Approximation with Kronecker Products

By Charles F. Van Loan, Nikos P. Pitsianis. 1992

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A literature survey of low‐rank tensor approximation techniques (2013, neutral): “An early technique is based on best approximation in the Frobenius norm by a Kronecker product [251, 256], by a short sum of Kronecker products [203], or by a more general low-rank tensor decomposition.”
Natural continual learning: success is a journey, not (just) a destination (2021, methodology): “…π : As in (Martens and Grosse, 2015), we use a trace norm in bounding the approximation error, and noting that Tr( X ⊗ Y ) = Tr( X )Tr( Y ), we can compute the optimal π as: Minimal mean-squared error The second approximation we consider was originally proposed by Van Loan and Pitsianis (1993).”
Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions (2009, background): “In the two-dimensional case this observation probably belongs to Van Loan and Pitsianis [18] and the generalization to the d-dimensional case is easy (it was considered, for example, in [17, 12]).”
Tensor approximations of matrices generated by asymptotically smooth functions (2003, background): “For m = 2 a similar observation has been made in [19]; in that case the rank condition is also sufficient.”
A Kronecker Product Preconditioner for Stochastic Galerkin Finite Element Discretizations (2010, methodology): “The remaining question about the symmetry and positive definiteness of P 1 is also resolved in the work of Van Loan and Pitsianis .”
Determining the Number of Factors from Empirical Distribution of Eigenvalues (2010, background): “(For a general discussion of the quality of approximations with Kronecker products, see Van Loan & Pitsianis, 1993.)”
Separable nonlinear least squares: the variable projection method and its applications (2003, background): “An interesting paper that connects VP with the manipulation of tensor products [92] is [ 124 ], where the approximation in the Frobenius norm of a rectangular matrix by a Kronecker product is discussed.”
Kronecker-product approximations for some function-related matrices (2004, background): “As is noted in [23], the Kronecker-product approximations (1) can be reduced to approximations of low rank and vice versa.”
Tensor properties of multilevel Toeplitz and related matrices (2006, methodology): “Optimal approximations minimizing ||A−Ar||F can be obtained via the SVD algorithm due to the following observation [31].”
Tensor-Train Ranks for Matrices and Their Inverses (2011, background): “[22]) Ã(i1j1 , i2j2) = A(i1i2 , j1, j2), then A is a Kronecker product of two matrices if and only if Ã is a matrix of rank at most 1.”
Separable approximations of space‐time covariance matrices (2007, methodology): “For example, if n = m, then λ can be obtained in O(m3) flops (floating point operations) via the LU factorizations of S and T , see Van Loan and Pitsianis (1992) .”
Tensor-Train Decomposition (2011, background): “It is motivated by the following observation, first mentioned in [38] for the two-dimensional case.”
Algorithms for Numerical Analysis in High Dimensions (2005, methodology): “For the matrix A(j1, j ′ 1; j2, j ′ 2), we can again construct an optimal separated representation using an ordinary SVD algorithm, as was done in [44].”