arXiv subjects

Yurii Nesterov

Publications and source records attributed to Yurii Nesterov.

57 records · Page 4Linked to original sources

arXiv2015

Learning Supervised PageRank with Gradient-Free Optimization Methods

In this paper, we consider a problem of learning supervised PageRank models, which can account for some properties not considered by classical approaches such as the classical PageRank algorithm. Due to huge hidden dimension of the optimization problem we use random gradient-free methods to solve it. We prove a convergence theorem and estimate the number of arithmetic operations needed to solve it with a given accuracy. We find the best settings of the gradient-free optimization method in terms of the number of arithmetic operations needed to achieve given accuracy of the objective. In the paper, we apply our algorithm to the web page ranking problem. We consider a parametric graph model of users' behavior and evaluate web pages' relevance to queries by our algorithm. The experiments show that our optimization method outperforms the untuned gradient-free method in the ranking quality.

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arXiv2008

Generalized power method for sparse principal component analysis

In this paper we develop a new approach to sparse principal component analysis (sparse PCA). We propose two single-unit and two block optimization formulations of the sparse PCA problem, aimed at extracting a single sparse dominant principal component of a data matrix, or more components at once, respectively. While the initial formulations involve nonconvex functions, and are therefore computationally intractable, we rewrite them into the form of an optimization program involving maximization of a convex function on a compact set. The dimension of the search space is decreased enormously if the data matrix has many more columns (variables) than rows. We then propose and analyze a simple gradient method suited for the task. It appears that our algorithm has best convergence properties in the case when either the objective function or the feasible set are strongly convex, which is the case with our single-unit formulations and can be enforced in the block case. Finally, we demonstrate numerically on a set of random and gene expression test problems that our approach outperforms existing algorithms both in quality of the obtained solution and in computational speed.

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arXiv2004

Computationally efficient approximations of the joint spectral radius

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate. We introduce in this paper a procedure for approximating the joint spectral radius of a finite set of matrices with arbitrary high accuracy. Our approximation procedure is polynomial in the size of the matrices once the number of matrices and the desired accuracy are fixed.

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