systems coupled in the form of a complex networks have been exploited for modeling numerous real-world distributed systems. Due to the decentralized nature of dynamical networks, and inaccessibility or high control cost, in some applications, it is desired to employ pinning control techniques. A natural question that comes to mind is how one can select the proper pinned nodes and their feedback gains, (i.e., the performance metrics) to optimize the convergence rate to the homogeneous stationary state (i.e., the performance index) while the total pinning cost is kept below a given limit. This optimization problem has been converted into a semidefinite programming problem, and the optimal feedback gains have been derived analytically. Based on these results, for a given set of pinned nodes, an algorithm for determining the optimal feedback gains has been developed. Furthermore, it is shown that the edges between the pinned nodes do not impact the optimal results, and adding an edge between free nodes or between free and pinned nodes modifies the optimal convergence rate in nondecreasing order. Based on the derived analytical results, several interesting properties have been discovered. For symmetric networks, the nodes within a vertex orbit have the same optimal feedback gains, which are independent of the edges within each vertex orbit. For a number of topologies, closed-form formulas are provided for the optimal results. For a network of Chua systems, the impact of cut number is illustrated by comparing the optimal results for different topologies.

On path-greedy geometric spanners

A t-spanner is a subgraph of a graph G in which the length of the shortest path between two vertices never exceeds t times the length of the shortest path between them in G. A geometric graph is one whose vertices are points and whose edges are line segments between the corresponding points. Geometric t-spanners are t-spanners of the complete geometric graph on a given point set. Besides approximating the distance between points, we may ask a geometric t-spanner to be planar, have low degree, or low total edge length.One famous algorithm used to generate spanners is path-greedy, which scans pairs of vertices in non-decreasing order of edge length and adds the edge between them unless the current set of added edges already connects them with a path that t-approximates the edge length. Graphs from this algorithm are called path-greedy spanners. This work analyzes properties of path-greedy geometric spanners under different conditions. Specifically, we answer an open problem regarding the planarity and degree of path-greedy t-spanners for points in convex position in 2D. Further, we show a simple and efficient way to reduce the degree of a geometric spanner by adding extra points.

The soft-margin Support Vector Machine with ordered weighted average

This paper deals with a cost sensitive extension of the standard Support Vector Machine (SVM) using an ordered weighted sum of the deviations of misclassified individuals with respect to their corresponding supporting hyperplanes. In contrast with previous heuristic approaches, an exact method that applies the ordered weighted average operator in the classical SVM model is proposed. Specifically, when weights are sorted in non-decreasing order, a quadratic continuous formulation is developed. For general weights, a mixed integer quadratic formulation is proposed. In addition, our results prove that nonlinear kernel functions can be also applied to these new models extending its applicability beyond the linear case. Extensive computational results reported in the paper show that the predictive performance provided by the proposed exact solution approaches are better than the ones provided by the classical models (linear and nonlinear kernel) and similar or better than the previous ones provided by the heuristic solution by Maldonado et al. (2018).

Generalised online colouring problems in overlap graphs

In this paper we consider an online version of different colouring problems in overlap graphs, motivated by some stacking problems. The instance is a system of time intervals presented in non-decreasing order of the left endpoint. We consider the usual colouring problem as well as b-bounded colouring (colour class have a maximum capacity b) and the same problems in the complement graph. We also consider the case where at most b intervals of the same colour can intersect. For all these versions we obtain a logarithmic competitive ratio w.r.t. the maximum ratio of interval lengths while the best known ratio for the usual colouring was linear. To our knowledge it is the first time the other variants are considered in online overlap graphs. Moreover, in the offline case, a pre-processing allows us to deduce a logarithmic approximation ratio w.r.t. the maximum number of pairwise disjoint intervals in the system. Our method is based on a partition of the overlap graph into permutation graphs, leading to a competitive-preserving reduction of the problem in overlap graphs to the same problem in permutation graphs. We think that this new partition problem by itself is of interest for future work.

Counting Stirling permutations by number of pushes

Abstract Let 𝒯( k ) n denote the set of k-Stirling permutations having n distinct letters. Here, we consider the number of steps required (i.e., pushes) to rearrange the letters of a member of 𝒯( k ) n so that they occur in non-decreasing order. We find recurrences for the joint distribution on 𝒯( k ) n for the statistics recording the number of levels (i.e., occurrences of equal adjacent letters) and pushes. When k = 2, an explicit formula for the ordinary generating function of this distribution is also found. In order to do so, we determine the LU-decomposition of a certain infinite matrix having polynomial entries which enables one to compute explicitly the inverse matrix.

