Matrix Differential Calculus with Applications in Statistics and Econometrics

By R. Iyer, J. R. Magnus, H. Neudecker. 1991

Subfields: Statistics and Probability, Mathematical Optimization

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Citations in the corpus

Gradient descent GAN optimization is locally stable (2017, background): “Since the system consists of parameters arranged in the form of matrices, we will need vectorization calculus [Magnus et al., 1995] to arrange these parameters as a vector and differentiate them/with respect to them.”
Gradient of mutual information in linear vector Gaussian channels (2006, background): “The first-order expansion of the mutual information as a function of the transmit (positive definite) covariance matrix , denoted by , is obtained from the gradient (Theorem 2) as [15]”
Semiparametric estimation of fractional cointegrating subspaces (2006, methodology): “We will use the inequality of Exercise 19 on page 238 of Magnus and Neudecker [12].”
Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis (2012, methodology): “To use modern differential notations of vectors, we follow Magnus and Neudecker (1988) and denote”
Applied Multivariate Analysis (2002, background): “…21σ 12 σ 22σ 11 σ 22σ 12 σ 21σ 21 σ 21σ 22 σ 22σ 21 σ 22σ 22 = σ 11σ 11 + σ 11σ 11 σ 11σ 21 + σ 21σ 11 σ 11σ 21 + σ 21σ 11 σ 21σ 21 + σ 21σ 21 · · · · · · · · · · · · σ 12σ 12 + σ 12σ 12 σ 12σ 22 + σ 22σ 12 σ 12σ 22 + σ 22σ 12 σ 22σ 22 + σ 22σ 22 = [σ ikσ jm + σ imσ jk] (3.3.5) See Magnus and…”
Multilinear Analysis of Image Ensembles: TensorFaces (2002, methodology): “The 2-mode analysis technique for analyzing (statistical) data matrices of scalar entries is described by Magnus and Neudecker [8].”
Matrix Differential Calculus (2005, background): “For more details please also see [See Magnus and Neudecker, 1999, Table 2, pp. 176], from which we know that if f is a scalar function of a matrix variable X thendf(X) = Tr(ATdX) = vec(A)Td vecX ⇒ ∂ Tr(ATX) ∂X = A. Lemma 3.1 (Product rule).”
QML ESTIMATION OF A CLASS OF MULTIVARIATE ASYMMETRIC GARCH MODELS (2011, methodology): “We shall use standard results on the differentiation of matrices that can be found in Magnus and Neudecker (1988) and Harville (1997), in particular the chain rule for differentiation of composed functions: ∂ f (A)/∂x = Tr { (∂ f (A)/∂ A′)(∂ A/∂x) } .”
MIMO Capacity Characterization for Movable Antenna Systems (2022, methodology): “Thus, based on Taylor’s theorem, we can find a quadratic surrogate function to globally lower-bound the objective function ¯ g ( r m ) as [33] Thus, maximizing ¯ g ( r m ) can be converted to maximizing To this end, in the i th iteration of SCA, the optimization problem of the m th receive MA…”
Taxonomic Prediction with Tree-Structured Covariances (2013, unknown): “3 of [27]”