Fast Unimodular Counting

2000 ◽  
Vol 9 (3) ◽  
pp. 277-285 ◽  
Author(s):  
JOHN MOUNT
Keyword(s):  
High Dimension ◽  
Nonnegative Integer ◽  
Unit Cost ◽  
Fixed Number ◽  
Integral Vector ◽  
Fixed Dimension ◽  
Unimodular Matrix ◽  
Low Dimension ◽  
Integer Solutions ◽  
Special Case

This paper describes methods for counting the number of nonnegative integer solutions of the system Ax = b when A is a nonnegative totally unimodular matrix and b an integral vector of fixed dimension. The complexity (under a unit cost arithmetic model) is strong in the sense that it depends only on the dimensions of A and not on the size of the entries of b. For the special case of ‘contingency tables’ the run-time is 2O(√dlogd) (where d is the dimension of the polytope). The method is complementary to Barvinok's in that our algorithm is effective on problems of high dimension with a fixed number of (non-sign) constraints, whereas Barvinok's algorithms are effective on problems of low dimension and an arbitrary number of constraints.

scholarly journals Top Coefficients of the Denumerant

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Velleda Baldoni ◽  
Nicole Berline ◽  
Brandon Dutra ◽  
Matthias Köppe ◽  
Michele Vergne ◽  
...  
Keyword(s):  
Nonnegative Integer ◽  
Fixed Number ◽  
Lattice Points ◽  
Combinatorial Number Theory ◽  
International Audience ◽  
Integer Solutions ◽  
Combinatorial Function ◽  
Rational Generating Function ◽  
Rational Generating Functions ◽  
The Right

International audience For a given sequence $\alpha = [\alpha_1,\alpha_2,\ldots , \alpha_N, \alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\alpha)(t)$ that counts the nonnegative integer solutions of the equation $\alpha_1x_1+\alpha_2 x_2+ \ldots+ \alpha_Nx_N+ \alpha_{N+1}x_{N+1}=t$, where the right-hand side $t$ is a varying nonnegative integer. It is well-known that $E(\alpha)(t)$ is a quasipolynomial function of $t$ of degree $N$. In combinatorial number theory this function is known as the $\textit{denumerant}$. Our main result is a new algorithm that, for every fixed number $k$, computes in polynomial time the highest $k+1$ coefficients of the quasi-polynomial $E(\alpha)(t)$ as step polynomials of $t$. Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for $E(\alpha)(t)$ and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Experiments using a $\texttt{MAPLE}$ implementation will be posted separately. Considérons une liste $\alpha = [\alpha_1,\alpha_2,\ldots , \alpha_N, \alpha_{N+1}]$ de $N+1$ entiers positifs. Le dénumérant $E(\alpha)(t)$ est lafonction qui compte le nombre de solutions en entiers positifs ou nuls de l’équation $\sum^{N+1}_{i=1}x_i\alpha_i=t$, où $t$ varie dans les entiers positifs ou nuls. Il est bien connu que cette fonction est une fonction quasi-polynomiale de $t$, de degré $N$. Nous donnons un nouvel algorithme qui calcule, pour chaque entier fixé $k$ (mais $N$ n’est pas fixé, les $k+1$ plus hauts coefficients du quasi-polynôme $E(\alpha)(t)$ en termes de fonctions en dents de scie. Notre algorithme utilise la structure d’ensemble partiellement ordonné des pôles de la fonction génératrice de $E(\alpha)(t)$. Les $k+1$ plus hauts coefficients se calculent à l’aide de fonctions génératrices de points entiers dans des cônes polyèdraux de dimension inférieure ou égale à $k$.

LEARNING BY CRITERION OPTIMIZATION ON A UNITARY UNIMODULAR MATRIX GROUP

2008 ◽  
Vol 18 (02) ◽  
pp. 87-103 ◽  
Author(s):  
SIMONE FIORI
Keyword(s):  
Optimization Problems ◽  
Learning Theories ◽  
Neural Systems ◽  
Matrix Group ◽  
Unimodular Group ◽  
Unimodular Matrix ◽  
Low Dimension ◽  
Low Dimensional ◽  
Complex Valued ◽  
Special Case

The present manuscript aims at illustrating fundamental challenges and solutions arising in the design of learning theories by optimization on manifolds in the context of complex-valued neural systems. The special case of a unitary unimodular group of matrices is dealt with. The unitary unimodular group under analysis is a low dimensional and easy-to-handle matrix group. Notwithstanding, it exhibits a rich geometrical structure and gives rise to interesting speculations about methods to solve optimization problems on manifolds. Also, its low dimension allows us to treat most of the quantities involved in computation in closed form as well as to render them in graphical format. Some numerical experiments are presented and discussed within the paper, which deal with complex-valued independent component analysis.

scholarly journals GBUO: “The Good, the Bad, and the Ugly” Optimizer

2021 ◽  
Vol 11 (5) ◽  
pp. 2042
Author(s):  
Hadi Givi ◽  
Mohammad Dehghani ◽  
Zeinab Montazeri ◽  
Ruben Morales-Menendez ◽  
Ricardo A. Ramirez-Mendoza ◽  
...  
Keyword(s):  
High Dimension ◽  
Essential Role ◽  
Optimization Problems ◽  
Optimization Algorithms ◽  
Stochastic Search ◽  
Objective Functions ◽  
Science And Engineering ◽  
Fixed Dimension ◽  
Dimension Functions ◽  
Simulation Results