Conjectures on optimal nested generalized group testing algorithm

AbstractConsider a finite population of N items, where item i has a probability pi to be defective. The goal is to identify all items by means of group testing. This is the generalized group testing problem (hereafter GGTP). In the case of p1 = … = pN = p, Yao and Hwang (1990) proved that the pairwise testing algorithm is the optimal nested algorithm, with respect to the expected number of tests, for all N if and only if (R‐range hereafter) (an optimal at the boundary values). In this note, we present a result that helps to define the generalized pairwise testing algorithm (hereafter GPTA) for the GGTP. We present two conjectures: (a) when all pi,i = 1,…,N belong to the R‐range, GPTA is the optimal procedure among nested procedures applied to pi of nondecreasing order; and (b) if all pi,i = 1,…,N belong to the R‐range, GPTA is the optimal nested procedure, that is, minimizes the expected total number of tests with respect to all possible testing orders in the class of nested procedures. Although these conjectures are logically reasonable, we were only able to empirically verify the first one up to a particular level of N. We also provide a short survey of GGTP.

Some Results on the Eccentric Sequence of Graphs

The distance d(u, v) from a vertex u of graph G to a vertex v is the length of a shortest u to v path. The degree of the vertex u is the number of vertices at distance one. The sequence of numbers of vertices having 0,1,2,3,... is called the degree sequence, which is the list of degrees of vertices of G arranged in non-decreasing order. The eccentricity e(a) of a is the distance of a farthest vertex from a. Let G be a connected graph. The Eccentric Sequence of G is the list of the eccentricities of its vertices arranged in non-decreasing order. In this paper, we characterize the eccentric sequence of some of the derived graphs namely the line graph of the integral graph , the eccentric sequence of Mycieleskian of a graph.

Eccentric Sequence of Graphs

The distance d(u, v) from a vertex u of graph G to a vertex v is the length of a shortest u to v path. The eccentric sequences were the first distance related sequences introduced for undirected graphs. The eccentricity e(v) of v is the distance of a farthest vertex from v. The eccentric sequence of a graph G is a list of the eccentricities of vertices of graph G arranged in non-decreasing order. In this paper we determine the eccentric sequence of join of an empty graph and path graph(ie fan graph) and the eccentric sequence of the Cartesian product of paths P2 and Pn (ie Ladder graph).

A Novel Threshold Detection Technique for the Automatic Construction of Attribute Profiles in Hyperspectral Images

Attribute profiles are well-acknowledged as one of the most significant techniques to characterize spectral-spatial properties of a hyperspectral image. The spectral-spatial content of an attribute profile is influenced by the threshold values considered during its construction. In this article, we propose a robust method to detect the threshold values automatically by overcoming the limitations of the existing techniques. The proposed method employs a tree structure representing the connected components of the image and evaluates attribute values at each node. Then, a total characteristic function (TCF) is defined that represents these attribute values in a nondecreasing order. The defined TCF is analyzed using a novel technique to detect a few informative thresholds for the construction of a low-dimensional attribute profile representing substantial spectral-spatial information. The proposed threshold detection method is computationally efficient. To assess the effectiveness of the proposed technique experiments are conducted on three real hyperspectral datasets using six different attributes and the results are compared to the recent state-of-the-art method. The results demonstrate that the proposed method has several advantages over the existing state-of-the-art method.

Pairs of a tree and a nontree graph with the same status sequence

The status of a vertex x in a graph is the sum of the distances between x and all other vertices. Let G be a connected graph. The status sequence of G is the list of the statuses of all vertices arranged in nondecreasing order. G is called status injective if all the statuses of its vertices are distinct. Let G be a member of a family of graphs ℱ and let the status sequence of G be s.G is said to be status unique in ℱ if G is the unique graph in ℱ whose status sequence is s. In 2011, J.L. Shang and C. Lin posed the following two conjectures. Conjecture 1: A tree and a nontree graph cannot have the same status sequence. Conjecture 2: Any status injective tree is status unique in all connected graphs. We settle these two conjectures negatively. For every integer n≥10, we construct a tree Tn and a unicyclic graph Un, both of order n, with the following two properties:(1) Tn and Un have the same status sequence; (2) for n≥15, if n is congruent to 3 modulo 4 then Tn is status injective and among any four consecutive even orders, there is at least one order n such that Tn is status injective.

