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3D shape matching and Teichmüller spaces of pointed Riemann surfaces [article]

Claudio Fontanari, Letizia Pernigotti
2011 arXiv   pre-print
geometry.  ...  Shape matching represents a challenging problem in both information engineering and computer science, exhibiting not only a wide spectrum of multimedia applications, but also a deep relation with conformal  ...  hyperbolic pants decomposition. (4) Compute the Fenchel-Nielsen coordinates.  ... 
arXiv:1101.5116v1 fatcat:3simqs5ys5f7bcjdkrdc4k3ygy

Preface

Xiao-Shan Gao, Deepak Kapur
2012 Journal of symbolic computation  
There were 70 oral and poster presentations on recent advances in symbolic computation, automated reasoning, computational topology, and the history of mathematics, among which the following twenty four  ...  Triangular Decomposition of Polynomial Systems: Algorithmic Advances and Remaining Challenges. Dongming Wang. On the Mechanization of Geometry: From Theorem Proving to Knowledge Management.  ...  Theorem Proving in Geometry and Tools for Polynomial System Solving. • Bang-He Li. A Method for Solving Algebraic Equations up to Multiplicities via the Ritt-Wu Characteristic Sets. • Hongbo Li.  ... 
doi:10.1016/j.jsc.2011.12.020 fatcat:b32i5tp2bfdpld5v2arslnac64

Numerical Local Irreducible Decomposition [chapter]

Daniel A. Brake, Jonathan D. Hauenstein, Andrew J. Sommese
2016 Lecture Notes in Computer Science  
Using numerical algebraic geometry, each irreducible component is represented using a witness set thereby yielding a numerical irreducible decomposition of the solution set.  ...  We define local witness sets and describe a numerical algebraic geometric approach for computing a numerical local irreducible decomposition for polynomial systems. Several examples are presented.  ...  The local parameterization theorem is a local version of the Noether Normalization Theorem.  ... 
doi:10.1007/978-3-319-32859-1_9 fatcat:cqjc4cqa7jgfrogaq6se7clknq

Geometric Compression Using Riemann Surface Structure

Xianfeng Gu, Yalin Wang, Shing-Tung Yau
2003 Communications in Information and Systems  
First we compute a global conformal parameterization of the surface.  ...  We present the proofs of the theorems and the compression examples.  ...  The number of zero points is computed using Riemann-Roch theorem. The decomposition structure can be proved using circle-valued Morse theorem. Previous Work.  ... 
doi:10.4310/cis.2003.v3.n3.a2 fatcat:fkloh43245ervp7gsnv34oed7y

Parameterized Complexity of PCA (Invited Talk)

Fedor V. Fomin, Petr A. Golovach, Kirill Simonov, Susanne Albers
2020 Scandinavian Workshop on Algorithm Theory  
We discuss some recent progress in the study of Principal Component Analysis (PCA) from the perspective of Parameterized Complexity.  ...  By the Eckart-Young theorem [10] , PCA is efficiently solvable via Singular Value Decomposition (SVD).  ...  The proof of the theorem requires ideas from kernelization, linear algebra and algebraic geometry. Thus Robust PCA is fixed-parameter tractable when parameterized by k + d.  ... 
doi:10.4230/lipics.swat.2020.1 dblp:conf/swat/FominGS20 fatcat:xpthkgfjwnhwvchic4phtdzvdm

Rotation Invariant Householder Parameterization for Bayesian PCA [article]

Rajbir S. Nirwan, Nils Bertschinger
2019 arXiv   pre-print
In particular, we avoid the need to compute the Jacobian determinant of the parameter transformation.  ...  Here, we propose a parameterization based on Householder transformations, which remove the rotational symmetry of the posterior.  ...  be employed at different points. removed via singular value decomposition (SVD).  ... 
arXiv:1905.04720v1 fatcat:b5qmwoh6anhhbirmwmglqpvtaq

Decomposing Solution Sets of Polynomial Systems Using Derivatives [chapter]

Daniel A. Brake, Jonathan D. Hauenstein, Alan C. Liddell
2016 Lecture Notes in Computer Science  
A core computation in numerical algebraic geometry is the decomposition of the solution set of a system of polynomial equations into irreducible components, called the numerical irreducible decomposition  ...  We present a new approach which replaces path tracking with local computations involving derivatives, called a "local trace test."  ...  In numerical algebraic geometry, irreducible components are represented via a witness set with a numerical irreducible decomposition consisting of a witness set for each irreducible component.  ... 
doi:10.1007/978-3-319-42432-3_16 fatcat:eugxstgp4bf6nklwk4ia53oeta

Parameterized Fine-Grained Reductions [article]

