The Eckart-Young Theorem. Suppose a matrix A∈Rm×n has an SVD-decomposition A=UΣVT . Let k<r=rank(A) and truncated matrix Ak=∑i=1kσiuiviT, then, for any matrix B of rank k , the minimal error is achieved with Ak : minrank(B)=k||A−B||2=||A−Ak||2=σk+1. Th...
Eckart–Young TheoremBest rank-k approximationGiven a tensor f f in a Euclidean tensor space, we are interested in the critical points of the distance function from f f to the set of tensors of rank at most k k , which we call the critical rank-at-most- k k tensors for f f . ...
We present a simple proof of the following theorem of Wiegmann, but in principle given earlier by Eckart and Young: THEOREM If {Ai) is a set of complex r – s matrices such thatA A andA A are Hermitian for all i andj, then there exist unitary matrices P and Q such that for each...
The Eckart-Young-Mirsky theorem solves the problem of approximating a matrix by one of lower rank. However, the approximation generally differs from the original in all its elements. In this paper it is shown how to obtain a best approximation of lower rank in which a specified set of ...
of Theorem 3 in [2] In this paper we intend to present a perturbation theory for the problems related to the EYM theory. For this purpose, in this section we will restate Theo- rem 3 of [2], according to the submatrices A, B, C and D in G3. ...
In matricial analysis, the theorem of Eckart and Young provides a best approximation of an arbitrary matrix by a matrix of rank at most r. In variational analysis or optimization, the Moreau envelopes are appropriate ways of approximating or regularizing the rank function. We prove here that we...
1. Perturbation theory for the Eckart-Young-Mirsky theorem and the constrained total least squares problem [J] . Wei MS. Linear Algebra and its Applications . 1998,第2a3期 机译:Eckart-Young-Mirsky定理的摄动理论和受约束的总最小二乘问题 2. PERTURBATION THEORY FOR ORTHOGONAL PROJECTION METHOD...
AGeneralizationoftheEckart-Young-MirskyMatrixApproximationTheoremG.H.Golub*DepartmentofComputerScienceStanfordUniversityStanford,Cali..
Johnson, Richard M. 1963. "On a Theorem Stated by Eckart and Young." Psychometrika, 28:259-263.R. M. Johnson "On a theorem stated by Eckart and Young", Psykometrika , vol. 28, no. 3, pp.259 -263 1963Johnson, R.M.: On a theorem stated by Eckart and Young....
In this paper, we present a general algebraic structure of the solution set to the Generalized Eckart-Young problem. By this algebraic structure, we can parametrize simply all the solutions to this extraordinary important problem. Moreover, as an application of the main theorem, we derive the ...