scholarly journals Euclidean Distance Degree and Mixed Volume

2021 ◽  
Author(s):  
P. Breiding ◽  
F. Sottile ◽  
J. Woodcock
Keyword(s):  
Computational Complexity ◽  
Lagrange Multiplier ◽  
Euclidean Distance ◽  
Mixed Volume ◽  
Rectangular Parallelepiped ◽  
Newton Polytope ◽  
Newton Polytopes ◽  
Sparse Polynomials

AbstractWe initiate a study of the Euclidean distance degree in the context of sparse polynomials. Specifically, we consider a hypersurface $$f=0$$ f = 0 defined by a polynomial f that is general given its support, such that the support contains the origin. We show that the Euclidean distance degree of $$f=0$$ f = 0 equals the mixed volume of the Newton polytopes of the associated Lagrange multiplier equations. We discuss the implication of our result for computational complexity and give a formula for the Euclidean distance degree when the Newton polytope is a rectangular parallelepiped.

scholarly journals Wi-Fi Backscatter System with Tag Sensors Using Multi-Antennas for Increased Data Rate and Reliability

Sensors ◽  
2020 ◽  
Vol 20 (5) ◽  
pp. 1314
Author(s):  
Taeoh Kim ◽  
Hyobeen Park ◽  
Yunho Jung ◽  
Seongjoo Lee
Keyword(s):  
Computational Complexity ◽  
Euclidean Distance ◽  
Multiple Antennas ◽  
Power Level ◽  
Low Complexity ◽  
Data Rate ◽  
Threshold Method ◽  
Channel State ◽  
Distance Method ◽  
Ber Performance

In this paper, we propose tag sensor using multi-antennas in a Wi-Fi backscatter system, which results in an improved data rate or reliability of the signal transmitted from a tag sensor to a reader. The existing power level modulation method, which is proposed to improve data rate in a Wi-Fi backscatter system, has low reliability due to the reduced distance between symbols. To address this problem, we propose a Wi-Fi backscatter system that obtains channel diversity by applying multiple antennas. Two backscatter methods are described for improving the data rate or reliability in the proposed system. In addition, we propose three low complexity demodulation methods to address the high computational complexity problem caused by multiple antennas: (1) SET (subcarrier energy-based threshold) method, (2) TCST (tag’s channel state-based threshold) method, and (3) SED (similar Euclidean distance) method. In order to verify the performance of the proposed backscatter method and low complexity demodulation schemes, the 802.11 TGn (task group n) channel model was utilized in simulation. In this paper, the proposed tag sensor structure was compared with existing methods using only sub-channels with a large difference in received CSI (channel state information) values or adopting power-level modulation. The proposed scheme showed about 10 dB better bit error rate (BER) performance and throughput. Also, proposed low complexity demodulation schemes were similar in BER performance with a difference of up to 1 dB and the computational complexity was reduced by up to 60% compared to the existing Euclidean distance method.

scholarly journals Newton Polytopes of Cluster Variables of Type $A_n$

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Adam Kalman
Keyword(s):  
Type A ◽  
Laurent Expansion ◽  
Cluster Algebras ◽  
Newton Polytope ◽  
Newton Polytopes ◽  
Affine Hull ◽  
Cluster Variable ◽  
Nous Montrons ◽  
The Face ◽  
International Audience

International audience We study Newton polytopes of cluster variables in type $A_n$ cluster algebras, whose cluster and coefficient variables are indexed by the diagonals and boundary segments of a polygon. Our main results include an explicit description of the affine hull and facets of the Newton polytope of the Laurent expansion of any cluster variable, with respect to any cluster. In particular, we show that every Laurent monomial in a Laurent expansion of a type $A$ cluster variable corresponds to a vertex of the Newton polytope. We also describe the face lattice of each Newton polytope via an isomorphism with the lattice of elementary subgraphs of the associated snake graph. Nous étudions polytopes de Newton des variables amassées dans les algèbres amassées de type A, dont les variables sont indexés par les diagonales et les côtés d’un polygone. Nos principaux résultats comprennent une description explicite de l’enveloppe affine et facettes du polytope de Newton du développement de Laurent de toutes variables amassées. En particulier, nous montrons que tout monôme Laurent dans un développement de Laurent de variable amassée de type A correspond à un sommet du polytope de Newton. Nous décrivons aussi le treillis des facesde chaque polytope de Newton via un isomorphisme avec le treillis des sous-graphes élémentaires du “snake graph” qui est associé.