Elli Anastasiadi, Antonis Antonopoulos, Aris Pagourtzis, Stavros Petsalakis
2019 arXiv   pre-print
During recent years the field of fine-grained complexity has bloomed to produce a plethora of results, with both applied and theoretical impact on the computer science community.  ...  We provide a parameterized viewpoint of these reductions (PFGR) in order to further analyze the structure of improvable problems and set the foundations of a unified methodology for extending algorithmic  ...  Geometry).  ... 
arXiv:1902.05529v1 fatcat:b2migtkxfvgajevtpkja7az75q

Computational information geometry: theory and practice [article]

Karim Anaya-Izquierdo and Frank Critchley and Paul Marriott and Paul W. Vos
2012 arXiv   pre-print
In so doing, it extends information geometry from a manifold based approach to one where the simplex is the fundamental geometrical object, thereby allowing applications to models which do not have a fixed  ...  Finally, it starts to build a computational framework which will act as a proxy for the 'space of all distributions' that can be used, in particular, to investigate model selection and model uncertainty  ...  The decomposition presented in part (b) of the theorem shows that these subsets are, in fact, contained in -1-affine subspaces. Theorem 2.1.  ... 
arXiv:1209.1988v1 fatcat:y47sva2hyvdzbmlgjt7nzaolqu

On the treewidth of triangulated 3-manifolds

Kristóf Huszár, Jonathan Spreer, Uli Wagner
2019 Journal of Computational Geometry  
such as pathwidth or treewidth for graphs, or the concept of thin position for 3-manifolds, play an important role when studying algorithmic problems; in particular, there is a variety of problems in computational  ...  topology of a 3-manifold M on the one hand and width-type parameters of the dual graphs of triangulations of M on the other hand, answering a question that had been raised repeatedly by researchers in computational  ...  We are grateful to Arnaud de Mesmay and Saul Schleimer for noticing a gap in an earlier proof of Theorem 4 and for pointing us towards key ingredients of the current proof.  ... 
doi:10.20382/jogc.v10i2a5 dblp:journals/jocg/HuszarSW19 fatcat:fxkmg2nkkbewvepedijgdr3rwi

Computational Information Geometry in Statistics: Foundations [chapter]

Karim Anaya-Izquierdo, Frank Critchley, Paul Marriott, Paul Vos
2013 Lecture Notes in Computer Science  
Computational information geometry explicitly uses the information geometry of ∆ k to numerically compute statistically important features of M.  ...  For more details of the geometry of the observed face see the paper [3] . Theorem 3.  ... 
doi:10.1007/978-3-642-40020-9_33 fatcat:f5ul2qekzbfgddf74m46b6bnuu

Multivariate Analysis of Orthogonal Range Searching and Graph Distances

Karl Bringmann, Thore Husfeldt, Måns Magnusson, Michael Wagner
2019 International Symposium on Parameterized and Exact Computation  
ACM Subject Classification Theory of computation → Shortest paths, Theory of computationParameterized complexity and exact algorithms, Theory of computationComputational geometry, Mathematics of  ...  We also investigate the same problems parameterized by vertex cover number: Theorem 4.  ...  the dependency on k as well as the time required for computing a tree decomposition.  ... 
doi:10.4230/lipics.ipec.2018.4 dblp:conf/iwpec/BringmannHM18 fatcat:xx3ermqaqfho7d7k2ndpmcvydm

Book Review: Toric varieties

Frank Sottile
2013 Bulletin of the American Mathematical Society  
This second notion of a toric variety via monomial parameterization is distinct from the first notion of toric varieties associated to fans in Z n .  ...  For example, consider the following decomposition of the 3 × 3 square in Z 2 .  ... 
doi:10.1090/s0273-0979-2013-01441-1 fatcat:duzkjpdybzgclisic6xttl5rvm

Floor decompositions of tropical curves : the planar case [article]

Erwan Brugalle, Grigory Mikhalkin
2019 arXiv   pre-print
We give here detailed proofs in the tropical geometry framework, in the case when the ambient variety is a complex surface, and give some examples of computations using floor diagrams.  ...  In a previous paper, we announced a formula to compute Gromov-Witten and Welschinger invariants of some toric varieties, in terms of combinatorial objects called floor diagrams.  ...  This paper is devoted to explain how floor diagrams can be used to compute the numbers N (∆, g) and W (∆, r) (Theorems 3.6 and 3.9), and to give some examples of concrete computations (section 6).  ... 
arXiv:0812.3354v3 fatcat:akbk5hnrafejdlunmkndtiuslq

Parametric shortest-path algorithms via tropical geometry [article]

Michael Joswig, Benjamin Schröter
2021 arXiv   pre-print
We study parameterized versions of classical algorithms for computing shortest-path trees. This is most easily expressed in terms of tropical geometry.  ...  With the same conditions as in Theorem 4, and if k is considered a fixed constant, all parameterized shortest paths can be computed in O(n 3 ) time, if they exist. Remark 6.  ...  Floyd and Warshall's algorithm reduces the complexity of computing D * to O(n 3 ) via dynamic programming.  ... 
arXiv:1904.01082v4 fatcat:mwt4xy4dkfd6lcmao7mf4hmun4
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