56 research outputs found
Revisiting the D-iteration method: from theoretical to practical computation cost
In this paper, we revisit the D-iteration algorithm in order to better
explain its connection to the Gauss-Seidel method and different performance
results that were observed. In particular, we study here the practical
computation cost based on the execution runtime compared to the theoretical
number of iterations. We also propose an exact formula of the error for
PageRank class of equations.Comment: 9 page
D-iteration method or how to improve Gauss-Seidel method
The aim of this paper is to present the recently proposed fluid diffusion
based algorithm in the general context of the matrix inversion problem
associated to the Gauss-Seidel method. We explain the simple intuitions that
are behind this diffusion method and how it can outperform existing methods.
Then we present some theoretical problems that are associated to this
representation as open research problems. We also illustrate some connected
problems such as the graph transformation and the PageRank problem.Comment: 7 page
D-iteration: application to differential equations
In this paper, we study how the D-iteration algorithm can be applied to
numerically solve the differential equations such as heat equation in 2D or 3D.
The method can be applied on the class of problems that can be addressed by the
Gauss-Seidel iteration, based on the linear approximation of the differential
equations.Comment: 5 page
Optimized on-line computation of PageRank algorithm
In this paper we present new ideas to accelerate the computation of the
eigenvector of the transition matrix associated to the PageRank algorithm. New
ideas are based on the decomposition of the matrix-vector product that can be
seen as a fluid diffusion model, associated to new algebraic equations. We show
through experiments on synthetic data and on real data-sets how much this
approach can improve the computation efficiency.Comment: 7 page
Fast ranking algorithm for very large data
In this paper, we propose a new ranking method inspired from previous results
on the diffusion approach to solve linear equation. We describe new
mathematical equations corresponding to this method and show through
experimental results the potential computational gain. This ranking method is
also compared to the well known PageRank model.Comment: 3 page
D-iteration: Evaluation of the Asynchronous Distributed Computation
The aim of this paper is to present a first evaluation of the potential of an
asynchronous distributed computation associated to the recently proposed
approach, D-iteration: the D-iteration is a fluid diffusion based iterative
method, which has the advantage of being natively distributive. It exploits a
simple intuitive decomposition of the matrix-vector product as elementary
operations of fluid diffusion associated to a new algebraic representation. We
show through experiments on real datasets how much this approach can improve
the computation efficiency when the parallelism is applied: with the proposed
solution, when the computation is distributed over virtual machines (PIDs),
the memory size to be handled by each virtual machine decreases linearly with
and the computation speed increases almost linearly with with a slope
becoming closer to one when the number of linear equations to be solved
increases.Comment: 8 page
Introducing One Step Back Iterative Approach to Solve Linear and Non Linear Fixed Point Problem
In this paper, we introduce a new iterative method which we call one step
back approach: the main idea is to anticipate the consequence of the iterative
computation per coordinate and to optimize on the choice of the sequence of the
coordinates on which the iterative update computations are done. The method
requires the increase of the size of the state vectors and one iteration step
loss from the initial vector. We illustrate the approach in linear and non
linear iterative equations.Comment: 2 page
D-iteration: evaluation of the update algorithm
The aim of this paper is to analyse the gain of the update algorithm
associated to the recently proposed D-iteration: the D-iteration is a fluid
diffusion based new iterative method. It exploits a simple intuitive
decomposition of the product matrix-vector as elementary operations of fluid
diffusion (forward scheme) associated to a new algebraic representation. We
show through experimentations on real datasets how much this approach can
improve the computation efficiency in presence of the graph evolution.Comment: 5 page
Understanding differential equations through diffusion point of view: non-symmetric discrete equations
In this paper, we propose a new adaptation of the D-iteration algorithm to
numerically solve the differential equations. This problem can be reinterpreted
in 2D or 3D (or higher dimensions) as a limit of a diffusion process where the
boundary or initial conditions are replaced by fluid catalysts. It has been
shown that pre-computing the diffusion process for an elementary catalyst case
as a fundamental block of a class of differential equations, the computation
efficiency can be greatly improved. Here, we explain how the diffusion point of
view can be applied to decompose the fluid diffusion process per direction and
how to handle non-symmetric discrete equations. The method can be applied on
the class of problems that can be addressed by the Gauss-Seidel iteration,
based on the linear approximation of the differential equations.Comment: 4 page
Statistical reliability and path diversity based PageRank algorithm improvements
In this paper we present new improvement ideas of the original PageRank
algorithm. The first idea is to introduce an evaluation of the statistical
reliability of the ranking score of each node based on the local graph property
and the second one is to introduce the notion of the path diversity. The path
diversity can be exploited to dynamically modify the increment value of each
node in the random surfer model or to dynamically adapt the damping factor. We
illustrate the impact of such modifications through examples and simple
simulations.Comment: 8 page
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