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Stochastic Discrete Clenshaw-Curtis Quadrature

Nico Piatkowski, Katharina Morik
2016 International Conference on Machine Learning  
We propose an approximation scheme that, for any discrete graphical model whose parameter vector has bounded norm, estimates the partition function with arbitrarily small error.  ...  Our algorithm relies on a near minimax optimal polynomial approximation to the potential function and a Clenshaw-Curtis style quadrature.  ...  Acknowledgments This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) within the collaborative research center SFB 876, project A1.  ... 
dblp:conf/icml/PiatkowskiM16 fatcat:ilfth6gswjdjxawr74hvn4q6b4

Page 8845 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
This basis has several useful properties— local minimal support, low degree of polynomials.  ...  We also present several problems that arise in lower-degree polynomials.” 2000m:65021 65D15 41A30 Kushpel, A. K.  ... 

Effective approximation of the solutions of algebraic equations [article]

Marcin Bilski, Peter Scheiblechner
2020 arXiv   pre-print
Let F be a holomorphic map whose components satisfy some polynomial relations. We present an algorithm for constructing Nash maps locally approximating F, whose components satisfy the same relations.  ...  For i = m + 1, . . . , m + s, the degree of P ν i in z i is bounded by the cardinality of the generic fiber in V over C m .  ...  Bound for the degree of output polynomials We prove the claim: there is a bound for the degree of P ν i in z i , i = 1, . . . ,m, independent of ν.  ... 
arXiv:1603.07298v2 fatcat:t5dzhucl7vb6bnpemnztrnyiqu

Reed-Muller Codes: Theory and Algorithms [article]

Emmanuel Abbe and Amir Shpilka and Min Ye
2020 arXiv   pre-print
weight codewords using lower degree polynomials.  ...  In particular, the paper discusses the recent connections established between RM codes, thresholds of Boolean functions, polarization theory, hypercontractivity, and the techniques of approximating low  ...  We first show that each polynomial of weight at most β can be approximated by an explicit function of some lower order derivatives.  ... 
arXiv:2002.03317v2 fatcat:oaeaq4yyhvdcth3g4djndgfypq

Algebraic complexities and algebraic curves over finite fields

D. V. Chudnovsky, G. V. Chudnovsky
1987 Proceedings of the National Academy of Sciences of the United States of America  
We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with linear coding theory and algebraic curves over finite fields  ...  For infinite fields minimal complexities are known [Winograd, S. (1977) Math. Syst. Theory 10, 169-180].  ...  on ,lF(n, m) of multiplicative complexity over Fq of multiplication of a polynomial of degree n -1 by a polynomial of degree m -1 for q = 2: pFq(n, m) 2 max{N(n, m), N(m, n)}  ... 
doi:10.1073/pnas.84.7.1739 pmid:16593816 pmcid:PMC304516 fatcat:bjn6jpkwhze45keyvqsjtdkf4e

Finding orthogonal vectors in discrete structures [chapter]

Ryan Williams, Huacheng Yu
2013 Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms  
Letting HOPCROFT R denote Hopcroft's problem over a ring R, we give randomized algorithms and almost matching lower bounds (modulo a breakthrough in SAT algorithms) for HOPCROFT R , when R is the ring  ...  The algorithms arise from studying the communication problem of determining whether two lists of vectors (one list held by Alice, one by Bob) contain an orthogonal pair of vectors over a discrete structure  ...  Just as studying Z m for non-prime-power m was the key to developing subexponential locally decodable codes, studying Hopcroft's problem over Z m may lead to faster SAT algorithms.  ... 
doi:10.1137/1.9781611973402.135 dblp:conf/soda/WilliamsY14 fatcat:qucl3dve3vhjxlralylbh7xupy

A Novel Root-Finding Algorithm with Engineering Applications and Its Dynamics Via Computer Technology

Amir Naseem, M. A. Rehman, Thabet Abdeljawad
2022 IEEE Access  
We consider various different-degrees complex polynomials for the graphical analysis and used a computer tool to create the polynomiographs of the proposed quartic-order algorithm and compare it to other  ...  We develop this algorithm by utilizing the forward-and finite-difference schemes on well-known Househ"older's method, resulting in an efficient and derivative-free algorithm with a low per iteration computing  ...  for the quadratic-degree polynomial z 2 − 1 having two distinct roots: 1 and −1.  ... 
doi:10.1109/access.2022.3150775 fatcat:t7ovazqedzf6bdfnzms2xopjfq

Lower Bounds for Constant Query Affine-Invariant LCCs and LTCs

Arnab Bhattacharyya, Sivakanth Gopi
2017 ACM Transactions on Computation Theory  
This then allows us to bound the number of codewords, using known decomposition theorems which approximate any bounded function in terms of a finite number of low-degree non-classical polynomials, upto  ...  In this work, we give lower bounds on the length of locally correctable and locally testable affine-invariant codes with constant query complexity.  ...  We thank Madhu Sudan for helpful pointers to previous work. The second author would like to thank his advisor, Zeev Dvir, for his guidance and encouragement.  ... 
doi:10.1145/3016802 fatcat:s5tvj25yqzhsnpv2icij2rhrlu