Optimization problems in various fields of science and engineering should be solved using appropriate methods. Stochastic search-based optimization algorithms are a widely used approach for solving optimization problems. In this paper, a new optimization algorithm called “the good, the bad, and the ugly” optimizer (GBUO) is introduced, based on the effect of three members of the population on the population updates. In the proposed GBUO, the algorithm population moves towards the good member and avoids the bad member. In the proposed algorithm, a new member called ugly member is also introduced, which plays an essential role in updating the population. In a challenging move, the ugly member leads the population to situations contrary to society’s movement. GBUO is mathematically modeled, and its equations are presented. GBUO is implemented on a set of twenty-three standard objective functions to evaluate the proposed optimizer’s performance for solving optimization problems. The mentioned standard objective functions can be classified into three groups: unimodal, multimodal with high-dimension, and multimodal with fixed dimension functions. There was a further analysis carried-out for eight well-known optimization algorithms. The simulation results show that the proposed algorithm has a good performance in solving different optimization problems models and is superior to the mentioned optimization algorithms.

Feature Selection in High Dimension

2010 ◽  
pp. 97-119
Author(s):  
Sébastien Gadat ◽  
Sébastien Gadat
Keyword(s):  
Image Analysis ◽  
Feature Selection ◽  
Stochastic Approximation ◽  
High Dimension ◽  
Small Proportion ◽  
Features Selection ◽  
Approximation Techniques ◽  
Whole Number ◽  
Special Case ◽  
Features Weighting

Variable selection for classification is a crucial paradigm in image analysis. Indeed, images are generally described by a large amount of features (pixels, edges …) although it is difficult to obtain a sufficiently large number of samples to draw reliable inference for classifications using the whole number of features. The authors describe in this chapter some simple and effective features selection methods based on filter strategy. They also provide some more sophisticated methods based on margin criterion or stochastic approximation techniques that achieve great performances of classification with a very small proportion of variables. Most of these “wrapper” methods are dedicated to a special case of classifier, except the Optimal features Weighting algorithm (denoted OFW in the sequel) which is a meta-algorithm and works with any classifier. A large part of this chapter will be dedicated to the description of the description of OFW and hybrid OFW algorithms. The authors illustrate also several other methods on practical examples of face detection problems.

scholarly journals New Results on Vector and Homing Vector Automata

2019 ◽  
Vol 30 (08) ◽  
pp. 1335-1361
Author(s):  
Özlem Salehi ◽  
Abuzer Yakaryılmaz ◽  
A. C. Cem Say
Keyword(s):  
Real Time ◽  
Nonnegative Integer ◽  
Diophantine Equations ◽  
Finite Automata ◽  
Separation Problem ◽  
Integer Solutions ◽  
Homing Vector

We present several new results and connections between various extensions of finite automata through the study of vector automata and homing vector automata. We show that homing vector automata outperform extended finite automata when both are defined over [Formula: see text] integer matrices. We study the string separation problem for vector automata and demonstrate that generalized finite automata with rational entries can separate any pair of strings using only two states. Investigating stateless homing vector automata, we prove that a language is recognized by stateless blind deterministic real-time version of finite automata with multiplication iff it is commutative and its Parikh image is the set of nonnegative integer solutions to a system of linear homogeneous Diophantine equations.

scholarly journals Nonnegative integer solutions of the equationFn

2019 ◽  
Vol 43 (3) ◽  
pp. 1115-1123 ◽  
Author(s):  
Fatih ERDUVAN ◽  
Refik KESKİN
Keyword(s):  
Nonnegative Integer ◽  
Integer Solutions

scholarly journals Vanishing Theorems on Compact Hyper-kähler Manifolds

Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Qilin Yang
Keyword(s):  
Line Bundle ◽  
Nonnegative Integer ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Holomorphic Line Bundle ◽  
Vanishing Theorems ◽  
Vanishing Theorem ◽  
Positive Holomorphic Line Bundle ◽  
Special Case

We prove that if B is a k-positive holomorphic line bundle on a compact hyper-kähler manifold M, then HpM,Ωq⊗B=0 for P>n+[k/2] with q a nonnegative integer. In a special case, k=0 and q=0, we recover a vanishing theorem of Verbitsky’s with a little stronger assumption.

Integer Farkas Lemma

2015 ◽  
Vol 17 (01) ◽  
pp. 1540003 ◽  
Author(s):  
R. Chandrasekaran
Keyword(s):  
Nonnegative Integer ◽  
Original System ◽  
Single Equation ◽  
Integer Solution ◽  
Polynomial Algorithms ◽  
Inequality System ◽  
Farkas Lemma ◽  
Single Constraint ◽  
Nonnegative Solutions ◽  
Integer Solutions

Farkas type results are available for solutions to linear systems. These can also include restrictions such as nonnegative solutions or integer solutions. They show that the unsolvability can be reduced to a single constraint that is not solvable and this condition is implied by the original system. Such a result does not exist for integer solution to inequality system because a single inequality is always solvable in integers. But a single equation that does not have nonnegative integer solution exists. We present some cases when polynomial algorithms to find nonnegative integer solutions exist.

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