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Linear map

About: Linear map is a research topic. Over the lifetime, 4293 publications have been published within this topic receiving 82572 citations. The topic is also known as: linear mapping & linear function.


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Journal Article
TL;DR: In this paper, it was shown that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space.
Abstract: The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard because it contains vector cardinality minimization as a special case. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to minimizing the nuclear norm and illustrate our results with numerical examples.

2,742 citations

Journal ArticleDOI
TL;DR: A linear map from M n to M m is completely positive iff it admits an expression Φ(A)=Σ i V ∗ i AV i where Vi are n×m matrices as mentioned in this paper.

2,534 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider the minimization of a second-degree polynomial subject to linear constraints and show that the Moore-Penrose inverse can be reduced to a linear transformation.
Abstract: Preface. - Matrices. - Submatrices and partitioned matricies. - Linear dependence and independence. - Linear spaces: row and column spaces. - Trace of a (square) matrix. - Geometrical considerations. - Linear systems: consistency and compatability. - Inverse matrices. - Generalized inverses. - Indepotent matrices. - Linear systems: solutions. - Projections and projection matrices. - Determinants. - Linear, bilinear, and quadratic forms. - Matrix differentiation. - Kronecker products and the vec and vech operators. - Intersections and sums of subspaces. - Sums (and differences) of matrices. - Minimzation of a second-degree polynomial (in n variables) subject to linear constraints. - The Moore-Penrose inverse. - Eigenvalues and Eigenvectors. - Linear transformations. - References. - Index.

1,987 citations

Journal ArticleDOI
TL;DR: The algorithms presented here are based on the recently developed theory of wavelets and are applicable to all Calderon-Zygmund and pseudo-differential operators, and indicate that many previously intractable problems become manageable with the techniques presented here.
Abstract: A class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors. Previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices. In contrast, the methods presented here are based on the recently developed theory of wavelets and are applicable to all Calderon-Zygmund and pseudo-differential operators. The algorithms of this paper require order O(N) or O(N log N) operations to apply an N × N matrix to a vector (depending on the particular operator and the version of the algorithm being used), and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here.

1,841 citations

Book
01 Jan 1979
TL;DR: In this article, the Moore-Penrose or generalized inverse has been applied to the theory of finite Markov chains, and applications of the Drazin inverse have been discussed.
Abstract: Preface to the Classics edition Preface Introduction and other preliminaries 1. The Moore-Penrose or generalized inverse 2. Least squares solutions 3. Sums, partitioned matrices and the constrained generalized inverse 4. Partial isometries and EP matrices 5. The generalized inverse in electrical engineering 6. (i, j, k)-Generalized inverses and linear estimation 7. The Drazin inverse 8. Applications of the Drazin inverse to the theory of finite Markov chains 9. Applications of the Drazin inverse 10. Continuity of the generalized inverse 11. Linear programming 12. Computational concerns Bibliography Index.

1,676 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202327
202293
2021177
2020209
2019201
2018174