威尼斯人

11月14日 数值代数及其应用研讨会

腾讯会议：883-890-341

时间：9:00-15:00

组织人：张雷洪、黄金枝

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Symmetric tensors and anti-symmetric tensors

徐常青 教授（苏州科技大学）

摘要：In this talk, I will introduce some basic on tensors, and initialze the S-products of tensors to unify the outer product, contractive product, and the inner product of tensors. The symmetry tensors and anti-symmetry tensors are also introduced with investigation on their properties. 

A Column-Wise Update Algorithm for Sparse Stochastic Matrix Factorization

白正简 教授（厦门大学）

摘要：Nonnegative matrix factorization arises widely in machine learning and data analysis. In this talk, for a given factorization of rank r, we consider the sparse stochastic matrix factorization (SSMF) of decomposing a prescribed m-by-n stochastic matrix V into a product of an m-by-r stochastic matrix W and an r-by-n stochastic matrix H, where both W and H are required to be sparse. With the prescribed sparsity level, we reformulate the SSMF as an unconstrained nonconvex-nonsmooth minimization problem and introduce a column-wise update algorithm for solving the minimization problem. We show that our algorithm converges globally. The main advantage of our algorithm is that the generated sequence converges to a special critical point of the cost function, which is nearly a global minimizer over each column vector of the W-factor and is a global minimizer over the H-factor as a whole if there is no sparsity requirement on H. Numerical experiments on both synthetic and real data sets are given to demonstrate the effectiveness of our proposed algorithm.

A Restarted and Randomized Algorithm for Evaluating Energy of Extremely Large-Scale Graphs

吴钢 教授（中国矿业大学）

摘要： Graph energy is a spectrum-based graph invariant that has been studied extensively. However, as far as we are aware, most of the existing works try to establish theoretical bounds, and there are few efficient algorithms for computing energy of extremely large-scale graph. To fill-in this gap, by using the tool of randomized singular value decomposition, we propose a randomized algorithm for evaluating energy of large-scale graphs. However, the number of sampling used in this algorithm is difficult to determine in advance, and the graph energy is often underestimated by using this algorithm. In order to improve the quality of the evaluation, we then propose a {\it non-restarted} randomized algorithm that updates the columns of the search basis incrementally. The error analysis and the convergence of this algorithm are established. However, the non-restarted algorithm may suffer from heavily computational cost and large storage requirements as the iteration proceeds. So as to release the overhead of the non-restarted algorithm, we then propose a {\it restarted} randomized algorithm for evaluating energy of extremely large-scale graphs. The rationality of the restarted algorithm is given. Extensive numerical experiments are performed on extremely large-scale real-world and synthetic graphs, to show the effectiveness of our strategies and efficiency of the proposed algorithms.

Confluent Vandermonde with Arnoldi

牛强 教授（西交利物浦大学）

摘要: In this work, we extend the Vandermonde with Arnoldi method recently advocated by P. D. Brubeck, Y. Nakatsukasa and L. N. Trefethen [SIAM Review, 63 (2021) 405-415] to dealing with the confluent Vandermonde matrix. To apply the Arnoldi process, it is critical to find a Krylov subspace which generates the column space of the confluent Vandermonde matrix. A theorem is established for such Krylov subspaces for any order derivatives. This enables us to compute the derivatives of high degree polynomials to high precision. It also makes many applications involving derivatives possible, as illustrated by numerical examples.

Nonlinear power-like iteration by polar decomposition and its application to tensor approximation

董波 教授（大连理工大学）

摘要：Low rank tensor approximation is an important subject with a wide range of appli- cations. Most prevailing techniques for computing the low rank approximation in the Tucker format often first assemble relevant factors into matrices and then update by turns one factor matrix at a time. In order to improve two factor matrices simultaneously, a special system of nonlinear matrix equations over a certain product Stiefel manifold must be resolved at every update. The solution to the system consists of orbit varieties invariant under the orthogonal group action, which thus imposes challenges on its analysis. In this talk, we will propose a scheme similar to the power method for subspace iterations except that the polar decomposition is used as the normalization process and that the iteration can be applied to both the orbits and the cross-sections. The notion of quotient manifold is employed to factor out the effect of orbital solutions. The dynamics of the iteration is completely characterized. An isometric isomorphism between the tangent spaces of two properly identified Riemannian manifolds is established to lend a hand to the proof of convergence.