Dohy Hong, Philippe Jacquet
In this paper, we apply the ideas of the matrix column based diffusion approach to define a new eigenvector computation algorithm of a stationary probability of a Markov chain.
Dohy Hong
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.
Dohy Hong
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.
Dohy Hong
The aim of this paper is to explain how the D-iteration can be used for an efficient asynchronous distributed computation. We present the main ideas of the method and illustrate them through very simple examples.
Dohy Hong
The aim of this paper is to present a first evaluation of a dynamic partition strategy associated to the recently proposed asynchronous distributed computation scheme based on the D-iteration approach. The D-iteration is a fluid diffusion point of view based iteration method to solve numerically linear equations. Using a simple static partition strategy, it has been shown that, when the computation is distributed over K virtual machines (PIDs), the memory size to be handled by each virtual machine decreases linearly with K and the computation speed increases almost linearly with K with a slope becoming closer to one when the number N of linear equations to be solved increases. Here, we want to evaluate how further those results can be improved when a simple dynamic partition strategy is deployed and to show that the dynamic partition strategy allows one to control and equalize the computation load between PIDs without any deep analysis of the matrix or of the underlying graph structure.
Dohy Hong
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 $K$ virtual machines (PIDs), the memory size to be handled by each virtual machine decreases linearly with $K$ and the computation speed increases almost linearly with $K$ with a slope becoming closer to one when the number $N$ of linear equations to be solved increases.
Dohy Hong, Fabien Mathieu, Gérard Burnside
In this paper, we define the general framework to describe the diffusion operators associated to a positive matrix. We define the equations associated to diffusion operators and present some general properties of their state vectors. We show how this can be applied to prove and improve the convergence of a fixed point problem associated to the matrix iteration scheme, including for distributed computation framework. The approach can be understood as a decomposition of the matrix-vector product operation in elementary operations at the vector entry level.
Dohy Hong
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.
Dohy Hong, Gérard Burnside
In this paper, we describe the general framework to describe the diffusion operators associated to a positive matrix. We define the equations associated to diffusion operators and present some general properties of their state vectors. We show how this can be applied to prove and improve the convergence of a fixed point problem associated to the matrix iteration scheme. The approach can be understood as a decomposition of the matrix-vector product operation in elementary operation at the vector entry level.
Dohy Hong
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.
Dohy Hong
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.
Dohy Hong
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.
Dohy Hong, Gérard Burnside, Philippe Raoult
In this paper, we revisit the D-iteration algorithm in order to better explain different performance results that were observed for the numerical computation of the eigenvector associated to the PageRank score. We revisit here the practical computation cost based on the execution runtime compared to the theoretical number of iterations.
Dohy Hong
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. Pre-computing the diffusion process for an elementary catalyst case as a fundamental block of a class of differential equations, we show that the computation efficiency can be greatly improved. 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.
Dohy Hong
In this paper, we explain the convergence speed of different iteration schemes with the fluid diffusion view when solving a linear fixed point problem. This interpretation allows one to better understand why power iteration or Jacobi iteration may converge faster or slower than Gauss-Seidel iteration.
Dohy Hong
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.
Dohy Hong
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.