On the geometric interpretation of the nonnegative rank

The nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank-one factors needed to reconstruct it exactly. The problem of determining this rank and computing the corresponding nonnegative factors is difficult; however it has many potential applications, e.g., in data mining, graph theory and computational geometry. In particular, it can be used to characterize the minimal size of any extended reformulation of a given combinatorial optimization program. In this paper, we introduce and study a related quantity, called the restricted nonnegative rank. We show that computing this quantity is equivalent to a problem in polyhedral combinatorics, and fully characterize its computational complexity. This in turn sheds new light on the nonnegative rank problem, and in particular allows us to provide new improved lower bounds based on its geometric interpretation. We apply these results to slack matrices and linear Euclidean distance matrices and obtain counter-examples to two conjectures of Beasly and Laffey, namely we show that the nonnegative rank of linear Euclidean distance matrices is not necessarily equal to their dimension, and that the rank of a matrix is not always greater than the nonnegative rank of its square.nonnegative rank, restricted nonnegative rank, nested polytopes, computational complexity, computational geometry, extended formulations, linear Euclidean distance matrices.

Regression on fixed-rank positive semidefinite matrices: a Riemannian approach

The paper addresses the problem of learning a regression model parameterized by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear nature of the search space and on scalability to high-dimensional problems. The mathematical developments rely on the theory of gradient descent algorithms adapted to the Riemannian geometry that underlies the set of fixed-rank positive semidefinite matrices. In contrast with previous contributions in the literature, no restrictions are imposed on the range space of the learned matrix. The resulting algorithms maintain a linear complexity in the problem size and enjoy important invariance properties. We apply the proposed algorithms to the problem of learning a distance function parameterized by a positive semidefinite matrix. Good performance is observed on classical benchmarks

On the Geometric Interpretation of the Nonnegative Rank

The nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank-one factors needed to reconstruct it exactly. The problem of determining this rank and computing the corresponding nonnegative factors is difficult; however it has many potential applications, e.g., in data mining, graph theory and computational geometry. In particular, it can be used to characterize the minimal size of any extended reformulation of a given combinatorial optimization program. In this paper, we introduce and study a related quantity, called the restricted nonnegative rank. We show that computing this quantity is equivalent to a problem in polyhedral combinatorics, and fully characterize its computational complexity. This in turn sheds new light on the nonnegative rank problem, and in particular allows us to provide new improved lower bounds based on its geometric interpretation. We apply these results to slack matrices and linear Euclidean distance matrices and obtain counter-examples to two conjectures of Beasly and Laffey, namely we show that the nonnegative rank of linear Euclidean distance matrices is not necessarily equal to their dimension, and that the rank of a matrix is not always greater than the nonnegative rank of its square

Monotonicity of Fitness Landscapes and Mutation Rate Control

A common view in evolutionary biology is that mutation rates are minimised. However, studies in combinatorial optimisation and search have shown a clear advantage of using variable mutation rates as a control parameter to optimise the performance of evolutionary algorithms. Much biological theory in this area is based on Ronald Fisher's work, who used Euclidean geometry to study the relation between mutation size and expected fitness of the offspring in infinite phenotypic spaces. Here we reconsider this theory based on the alternative geometry of discrete and finite spaces of DNA sequences. First, we consider the geometric case of fitness being isomorphic to distance from an optimum, and show how problems of optimal mutation rate control can be solved exactly or approximately depending on additional constraints of the problem. Then we consider the general case of fitness communicating only partial information about the distance. We define weak monotonicity of fitness landscapes and prove that this property holds in all landscapes that are continuous and open at the optimum. This theoretical result motivates our hypothesis that optimal mutation rate functions in such landscapes will increase when fitness decreases in some neighbourhood of an optimum, resembling the control functions derived in the geometric case. We test this hypothesis experimentally by analysing approximately optimal mutation rate control functions in 115 complete landscapes of binding scores between DNA sequences and transcription factors. Our findings support the hypothesis and find that the increase of mutation rate is more rapid in landscapes that are less monotonic (more rugged). We discuss the relevance of these findings to living organisms

Modeling and optimal design of shorting vias to suppress radiated emission in high-speed alternating PCB planes

An analytical mode analysis of vias in the multilayered printed-circuit-board periphery is developed to suppress the electromagnetic radiation induced by ground bounce. After separating the even and odd modes in alternating planes, the far-field radiation of parallel plates is derived using Huygens' principle. It is mainly contributed by the odd mode excitation, while the even mode sets a lower bound on the radiation level from the system when shorting vias are inserted between alternating ground plates. For the odd-mode radiation, a canonical problem is then constructed and analytically solved by applying image theory. Based on that, a systematic approach to achieve the optimum suppression design is developed for the various geometry parameters of the shorting vias, including the pitch, radius, and distance to the board edge. Full-wave simulation and measurement are also presented and the good agreement with the theoretical prediction validates the correctness and efficiency of the present analysis and design