scholarly journals Criteria for strict monotonicity of the mixed volume of convex polytopes

2019 ◽  
Vol 19 (4) ◽  
pp. 527-540 ◽  
Author(s):  
Frédéric Bihan ◽  
Ivan Soprunov
Keyword(s):  
Convex Hull ◽  
Convex Polytopes ◽  
Mixed Volume ◽  
Polynomial Systems ◽  
Strict Monotonicity ◽  
Newton Polytopes ◽  
Sparse Polynomial ◽  
Geometric Result ◽  
Cramer’S Rule ◽  
Sparse Polynomial Systems

Abstract Let P1, …, Pn and Q1, …, Qn be convex polytopes in ℝn with Pi ⊆ Qi. It is well-known that the mixed volume is monotone: V(P1, …, Pn) ≤ V(Q1, …, Qn). We give two criteria for when this inequality is strict in terms of essential collections of faces as well as mixed polyhedral subdivisions. This geometric result allows us to characterize sparse polynomial systems with Newton polytopes P1, …, Pn whose number of isolated solutions equals the normalized volume of the convex hull of P1 ∪ … ∪ Pn. In addition, we obtain an analog of Cramer’s rule for sparse polynomial systems.

scholarly journals AN ORACLE-BASED, OUTPUT-SENSITIVE ALGORITHM FOR PROJECTIONS OF RESULTANT POLYTOPES

2013 ◽  
Vol 23 (04n05) ◽  
pp. 397-423 ◽  
Author(s):  
IOANNIS Z. EMIRIS ◽  
VISSARION FISIKOPOULOS ◽  
CHRISTOS KONAXIS ◽  
LUIS PEÑARANDA
Keyword(s):  
Orthogonal Projection ◽  
Geometric Modeling ◽  
Tropical Geometry ◽  
Newton Polytope ◽  
Newton Polytopes ◽  
Outer Approximations

We design an algorithm to compute the Newton polytope of the resultant, known as resultant polytope, or its orthogonal projection along a given direction. The resultant is fundamental in algebraic elimination, optimization, and geometric modeling. Our algorithm exactly computes vertex- and halfspace-representations of the polytope using an oracle producing resultant vertices in a given direction, thus avoiding walking on the polytope whose dimension is α - n - 1, where the input consists of α points in ℤn. Our approach is output-sensitive as it makes one oracle call per vertex and per facet. It extends to any polytope whose oracle-based definition is advantageous, such as the secondary and discriminant polytopes. Our publicly available implementation uses the experimental CGAL package triangulation. Our method computes 5-, 6- and 7- dimensional polytopes with 35K, 23K and 500 vertices, respectively, within 2hrs, and the Newton polytopes of many important surface equations encountered in geometric modeling in < 1sec, whereas the corresponding secondary polytopes are intractable. It is faster than tropical geometry software up to dimension 5 or 6. Hashing determinantal predicates accelerates execution up to 100 times. One variant computes inner and outer approximations with, respectively, 90% and 105% of the true volume, up to 25 times faster.

scholarly journals Blind Signal Processing Algorithmsbased based on Recursive Gradient Estimation

