# Convergence in Variance of Chebyshev Accelerated Gibbs Samplers

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^{1}, Antti Solonen

^{2}, Tiangang Cui

^{1}, James Martin

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104 citations

62 citations

19 citations

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### Cites background or methods from "Convergence in Variance of Chebyshe..."

...The Chebyshev accelerated SSOR solver and corresponding Chebyshev accelerated SSOR sampler (Fox and Parker, 2014) are depicted in panels C and D of Figure 1....

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...But even sooner, after k∗∗ = k∗/2 iterations, the Chebyshev error reduction for the variance is predicted to be smaller than ε (Fox and Parker [19])....

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...Using Theorem 5, we derived the Chebyshev accelerated SSOR sampler (Fox and Parker, 2014) by iteratively updating parameters via (13) and then generating a sampler via (17)....

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...But even sooner, after k∗∗ = k∗/2 iterations, the Chebyshev error reduction for the variance is predicted to be smaller than ε (Fox and Parker, 2014). imsart-bj ver....

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...Fox and Parker (2014) considered point-wise convergence of the mean and variance of a Gibbs SSOR sampler accelerated by Chebyshev polynomials....

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14 citations

### Cites background from "Convergence in Variance of Chebyshe..."

...Alternative techniques for sampling from high-dimensional Gaussian distributions can be found, for instance, in [211, 100]....

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##### References

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34,706 citations

### "Convergence in Variance of Chebyshe..." refers methods in this paper

...A standard method of reducing the asymptotic average reduction factor is by polynomial acceleration, particularly using Chebyshev polynomials [1, 6, 10, 23]....

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...Analogous to the SSOR solver algorithms in [10, 23], the Chebyshev sampler implements sequential forward and backward sweeps of an SOR sampler [7, 20] (i....

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...Since a symmetric splitting is required, Chebyshev acceleration in a linear solver is commonly implemented with a symmetric successive overrelaxation (SSOR) splitting A = MSSOR−NSSOR, with algorithms to be found, for example, in [10, 23]....

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...For example, they occur in the stationary linear iterative methods used to solve systems of linear equations [1, 10, 17, 23]....

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...We could have equally followed the excellent presentations of the same methods in Golub and Van Loan [10] or Saad [23]....

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18,328 citations

12,575 citations

### "Convergence in Variance of Chebyshe..." refers methods in this paper

...A standard method of reducing the asymptotic average reduction factor is by polynomial acceleration, particularly using Chebyshev polynomials [1, 6, 10, 23]....

[...]

...Analogous to the SSOR solver algorithms in [10, 23], the Chebyshev sampler implements sequential forward and backward sweeps of an SOR sampler [7, 20] (i....

[...]

...Since a symmetric splitting is required, Chebyshev acceleration in a linear solver is commonly implemented with a symmetric successive overrelaxation (SSOR) splitting A = MSSOR−NSSOR, with algorithms to be found, for example, in [10, 23]....

[...]

...For example, they occur in the stationary linear iterative methods used to solve systems of linear equations [1, 10, 17, 23]....

[...]

...We could have equally followed the excellent presentations of the same methods in Golub and Van Loan [10] or Saad [23]....

[...]

8,491 citations

6,854 citations

### "Convergence in Variance of Chebyshe..." refers background in this paper

...84], [19], though distributions with nonconnected support for which Algorithm 1 fails are easy to find [19]....

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...Convergence in both cases is geometric, with the asymptotic average reduction factor given by ρ (G) (though this is called the “convergence rate” in the statistics literature [19])....

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