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Efficiently Approximating Weighted Sums with Exponentially Many Terms [chapter]

Deepak Chawla, Lin Li, Stephen Scott
2001 Lecture Notes in Computer Science  
This is useful when there are exponentially many such inputs and no apparent means to efficiently compute their weighted sum.  ...  These uses require exponentially many inputs, so we define Markov chains over the inputs to approximate the weighted sums.  ...  However, the exponential number of prunings motivates us to use an MCMC approach to approximate the weighted sum of the experts' predictions.  ... 
doi:10.1007/3-540-44581-1_6 fatcat:azrll3l4yrca3ownzuohcnwlvu

Maintaining time-decaying stream aggregates

Edith Cohen, Martin J. Strauss
2006 Journal of Algorithms  
We develop storage-efficient algorithms, and establish upper and lower bounds.  ...  of maintaining time-decaying aggregates and statistics of a data stream: the relative contribution of each data item to the aggregate is scaled down by a factor that depends on, and is non-increasing with  ...  This result shows that polynomially decaying sums can be tracked nearly as efficiently as Exponentially decaying sums.  ... 
doi:10.1016/j.jalgor.2005.01.006 fatcat:244bqng2hrakjpzzdwdxzowc3a

Maintaining time-decaying stream aggregates

Edith Cohen, Martin Strauss
2003 Proceedings of the twenty-second ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems - PODS '03  
We develop storage-efficient algorithms, and establish upper and lower bounds.  ...  of maintaining time-decaying aggregates and statistics of a data stream: the relative contribution of each data item to the aggregate is scaled down by a factor that depends on, and is non-increasing with  ...  This result shows that polynomially decaying sums can be tracked nearly as efficiently as Exponentially decaying sums.  ... 
doi:10.1145/773153.773175 dblp:conf/pods/CohenS03 fatcat:zwgqbpxbafhehor34jqz5qoory

Efficient estimation of neural weights by polynomial approximation

G. Ritter
1999 IEEE Transactions on Information Theory  
We design here algorithms for efficient uniform approximation by a certain class of neural networks with one hidden layer which we call nearly exponential.  ...  Index Terms-Approximation algorithms, complexity problem for neural networks, nearly exponential activation function, neural network with one hidden layer, order of uniform approximation.  ...  ACKNOWLEDGMENT The author thanks Herr Christoph Pesch for his help with the SNNS and a referee for pointing out [2] and [8] and a weakness of Algorithm 4.1 that led to the formulation of Algorithm  ... 
doi:10.1109/18.771153 fatcat:tvp5brxe2fcepdbennkblomyfe

Numerical Error in Weighting Function-Based Unsteady Friction Models for Pipe Transients

John Vítkovský, Mark Stephens, Anton Bergant, Angus Simpson, Martin Lambert
2006 Journal of Hydraulic Engineering  
Approximation of the weighting function with a sum of exponential terms allows for a considerable increase in computation speed using recursive algorithms.  ...  In addition, the numerical algorithm used for unsteady friction should be highly efficient, as inverse analysis requires the transient model to be run many times.  ...  The recursive convolution method relies on the approximation of the weighting function by a sum of exponential terms.  ... 
doi:10.1061/(asce)0733-9429(2006)132:7(709) fatcat:f7wwtjlgv5d3hduzdida7ycfca

Efficient Mean Shift Clustering Using Exponential Integral Kernels

S. Sutor, R. Röhr, G. Pujolle, R. Reda
2008 Zenodo  
Novel corresponding exponential integral kernels are introduced to allow the application of nonuniform kernels for clustering, which dramatically increases robustness without giving up the efficiency of  ...  For efficiency, all calculations are performed on integral images.  ...  In this paper, exponential integral kernels are introduced which allow mean shift to be calculated on integral images with weighted non-uniform kernels.  ... 
doi:10.5281/zenodo.1073240 fatcat:jtqslzxokbelpnmvwsshrqhcle

A Novel Approach for the Efficient Computation of 1-D and 2-D Summations

E. Pinar Karabulut, Vakur B. Erturk, Lale Alatan, S. Karan, Burak Alisan, M. I. Aksun
2016 IEEE Transactions on Antennas and Propagation  
The method is based on applying a subspace algorithm to the samples of partial sums and approximating them in terms of complex exponentials.  ...  For a convergent summation, the residue of the exponential term with zero complex pole of this approximation corresponds to the result of the summation.  ...  Once the S(m) values are computed, they can be approximated in terms of complex exponentials, and similarly, the residue of the exponential term with zero complex pole corresponds to the result of the  ... 
doi:10.1109/tap.2016.2521860 fatcat:iw7h5pmiwbfdhcbj7j4zsbmxje

