We propose a dual Markov chain model to accommodate probabilities as well as perturbation,error bounds,or variances,in the Markov chain process.This motivates us to extend the Perron-Frobenius theory to dual number ma...We propose a dual Markov chain model to accommodate probabilities as well as perturbation,error bounds,or variances,in the Markov chain process.This motivates us to extend the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts.It is shown that such a dual number matrix always has a positive dual number eigenvalue with a positive dual number eigenvector.The standard part of this positive dual number eigenvalue is larger than or equal to the modulus of the standard part of any other eigenvalue of this dual number matrix.An explicit formula to compute the dual part of this positive dual number eigenvalue is presented.The Collatz minimax theorem also holds here.The results are nontrivial as even a positive dual number matrix may have no eigenvalue at all.An algorithm based upon the Collatz minimax theorem is constructed.The convergence of the algorithm is studied.An upper bound on the distance of stationary states between the dual Markov chain and the perturbed Markov chain is given.Numerical results on both synthetic examples and the dual Markov chain including some real world examples are reported.展开更多
In this paper, we present a useful result on the structures of circulant inverse Mmatrices. It is shown that if the n × n nonnegative circulant matrix A = Circ[c0, c1,… , c（n- 1）] is not a positive matrix and ...In this paper, we present a useful result on the structures of circulant inverse Mmatrices. It is shown that if the n × n nonnegative circulant matrix A = Circ[c0, c1,… , c（n- 1）] is not a positive matrix and not equal to c0I, then A is an inverse M-matrix if and only if there exists a positive integer k, which is a proper factor of n, such that cjk 〉 0 for j=0,1…, [n-k/k], the other ci are zero and Circ[co, ck,… , c（n-k）] is an inverse M-matrix. The result is then extended to the so-called generalized circulant inverse M-matrices.展开更多
In this paper,we study Riemannian optimization methods for the problem of nonnegative matrix completion that is to recover a nonnegative low rank matrix from its partial observed entries.With the underlying matrix inc...In this paper,we study Riemannian optimization methods for the problem of nonnegative matrix completion that is to recover a nonnegative low rank matrix from its partial observed entries.With the underlying matrix incohence conditions,we show that when the number m of observed entries are sampled independently and uniformly without replacement,the inexact Riemannian gradient descent method can recover the underlying n_(1)-by-n_(2)nonnegative matrix of rank r provided that m is of O(r^(2)slog^(2)s),where s=max{n_(1),n_(2)}.Numerical examples are given to illustrate that the nonnegativity property would be useful in the matrix recovery.In particular,we demonstrate the number of samples required to recover the underlying low rank matrix with using the nonnegativity property is smaller than that without using the property.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.12126608,12131004)the R&D project of Pazhou Lab(Huangpu)(Grant No.2023K0603)the Fundamental Research Funds for the Central Universities(Grant No.YWF-22-T-204).
文摘We propose a dual Markov chain model to accommodate probabilities as well as perturbation,error bounds,or variances,in the Markov chain process.This motivates us to extend the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts.It is shown that such a dual number matrix always has a positive dual number eigenvalue with a positive dual number eigenvector.The standard part of this positive dual number eigenvalue is larger than or equal to the modulus of the standard part of any other eigenvalue of this dual number matrix.An explicit formula to compute the dual part of this positive dual number eigenvalue is presented.The Collatz minimax theorem also holds here.The results are nontrivial as even a positive dual number matrix may have no eigenvalue at all.An algorithm based upon the Collatz minimax theorem is constructed.The convergence of the algorithm is studied.An upper bound on the distance of stationary states between the dual Markov chain and the perturbed Markov chain is given.Numerical results on both synthetic examples and the dual Markov chain including some real world examples are reported.
基金This work is supported by National Natural Science Foundation of China （No. 10531080）.
文摘In this paper, we present a useful result on the structures of circulant inverse Mmatrices. It is shown that if the n × n nonnegative circulant matrix A = Circ[c0, c1,… , c（n- 1）] is not a positive matrix and not equal to c0I, then A is an inverse M-matrix if and only if there exists a positive integer k, which is a proper factor of n, such that cjk 〉 0 for j=0,1…, [n-k/k], the other ci are zero and Circ[co, ck,… , c（n-k）] is an inverse M-matrix. The result is then extended to the so-called generalized circulant inverse M-matrices.
基金supported in part by the National Natural Science Foundation of China under Grant No.12171369Key NSF of Shandong Province under Grant No.ZR2020KA008+1 种基金supported in part by HKRGC GRF 12300519,17201020 and 17300021,HKRGC CRF C1013-21GF and C7004-21GFJoint NSFC and RGC N-HKU769/21。
文摘In this paper,we study Riemannian optimization methods for the problem of nonnegative matrix completion that is to recover a nonnegative low rank matrix from its partial observed entries.With the underlying matrix incohence conditions,we show that when the number m of observed entries are sampled independently and uniformly without replacement,the inexact Riemannian gradient descent method can recover the underlying n_(1)-by-n_(2)nonnegative matrix of rank r provided that m is of O(r^(2)slog^(2)s),where s=max{n_(1),n_(2)}.Numerical examples are given to illustrate that the nonnegativity property would be useful in the matrix recovery.In particular,we demonstrate the number of samples required to recover the underlying low rank matrix with using the nonnegativity property is smaller than that without using the property.