Властивості Евклідових задач лексикографічної комбінаторної оптимізації на розміщеннях

In the paper Euclidean problems of lexicographic combinatorial; optimization are discussed. These problems are to find lexicographically minimal (for minimization problems) or lexicographically maximal (for maximization problems) points among those that give the extremum of the objective function on a given Euclidean combinatorial set. The properties of linear and linear-fractional problems of lexicographic combinatorial optimization on a general set of arrangements are substantiated. The results obtained in the work are based on the previously known criteria of extremals of linear and linear-fractional functions on arrangements: any extremal is an element of certain set of polyarrangements (for linear problems the form of the extremal set is established explicitly, for linear-fractional problems the polyarrangement set is formed on the basis of some known extremal). In the paper we substantiate the form of points that are an lexicographic minimal and lexicographic maximal of linear function on the general set of arrangements. In particular if elements of multiset are in nondecreasing order, coefficients of objective function are in nonincreasing order and s is the least index such that corresponding coefficient of objective function is negative, tehn lexicographic minimal if formed as s – 1 first and k – s + 1 ( k is the dimension of space) last elements of multiset which are in nondecreasing order. For problems with linear-fractional function we obtain the method of forming solution of lexicographic combinatorial problem on arrangements, if any minimal (for minimization problems) or any maximal (for maximization problems) of objective function on given set of arrangements is known. In this case ordering of components of the extremal is carried out taking into account ordering for nonincreasing of coefficients of special linear function

Calculating the cardinality of some classes of binary matrices using bitwise operations — A polynomial algorithm

Let [Formula: see text] be the set of all [Formula: see text] binary matrices with exactly [Formula: see text] units in each row and each column, [Formula: see text]. A matrix [Formula: see text] will be called primitive, if there is no [Formula: see text] submatrix of [Formula: see text] that belongs to the set [Formula: see text], [Formula: see text]. The article describes a polynomial algorithm, which works in time [Formula: see text] for verifying whether a [Formula: see text]-matrix is primitive. The work applies this algorithm for finding all primitive [Formula: see text]-matrices which rows and columns are in lexicographically nondecreasing order (semi-canonical binary matrices) for some integers [Formula: see text] and [Formula: see text].

Efficient Recovery of Low-Rank Matrix via Double Nonconvex Nonsmooth Rank Minimization.

Recently, there is a rapidly increasing attraction for the efficient recovery of low-rank matrix in computer vision and machine learning. The popular convex solution of rank minimization is nuclear norm-based minimization (NNM), which usually leads to a biased solution since NNM tends to overshrink the rank components and treats each rank component equally. To address this issue, some nonconvex nonsmooth rank (NNR) relaxations have been exploited widely. Different from these convex and nonconvex rank substitutes, this paper first introduces a general and flexible rank relaxation function named weighted NNR relaxation function, which is actually derived from the initial double NNR (DNNR) relaxations, i.e., DNNR relaxation function acts on the nonconvex singular values function (SVF). An iteratively reweighted SVF optimization algorithm with continuation technology through computing the supergradient values to define the weighting vector is devised to solve the DNNR minimization problem, and the closed-form solution of the subproblem can be efficiently obtained by a general proximal operator, in which each element of the desired weighting vector usually satisfies the nondecreasing order. We next prove that the objective function values decrease monotonically, and any limit point of the generated subsequence is a critical point. Combining the Kurdyka-Łojasiewicz property with some milder assumptions, we further give its global convergence guarantee. As an application in the matrix completion problem, experimental results on both synthetic data and real-world data can show that our methods are competitive with several state-of-the-art convex and nonconvex matrix completion methods.

Batching and resource allocation decisions on an m-machine proportionate flowshop

This article considers an m-machine proportionate flowshop scheduling problem where each stage of production consists of a batching operation. Moreover, we assume that the job processing times are controllable through the allocation of a non-renewable resource. The objective consists of minimising the makespan. The scheduler’s task consists of (1) allocating jobs to batches; (2) scheduling batches on the m-machine flowshop; (3) allocating resources to batches; and (4) allocating the resources within each batch to jobs. We show that there exists an optimal solution to the problem that consists of sequencing the jobs in Λ-shape order based on their workloads. Furthermore, the jobs that are sorted in non-decreasing order of workload are scheduled in batches of equal size, whereas the remaining jobs are allocated to batches of different sizes. We present an time algorithm based on the observation that all jobs scheduled in the first segment of the Λ-shape have workloads that are smaller than (or equal to) that of the last job in the sequence.

Multiple Due Time Surgical Scheduling With Truncated Learning and Deteriorating Effect

The surgical scheduling problem that incorporates the surgical duration, the setup time, the turnover time and the due time is studied in this paper. The actual surgical duration is affected by the normal duration, the surgical sequence, the accumulated experience of surgical teams and a controlling parameter. Besides, the surgical duration, the setup time and the turnover time are affected by the deteriorating effect, which means when the start time, the setup time or the turnover time of a surgery is postponed, the actual duration, the actual setup time or the actual turnover time will be prolonged. A schedule problem is formulated to minimize the maximum surgical tardiness. By building and analyzing the surgical scheduling model, each surgical team should operate surgeries according to a non-decreasing order of patients' normal surgical duration. Furthermore, a branch-and-bound algorithm is provided to solve the surgical teams scheduling problem. Our experimental results show the effectiveness and stability of our proposed algorithm.