Matrix Computations

By Andrzej Chrzeszczyk, J. Kochanowski. 2011

14,206 citations recorded by the source corpus.

Subfields: Numerical Methods & Scientific Computing

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In Numerical Methods & Scientific Computing

Citations in the corpus

Analyzing and Improving the Image Quality of StyleGAN (2019, neutral): “This is seen using the Singular Value Decomposition J T w = U ˜ ΣV T , where U ∈ R L × L and V ∈ R M × M are orthogonal matrices, and ˜ Σ = [ Σ 0 ] is a horizontal concatenation of a diagonal matrix Σ ∈ R L × L and a zero matrix 0 ∈ R L × ( M − L ) [15].”
Beyond CCA: Moment Matching for Multi-View Models (2016, neutral): “A (cid:62) A = AA (cid:62) with a symmetric matrix as a particular case, the upper triangular matrix in its Schur decomposition is zero (Golub & Van Loan, 1996, Chapter 7) and then the Schur vectors correspond to the (orthogonal in this case) eigenvectors of this matrix.”
From Frequency to Meaning: Vector Space Models of Semantics (2010, background): “Minimizing the the Frobenius norm ‖X̂−X‖F will minimize the noise, if the noise has a Gaussian distribution.”
Recasting Gradient-Based Meta-Learning as Hierarchical Bayes (2018, neutral): “, n (Theorem 8.7.1 in Golub & Van Loan, 1983).”
Accelerating Very Deep Convolutional Networks for Classification and Detection (2015, methodology): “A simple decomposition is the Singular Value Decomposition (SVD) [31]: M = Ud′Sd′Vd′, where Ud′ and Vd′ are d-by-d′ column-orthogonal matrices and Sd′ is a d′-by-d′ diagonal matrix.”
Efficient and accurate approximations of nonlinear convolutional networks (2014, methodology): “This problem can be solved by SVD [5] How good is the low-rank assumption of the responses?”
The Uncertainty Bellman Equation and Exploration (2017, methodology): “by the Sherman-Morrison-Woodbury formula [8], the cost of this update is one matrix multiply and one matrix-matrix subtraction per step.”
On the interpretation of weight vectors of linear models in multivariate neuroimaging (2014, background): “The problem of “ optimal ” factor extraction under various conditions, however, is its own ﬁ eld of research treated in an extensive existing body on linear decomposition methods and their underlying assumptions (for some overview see Comon & Jutten, 2010; Golub & Van Loan, 1996; Haufe, 2011).”
Making AI Forget You: Data Deletion in Machine Learning (2019, methodology): “Generally speaking, the Sherman-Morrison-Woodbury matrix identity and matrix factorization techniques can be used to derive fast and explicit formulas for updating linear models [55, 27, 83, 43].”
Community detection in graphs (2009, methodology): “The time required to compute the first k eigenvectors of a matrix with the Lanczos method depends on the size of the eigengap |λk+1 − λk| (Golub and Loan, 1989).”
Fast, Exact and Multi-scale Inference for Semantic Image Segmentation with Deep Gaussian CRFs (2016, methodology): “In this work, we considered the Jacobi, Gauss-Seidel, Conjugate Gradient, and Generalized Minimal Residual (GMRES) methods [20], as candidates for iterative solvers.”
Online Learning for Matrix Factorization and Sparse Coding (2009, neutral): “For more details and properties of the Kronecker product, see Golub and Van Loan (1996), and Magnus and Neudecker (1999).”
Learning with Submodular Functions: A Convex Optimization Perspective (2011, methodology): “6 Spectral functions of submatrices Given a positive semidefinite matrix Q ∈ Rp×p and a real-valued function h from R+ to R, one may define the matrix function Q 7→ h(Q) defined on positive semi-definite matrices by leaving unchanged the eigenvectors of Q and applying h to each of the eigenvalues [8”