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Register a new userBarzilai and Borwein conjugate gradient method equipped with a non-monotone line search technique and its application on non-negative matrix factorization
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Added by
Sajad Fathi Hafshejani
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this paper, we propose a new non-monotone conjugate gradient method for solving unconstrained nonlinear optimization problems. We first modify the non-monotone line search method by introducing a new trigonometric function to calculate the non-monotone parameter, which plays an essential role in the algorithms efficiency. Then, we apply a convex combination of the Barzilai-Borwein method for calculating the value of step size in each iteration. Under some suitable assumptions, we prove that the new algorithm has the global convergence property. The efficiency and effectiveness of the proposed method are determined in practice by applying the algorithm to some standard test problems and non-negative matrix factorization problems.
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The Barzilai-Borwein (BB) method has demonstrated great empirical success in nonlinear optimization. However, the convergence speed of BB method is not well understood, as the known convergence rate of BB method for quadratic problems is much worse than the steepest descent (SD) method. Therefore, there is a large discrepancy between theory and practice. To shrink this gap, we prove that the BB method converges $R$-linearly at a rate of $1-1/kappa$, where $kappa$ is the condition number, for strongly convex quadratic problems. In addition, an example with the theoretical rate of convergence is constructed, indicating the tightness of our bound.
In this paper we explore avenues for improving the reliability of dimensionality reduction methods such as Non-Negative Matrix Factorization (NMF) as interpretive exploratory data analysis tools. We first explore the difficulties of the optimization problem underlying NMF, showing for the first time that non-trivial NMF solutions always exist and that the optimization problem is actually convex, by using the theory of Completely Positive Factorization. We subsequently explore four novel approaches to finding globally-optimal NMF solutions using various ideas from convex optimization. We then develop a new method, isometric NMF (isoNMF), which preserves non-negativity while also providing an isometric embedding, simultaneously achieving two properties which are helpful for interpretation. Though it results in a more difficult optimization problem, we show experimentally that the resulting method is scalable and even achieves more compact spectra than standard NMF.
Dimensionality reduction is considered as an important step for ensuring competitive performance in unsupervised learning such as anomaly detection. Non-negative matrix factorization (NMF) is a popular and widely used method to accomplish this goal. But NMF do not have the provision to include the neighborhood structure information and, as a result, may fail to provide satisfactory performance in presence of nonlinear manifold structure. To address that shortcoming, we propose to consider and incorporate the neighborhood structural similarity information within the NMF framework by modeling the data through a minimum spanning tree. We label the resulting method as the neighborhood structure assisted NMF. We further devise both offline and online algorithm
In this article, we study algorithms for nonnegative matrix factorization (NMF) in various applications involving streaming data. Utilizing the continual nature of the data, we develop a fast two-stage algorithm for highly efficient and accurate NMF. In the first stage, an alternating non-negative least squares (ANLS) framework is used, in combination with active set method with warm-start strategy for the solution of subproblems. In the second stage, an interior point method is adopted to accelerate the local convergence. The convergence of the proposed algorithm is proved. The new algorithm is compared with some existing algorithms in benchmark tests using both real-world data and synthetic data. The results demonstrate the advantage of our algorithm in finding high-precision solutions.
The Conditional Gradient Method is generalized to a class of non-smooth non-convex optimization problems with many applications in machine learning. The proposed algorithm iterates by minimizing so-called model functions over the constraint set. Complemented with an Amijo line search procedure, we prove that subsequences converge to a stationary point. The abstract framework of model functions provides great flexibility for the design of concrete algorithms. As special cases, for example, we develop an algorithm for additive composite problems and an algorithm for non-linear composite problems which leads to a Gauss--Newton-type algorithm. Both instances are novel in non-smooth non-convex optimization and come with numerous applications in machine learning. Moreover, we obtain a hybrid version of Conditional Gradient and Proximal Minimization schemes for free, which combines advantages of both. Our algorithm is shown to perform favorably on a sparse non-linear robust regression problem and we discuss the flexibility of the proposed framework in several matrix factorization formulations.
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