Orthogonal Fourier–Mellin moments for invariant pattern recognition

Yunlong Sheng, Lixin Shen
1994 Optical Society of America. Journal A: Optics, Image Science, and Vision  
For small images, the description by the orthogonal Fourier-Mellin moments is better than that by the Zernike moments in terms of image-reconstruction errors and signal-to-noise ratio.  ...  The new orthogonal radial polynomials have more zeros than do the Zernike radial polynomials in the region of small radial distance.  ...  and If(r, 0)12 are integrable and the polynomials Qn(r) are bounded over (0,1) as n becomes infinite. 8 These conditions are, in general, satisfied.  ... 
doi:10.1364/josaa.11.001748 fatcat:jo3oickil5fhrfqppjnbcbtjdq

Improved Bounds on the Sign-Rank of AC^0

Mark Bun, Justin Thaler, Marc Herbstritt
2016 International Colloquium on Automata, Languages and Programming  
Specifically, they exhibited a matrix A = [F (x, y)] x,y for a specific function We prove a generalization of Razborov and Sherstov's result, yielding exponential sign-rank lower bounds for a non-trivial  ...  class of functions (that includes the function used by Razborov and Sherstov).  ...  This can be viewed as a dual formulation of a bound on the growth of low-degree polynomials.  ... 
doi:10.4230/lipics.icalp.2016.37 dblp:conf/icalp/BunT16 fatcat:jhf5tpfjmbfkllrbxqh5nkgcmi

Gravitational recoil from binary black hole mergers: The close-limit approximation

Carlos F. Sopuerta, Nicolás Yunes, Pablo Laguna
2006 Physical Review D  
This is a lower bound because it neglects the initial merger phase. We can however obtain a rough estimate by using PN methods or the close-limit approximation.  ...  We also provide non-linear fits to these estimated upper and lower bounds.  ...  Acknowledgments The authors acknowledge the support of the Center for Gravitational Wave Physics funded by the National Science Foundation under Cooperative Agreement PHY-0114375.  ... 
doi:10.1103/physrevd.74.124010 fatcat:fileqqackza7lbcaegtkhuo7y4

Homomorphic fingerprints under misalignments

Alexandr Andoni, Assaf Goldberger, Andrew McGregor, Ely Porat
2013 Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13  
This lower bound addresses a long-standing open problem on the low distor- *  ...  Fingerprinting is a widely-used technique for efficiently verifying that two files are identical.  ...  The authors would like to thank Robi Krauthgamer, Hossein Jowhari, and Atri Rudra for early conversations about the problem, as well as the anonymous referees for their valueable comments.  ... 
doi:10.1145/2488608.2488726 dblp:conf/stoc/AndoniGMP13 fatcat:my6bdailrre4hfzhj2ksvoukdm


2000 Applied Physics Letters  
We define a measure of complexity for data dependent hypothesis classes and provide data dependent versions of bounds on error deviance and estimation error.  ...  We also provide a structural risk minimization procedure over data dependent hierarchies and prove consistency.  ...  Acknowledgments We would like to thank Leonid Gurvits for a very helpful conversation. Clint Scovel gratefully acknowledges support from the Los Alamos DOE Program in Applied Mathematics.  ... 
doi:10.1162/153244302760200650 fatcat:yfeyp2m3pjb2dbuyu5m4dnm3zy

On the Complexity of Familiar Functions and Numbers

J. M. Borwein, P. B. Borwein
1988 SIAM Review  
This paper examines low-complexity approximations to familiar functions and numbers.  ...  For most functions, provably optimal methods are not known; however the gap between what is known and what is possible is often small.  ...  The subscript on the order symbol is for emphasis. We will sometimes use 2 and rat as the lower bound order symbols.  ... 
doi:10.1137/1030134 fatcat:dyuvva7n6ffhrf7w6fr5pipvbi

Bayesian Kernel Shaping for Learning Control

Jo-Anne Ting, Mrinal Kalakrishnan, Sethu Vijayakumar, Stefan Schaal
2008 Neural Information Processing Systems  
It can be used for nonparametric regression with local polynomials or as a novel method to achieve nonstationary regression with Gaussian processes.  ...  We introduce a Bayesian formulation of nonparametric regression that, with the help of variational approximations, results in an EM-like algorithm for simultaneous estimation of regression and kernel parameters  ...  In this paper, we consider local kernel shaping by averaging over data samples with the help of locally polynomial models and formulate this approach, in a Bayesian framework, for both function approximation  ... 
dblp:conf/nips/TingKVS08 fatcat:qxnrz6fv7ncqdkvsmzr7bcrtt4
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