An algorithm to enumerate all possible protein conformations verifying a set of distance constraints

International audienceBackground: The determination of protein structures satisfying distance constraints is an important problem in structural biology. Whereas the most common method currently employed is simulated annealing, there have been other methods previously proposed in the literature. Most of them, however, are designed to find one solution only. Results: In order to explore exhaustively the feasible conformational space, we propose here an interval Branch-and-Prune algorithm (iBP) to solve the Distance Geometry Problem (DGP) associated to protein structure determination. This algorithm is based on a discretization of the problem obtained by recursively constructing a search space having the structure of a tree, and by verifying whether the generated atomic positions are feasible or not by making use of pruning devices. The pruning devices used here are directly related to features of protein conformations. Conclusions: We described the new algorithm iBP to generate protein conformations satisfying distance constraints, that would potentially allows a systematic exploration of the conformational space. The algorithm iBP has been applied on three α-helical peptides

Aspects of electrostatics in BTZ geometries

In the present paper the electrostatics of charges in nonrotating BTZ black hole and wormhole spacetimes is studied. Our attention is focused on the self-force of a point charge in the geometry, for which a regularization prescription based on the Haddamard Green function is employed. The differences between the self-force in both cases is a theoretical experiment for distinguishing both geometries, which otherwise are locally indistinguishable. This idea was applied before to four and higher-dimensional black holes by the present and other authors. However, the particularities of the BTZ geometry makes the analysis considerable more complicated than those. First, the BTZ spacetimes are not asymptotically flat but instead asymptotically AdS. In addition, the relative distance d ( r , r + 1 ) between two particles located at a radius r and r + 1 in the geometry tends to zero when r → ∞ . This behavior, which is radically different in a flat geometry, changes the analysis of the asymptotic conditions for the electrostatic field. The other problem is that there exist several regularization methods other than the one we are employing, and there does not exist a proof in three dimensions that they are equivalent. However, we focus on the Haddamard method and obtain an expression for the hypothetical self-force in series, and the resulting expansion is convergent to the real solution. We suspect that the convergence is not uniform, and furthermore there are no summation formulas at our disposal. It appears, for points that are far away from the black hole the calculation of the Haddamard self-force requires higher-order summation. These subtleties are carefully analyzed in the paper, and it is shown that they lead to severe problems when calculating the Haddamard self-force for asymptotic points in the geometry.Fil: Herrera, Yago. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Hurovich, Valeria Laura. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Santillán, Osvaldo Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas ; ArgentinaFil: Simeone, Claudio Mauricio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentin

Applying Binomial Statistics to Weighted Monte Carlo

Weighted Monte Carlo calculations requiring a uniform sampling of the problem-space can suffer from diminished statistical significance because many, if not most, of the randomly-chosen sampling points contribute only slightly to the desired result. Their contribution is reduced in size due the variable-size of the weighting terms. In contrast, none of the randomly-chosen points which are favored by variable size weighting terms will have their statistical significance enhanced beyond that of just one random point in the Monte Carlo sampling. A Monte Carlo analysis was used in earlier work to verify both Gauss\u27 Law and Newton\u27s Shell Theorem. Both examples suffered from statistical difficulties since each Monte Carlo sampling point has a weight inversely proportional to the square of the distance between source and field points. The present work analyzes the diminished significance in weighted Monte Carlo for the specific example of Newton\u27s Shell Theorem, describing the geometry in terms of closest approach distance of the spherical mass shell to the field point. Binomial Statistics is used to remedy this diminished statistical significance by providing a prescription for increasing the value of the Monte Carlo sample size needed to assure that the chosen precision remains invariant as the mass-shell geometry is changed

Instability of a supersonic shock free elliptic jet

This paper presents a comparison of the measured and the computed spatial stability properties of an aspect ratio 2 supersonic shock free elliptic jet. The shock free nature of the elliptic jet provides an ideal test of validity of modeling the large scale coherent structures in the initial mixing region of noncircular supersonic jets with linear hydrodynamic stability theory. Both aerodynamic and acoustic data were measured. The data are used to compute the mean velocity profiles and to provide a description of the spatial composition of pressure waves in the elliptic jet. A hybrid numerical scheme is applied to solve the Rayleigh problem governing the inviscid linear spatial stability of the jet. The measured mean velocity profiles are used to provide a qualitative model for the cross sectional geometry and the smooth velocity profiles used in the stability analysis. Computational results are presented for several modes of instability at two jet cross sections. The acoustic measurements show that a varicose instability is the jet's perferred mode of motion. The stability analysis predicts that the Strouhal number varies linearly as a function of axial distance in the jet's initial mixing region, which is in good qualitative agreement with previous measurements