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
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
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.
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.
