Kronecker Products and Matrix Calculus: With Applications

By A. Graham. 1981

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Citations in the corpus

Training Networks in Null Space of Feature Covariance for Continual Learning (2021, background): “(1) We provide novel theoretical conditions for the stability and plasticity of network based on feature covariance.”
Classes of Kernels for Machine Learning: A Statistics Perspective (2002, background): “Separable nonstationary kernels possess the property that their Gram matrix G , whose ij - th element is G ij = K ( x i , x j ) , can be written as a tensor product ( also called Kronecker product , see Graham , 1981 ) of two vectors defined by K 1 and K 2 respectively .”
Consensus Tracking Under Directed Interaction Topologies: Algorithms and Experiments (2008, background): “where , , denotes the Kronecker product (see [21] for its properties), denotes the identity matrix, and is an matrix”
A Jacobi spectral Galerkin method for the integrated forms of fourth‐order elliptic differential equations (2009, unknown): “We can also rewrite the above equation in the following form using the Kronecker matrix algebra (see, Graham [48]): Mv ≡ [A ⊗ C + 2B ⊗ B + C ⊗ A + β1(B ⊗ C + C ⊗ B) + β2C ⊗ C]v = f , (4.8) where f and v are F and U written in a column vector, i.e., f = (f44, f54, . . . , fN ,4; f45, f55, . . . , fN ”
Distributing the Kalman Filter for Large-Scale Systems (2007, unknown): “where B = μI + βhA, and C = βvI, the constants μ, βh, βv are in terms of α, β in (1), and ⊗ is the Kronecker product [27].”
Electricity consumption prediction with functional linear regression using spline estimators (2010, unknown): “Using the Kronecker product notation ⊗ [10], the vectorial form of the solution ̂ can be expressed as vec ̂ = [Ĉ + P(m)]−1vec D̂, (9) where Ĉ = P(0) 2 ⊗ (Ĉ + P 1 ), P = m−1 ∑ m1=0 ( m m1 ) P(m−m1) ′ 2 ⊗ P1 1 with D̂ = (d̂kl) ∈ R1 × R2 , d̂kl = 1 n n ∑ i=1 〈Xi, B1k〉〈Yi, B2l〉, Ĉ = (̂ckk′) ∈ R1 × R1 ”
The Graph Neural Network Model (2009, unknown): “By the Kronecker product properties, holds for matrices , , and having compatible dimensions [72].”
A decentralized approach to formation maneuvers (2003, methodology): “2are simplified by the use of Kronecker product notation [27].”
SPIRAL: Code Generation for DSP Transforms (2005, background): “Short vector instructions have the potential to speed up a program considerably, provided the program’s data ﬂow exhibits the ﬁne-grain parallelism necessary for their application.”
End-to-End attention based text-dependent speaker verification (2016, background): “where⊗ is the Kronecker product [32], ht is a 64-dim feature extracted from the CNN, and the posterior feature γ t has 10 dimensions, each represents a probability of phoneme in “Hey Cortana”1,”