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Register a new userAccelerating Block Coordinate Descent for Nonnegative Tensor Factorization
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Added by
Man Shun Ang
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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This paper is concerned with improving the empirical convergence speed of block-coordinate descent algorithms for approximate nonnegative tensor factorization (NTF). We propose an extrapolation strategy in-between block updates, referred to as heuristic extrapolation with restarts (HER). HER significantly accelerates the empirical convergence speed of most existing block-coordinate algorithms for dense NTF, in particular for challenging computational scenarios, while requiring a negligible additional computational budget.
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Nonnegative CANDECOMP/PARAFAC (NCP) decomposition is an important tool to process nonnegative tensor. Sometimes, additional sparse regularization is needed to extract meaningful nonnegative and sparse components. Thus, an optimization method for NCP that can impose sparsity efficiently is required. In this paper, we construct NCP with sparse regularization (sparse NCP) by l1-norm. Several popular optimization methods in block coordinate descent framework are employed to solve the sparse NCP, all of which are deeply analyzed with mathematical solutions. We compare these methods by experiments on synthetic and real tensor data, both of which contain third-order and fourth-order cases. After comparison, the methods that have fast computation and high effectiveness to impose sparsity will be concluded. In addition, we proposed an accelerated method to compute the objective function and relative error of sparse NCP, which has significantly improved the computation of tensor decomposition especially for higher-order tensor.
The method of block coordinate gradient descent (BCD) has been a powerful method for large-scale optimization. This paper considers the BCD method that successively updates a series of blocks selected according to a Markov chain. This kind of block selection is neither i.i.d. random nor cyclic. On the other hand, it is a natural choice for some applications in distributed optimization and Markov decision process, where i.i.d. random and cyclic selections are either infeasible or very expensive. By applying mixing-time properties of a Markov chain, we prove convergence of Markov chain BCD for minimizing Lipschitz differentiable functions, which can be nonconvex. When the functions are convex and strongly convex, we establish both sublinear and linear convergence rates, respectively. We also present a method of Markov chain inertial BCD. Finally, we discuss potential applications.
Matrix factorization techniques compute low-rank product approximations of high dimensional data matrices and as a result, are often employed in recommender systems and collaborative filtering applications. However, many algorithms for this task utilize an exact least-squares solver whose computation is time consuming and memory-expensive. In this paper we discuss and test a block Kaczmarz solver that replaces the least-squares subroutine in the common alternating scheme for matrix factorization. This variant trades a small increase in factorization error for significantly faster algorithmic performance. In doing so we find block sizes that produce a solution comparable to that of the least-squares solver for only a fraction of the runtime and working memory requirement.
Low-rank tensors are an established framework for high-dimensional least-squares problems. We propose to extend this framework by including the concept of block-sparsity. In the context of polynomial regression each sparsity pattern corresponds to some subspace of homogeneous multivariate polynomials. This allows us to adapt the ansatz space to align better with known sample complexity results. The resulting method is tested in numerical experiments and demonstrates improved computational resource utilization and sample efficiency.
In this paper, we present several descent methods that can be applied to nonnegative matrix factorization and we analyze a recently developped fast block coordinate method called Rank-one Residue Iteration (RRI). We also give a comparison of these different methods and show that the new block coordinate method has better properties in terms of approximation error and complexity. By interpreting this method as a rank-one approximation of the residue matrix, we prove that it emph{converges} and also extend it to the nonnegative tensor factorization and introduce some variants of the method by imposing some additional controllable constraints such as: sparsity, discreteness and smoothness.
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