Toeplitz and Circulant Matrices: A Review

By Robert M. Gray. 2005

2,631 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Statistical and Nonlinear Physics

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Citations in the corpus

Distinguishing Cause from Effect Using Observational Data: Methods and Benchmarks (2014, background): “, 1994) sp6 ITE: Shannon spacing VKDE (Noughabi and Noughabi, 2013) KDP ITE: Shannon KDP (Stowell and Plumbley, 2009) PSD ITE: Shannon PSD SzegoT (Ramirez et al., 2009; Gray, 2006) (Grenander and Szego, 1958) EdE ITE: Shannon Edgeworth (van Hulle, 2005) Gau based on (9) ME1 ITE: Shannon MaxEnt1 (Hyv”
Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling (2010, neutral): “Known results on circulant matrices (see Chapter 3 of Gray [Gra06]) imply that it has m th eigenvalue For m = n − 1 and k = o ( n ), the last equation can be massaged into [BGPS06, Section VI.”
Learning to Relate Images (2011, background): “They can therefore also be thought of as a way to implement mapping units that encode relations.”
The Singular Values of Convolutional Layers (2018, background): “Overview of the Analysis: If the signal is 1D and there is a single input and output channel, then the linear transformation associated with a convolution is encoded by a circulant matrix , i.e., a matrix whose rows are circular shifts of a single row (Gray, 2006).”
Joint Spatial Division and Multiplexing—The Large-Scale Array Regime (2012, background): “ J [a;b], respectively. Then, the eigenvectors of Capproximate the eigenvectors of Rin the sense that lim M!1 1 M U [a;b ]U H [a;b F [a;b]F H 2 F = 0: (59) A well-known property of circulant matrices [34] is that their eigenvectors form a unitary DFT matrix, i.e., the matrix whose (‘;m)-th element i”
Upper bounds on algebraic connectivity via convex optimization (2006, background): “The eigenvalues of C are (see, for example, [10]) μj = a + 2b cos(2πj/q), j = 1, .”
Sharp Thresholds for High-Dimensional and Noisy Sparsity Recovery Using $\ell _{1}$ -Constrained Quadratic Programming (Lasso) (2009, background): “The maximum and minimum eigenvalues ( and ) can be bounded using standard asymptotic results on Toeplitz matrix families [19].”
Learning the MMSE Channel Estimator (2017, result): “This yields an asymptotically equivalent matrix due to the results from [16].”
On distributed averaging algorithms and quantization effects (2007, background): “The second largest eigenvalue of is ([9, eq.”
Nonlinear image representation using divisive normalization (2008, methodology): “In computing the gradients, we take advantage of the fact that the matrix-vector product between a block-Toeplitz matrix and a vector is equivalent to a convolution [11], which may be efficiently implemented with fast Fourier transform.”
On Random Weights and Unsupervised Feature Learning (2011, neutral): “The matrix C is doubly block circulant (see Gray, 2006; Tee, 2007, for overviews), i.e., each row of blocks is a circular shift of the previous row, C =      C 1 C 2 · · · C n−1 C n C n C 1 · · · C n−2 C n−1 . . . . . . . . . . . . . . .”
Structured Transforms for Small-Footprint Deep Learning (2015, background): “Toeplitz and Hankel matrices are intimately related to one-dimensional discrete convolutions [10], and arise naturally in time series analysis and dynamical systems.”