2015 ◽  
Vol 5 (3) ◽  
pp. 548
Author(s):  
Namyong Kim ◽  
Mingoo Kang
Keyword(s):  
Computational Complexity ◽  
Euclidean Distance ◽  
Estimation Method ◽  
Processing Method ◽  
Gradient Estimation ◽  
Blind Signal Processing ◽  
Dirac Delta ◽  
Blind Signal ◽  
Gradient Calculation ◽  
Delta Functions

Blind algorithms based on the Euclidean distance (ED) between the output distribution function and a set of Dirac delta functions have a heavy computational burden of  due to some double summation operations for the sample size and symbol points. In this paper, a recursive approach to the estimation of the ED and its gradient is proposed to reduce the computational complexity for efficient implementation of the algorithm. The ED of the algorithm is comprised of information potentials (IPs), and the IPs at the next iteration can be calculated recursively based on the currently obtained IPs. Utilizing the recursively estimated IPs, the next step gradient for the weight update of the algorithm can be estimated recursively with the present gradient. With this recursive approach, the computational complexity of gradient calculation has only . The simulation results show that the proposed gradient estimation method holds significantly reduced computational complexity keeping the same performance as the block processing method

Bipolar Abstract Argumentation with Dual Attacks and Supports

2020 ◽  
Author(s):  
Nico Potyka
Keyword(s):  
Computational Complexity ◽  
Decision Problems ◽  
Abstract Argumentation ◽  
Stable Semantics ◽  
Support Relations ◽  
Support Relationships ◽  
Computational Properties ◽  
Inherent Tendency

Bipolar abstract argumentation frameworks allow modeling decision problems by defining pro and contra arguments and their relationships. In some popular bipolar frameworks, there is an inherent tendency to favor either attack or support relationships. However, for some applications, it seems sensible to treat attack and support equally. Roughly speaking, turning an attack edge into a support edge, should just invert its meaning. We look at a recently introduced bipolar argumentation semantics and two novel alternatives and discuss their semantical and computational properties. Interestingly, the two novel semantics correspond to stable semantics if no support relations are present and maintain the computational complexity of stable semantics in general bipolar frameworks.

The Earth Mover’s Distance as a Metric for the Space of Inorganic Compositions

2020 ◽  
Author(s):  
Cameron Hargreaves ◽  
Matthew Dyer ◽  
Michael Gaultois ◽  
Vitaliy Kurlin ◽  
Matthew J Rosseinsky
Keyword(s):  
Euclidean Distance ◽  
Nearest Neighbor ◽  
Nearest Neighbor Search ◽  
Inorganic Crystal Structure Database ◽  
Earth Mover’S Distance ◽  
Chemical Similarity ◽  
Earth Mover's Distance ◽  
Neighbor Search ◽  
The Earth ◽  
Binary Compounds

It is a core problem in any field to reliably tell how close two objects are to being the same, and once this relation has been established we can use this information to precisely quantify potential relationships, both analytically and with machine learning (ML). For inorganic solids, the chemical composition is a fundamental descriptor, which can be represented by assigning the ratio of each element in the material to a vector. These vectors are a convenient mathematical data structure for measuring similarity, but unfortunately, the standard metric (the Euclidean distance) gives little to no variance in the resultant distances between chemically dissimilar compositions. We present the Earth Mover’s Distance (EMD) for inorganic compositions, a well-defined metric which enables the measure of chemical similarity in an explainable fashion. We compute the EMD between two compositions from the ratio of each of the elements and the absolute distance between the elements on the modified Pettifor scale. This simple metric shows clear strength at distinguishing compounds and is efficient to compute in practice. The resultant distances have greater alignment with chemical understanding than the Euclidean distance, which is demonstrated on the binary compositions of the Inorganic Crystal Structure Database (ICSD). The EMD is a reliable numeric measure of chemical similarity that can be incorporated into automated workflows for a range of ML techniques. We have found that with no supervision the use of this metric gives a distinct partitioning of binary compounds into clear trends and families of chemical property, with future applications for nearest neighbor search queries in chemical database retrieval systems and supervised ML techniques.

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