New summing algorithm using ensemble computing

C D Helon, V Protopopescu
2002 Journal of Physics A: Mathematical and General  
From a practical point of view, the proposed algorithm may result in an exponential speedup, compared to known quantum and classical summing algorithms.  ...  The query complexity of the algorithm depends only on the scaling of the measurement sensitivity with the number of distinct spin sub-ensembles.  ...  Unfortunately, these results simply state that in the limit of infinite number of terms, the approximating sums coincide with the desired integrals.  ... 
doi:10.1088/0305-4470/35/42/102 fatcat:wtvwv5igwbafjjichzogcorsyq

Forward Decay: A Practical Time Decay Model for Streaming Systems

Graham Cormode, Vladislav Shkapenyuk, Divesh Srivastava, Bojian Xu
2009 Proceedings / International Conference on Data Engineering  
We provide efficient algorithms to compute a variety of aggregates and draw samples under forward decay, and show that these are easy to implement scalably.  ...  We argue that this is because the usual definitions of time decay are "backwards": the decayed weight of a tuple is based on its age, measured backward from the current time.  ...  Here, efficient approximate solutions are known.  ... 
doi:10.1109/icde.2009.65 dblp:conf/icde/CormodeSSX09 fatcat:qc4xkd4nt5hahli7eksp3hgdgm

Truncated Exponential Series based partial Euler Product calculations at quiescent regions of oscillatory divergence to produce approximations of the Riemann Siegel Z function

John Martin
2021 figshare.com  
Truncated exponential series based partial Euler Product calculations about the two quiescent regions at N ≈ ( t/π ) & sqrt(t/2/π) match finite Riemann Zeta Dirichlet series sum approximations of the Riemann  ...  Since the calculation is computationally expensive some approximations of the Euler product calculations are also explored.  ...  To reduce the residual high frequency terms, a 31 term sinc function is applied as a low pass averaging filter to equation (11) with the terms spaced by 1/log(t).  ... 
doi:10.6084/m9.figshare.14842803.v1 fatcat:ibswcvflczaxflhi3tunle7bgy

Optimal approximations of power laws with exponentials: application to volatility models with long memory

Thierry Bochud, Damien Challet
2007 Quantitative finance (Print)  
a power-law with a finite sum of weighted exponentials.  ...  CONCLUSIONS We have provided a simple method to use efficiently a sum of weighted exponentials as a parsimonious approximation of a power-law with any exponent.  ... 
doi:10.1080/14697680701278291 fatcat:4eydzh3evba5vpue4bbnbzxmom

The monomial method: Extensions, variations, and performance issues

Scott A. Burns
1994 International Journal for Numerical Methods in Engineering  
The monomial method solves systems of non-linear algebraic equations by constructing a sequence of approximating monomial (single-term polynomial) systems, much as Newton's method generates a sequence  ...  Two new versions ofthe algorithm are presented, both of which are simplified and computationally more efficient to implement in comparison to the original algorithm.  ...  Instead, a sum of products is exponentiated once for each term.  ... 
doi:10.1002/nme.1620371206 fatcat:sia33ar5w5acdgblhkcuar52ui

Solving Factored MDPs with Continuous and Discrete Variables [article]

Carlos E. Guestrin, Milos Hauskrecht, Branislav Kveton
2012 arXiv   pre-print
We present a new linear program approximation method that exploits the structure of the hybrid MDP and lets us compute approximate value functions more efficiently.  ...  Although many real-world stochastic planning problems are more naturally formulated by hybrid models with both discrete and continuous variables, current state-of-the-art methods cannot adequately address  ...  exponentially-large sums and complex integrals.  ... 
arXiv:1207.4150v1 fatcat:mtpzbz74yzebhcuke6ccc33e2q

Tensor Monte Carlo: particle methods for the GPU era [article]

Laurence Aitchison
2019 arXiv   pre-print
While the sum over exponentially many terms might seem to be intractable, in many cases it can be computed efficiently as a series of tensor inner-products.  ...  However, IWAEs scale poorly: as the latent dimensionality grows, they require exponentially many samples to retain the benefits of importance weighting.  ...  While the sum over exponentially many terms might seem to be intractable, in many cases it can be computed efficiently as a series of tensor inner-products.  ... 
arXiv:1806.08593v3 fatcat:rtlm6ux6yfgxvhnjow4gulai3q

On approximation of functions by exponential sums

Gregory Beylkin, Lucas Monzón
2005 Applied and Computational Harmonic Analysis  
We introduce a new approach, and associated algorithms, for the efficient approximation of functions and sequences by short linear combinations of exponential functions with complex-valued exponents and  ...  These approximations are obtained for a finite but arbitrary accuracy and typically have significantly fewer terms than Fourier representations.  ...  Brad Alpert (NIST) for pointing out connections with the steepest descent method and Dr. Martin Mohlenkamp (Ohio University) for many useful suggestions.  ... 
doi:10.1016/j.acha.2005.01.003 fatcat:6clifgejqrdqff4zpcxciz7fqa
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