We study the problem of variance estimation in general graph-structured problems. First, we develop a linear time estimator for the homoscedastic case that can consistently estimate the variance in general graphs. We show that our estimator attains minimax rates for the chain and 2D grid graphs when the mean signal has total variation with canonical scaling. Furthermore, we provide general upper bounds on the mean squared error performance of the fused lasso estimator in general graphs under a moment condition and a bound on the tail behavior of the errors. These upper bounds allow us to generalize for broader classes of distributions, such as sub-exponential, many existing results on the fused lasso that are only known to hold with the assumption that errors are sub-Gaussian random variables. Exploiting our upper bounds, we then study a simple total variation regularization estimator for estimating the signal of variances in the heteroscedastic case. We also provide lower bounds showing that our heteroscedastic variance estimator attains minimax rates for estimating signals of bounded variation in grid graphs, and $K$-nearest neighbor graphs, and the estimator is consistent for estimating the variances in any connected graph.

翻译：我们研究了一般图结构问题中的方差估计问题。首先，针对同方差情况，我们开发了一种线性时间估计器，能够一致地估计一般图中的方差。我们证明，当均值信号具有标准缩放下的全变差时，该估计器在链图和二维网格图中达到了极小化最优速率。此外，我们给出了矩条件及误差尾部行为有界条件下，融合套索估计器在一般图中均方误差性能的通用上界。这些上界使我们能够将许多仅在高斯次随机变量假设下成立的融合套索已有结果推广到更广泛的分布类（如次指数分布）。利用这些上界，我们进一步研究了一种简单的全变差正则化估计器，用于估计异方差情况下的方差信号。我们还提供了下界，证明我们提出的异方差方差估计器在网格图和K最近邻图中估计有界变差信号时达到了极小化最优速率，且在任意连通图中该估计器对方差估计具有一致性。