GVN: Graph Von Neumann Variance Estimator

View source: R/GVN.R

GVNR Documentation

Graph Von Neumann Variance Estimator


GVN computes graph equivalent of the Von Neummann variance estimator.


GVN(y, A, L)



Numeric vector that represents the noisy data.


Adjacency matrix of the graph.


Laplacian matrix of the graph.


In many real-world scenarios, the noise level \sigma^2 remains generally unknown. Given any function g : \mathbb R_+ \rightarrow \mathbb R_+, the expectation of the graph signal can be expressed as:

\mathbf E[\widetilde f^T g(L) \widetilde f] = f^T g(L) f + \mathbf E[\xi^T g(L) \xi] = f^T g(L) f + \sigma^2 \mathrm{Tr}(g(L))

A biased estimator of the variance \sigma^2 can be given by:

\hat \sigma^2_1 = \frac{\widetilde f^T g(L) \widetilde f}{\mathrm{Tr}(g(L))}

Assuming the original graph signal is smooth enough that f^T g(L) f is negligible compared to \mathrm{Tr}(g(L)), \hat \sigma^2 provides a reasonably accurate estimate of \sigma^2. For this function, a common choice is g(x) = x, leading to:

\hat \sigma^2_1 = \frac{\widetilde f^T L \widetilde f}{\mathrm{Tr}(L)} = \frac{\sum_{i,j \in V} w_{ij} |\widetilde f(i) - \widetilde f(j)|^2}{2 \mathrm{Tr}(L)}

This is the graph adaptation of the Von Neumann estimator, hence the term Graph Von Neumann estimator (GVN).


The Graph Von Neumann variance estimate for the given noisy data.


de Loynes, B., Navarro, F., Olivier, B. (2021). Data-driven thresholding in denoising with Spectral Graph Wavelet Transform. Journal of Computational and Applied Mathematics, Vol. 389.

von Neumann, J. (1941). Distribution of the ratio of the mean square successive difference to the variance. Ann. Math. Statistics, 35(3), 433–451.

See Also



## Not run: 
A <- minnesota$A
L <- laplacian_mat(A)
x <- minnesota$xy[ ,1]
n <- length(x)
f <- sin(x)
sigma <- 0.1
noise <- rnorm(n, sd = sigma)
y <- f + noise
GVN(y, A, L)

## End(Not run)

gasper documentation built on Oct. 27, 2023, 1:07 a.m.