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The graph total variation measures absolute variation of a vertex signal across edges. It is independent of the orientation used for the oriented incidence matrix. It vanishes exactly on signals constant on each connected component of a graph. A graph total variation denoising estimator combines this penalty with squared error, favouring signals that are piecewise constant on connected regions.

Let $h=\hat{\theta }-{\theta }^{\ast }$. The defining minimum for graph total variation denoising, compared with the feasible value ${\theta }^{\ast }$, gives $n^{-1}\Vert z-h{\Vert }_2^2+\lambda \Vert D\hat{\theta }{\Vert }_1\le n^{-1}\Vert z{\Vert }_2^2+\lambda \Vert D{\theta }^{\ast }{\Vert }_1$. Expand the squared Euclidean norm and cancel $\Vert z{\Vert }_2^2/n$ to obtain the basic inequality for a penalized least-squares estimator:

A penalized least-squares estimator balances squared error with a nonnegative penalty $J$. The Lasso uses the coefficient ${\ell }_1$ norm; graph total variation denoising uses the ${\ell }_1$ norm of endpoint differences. The choice of normalization of the squared loss changes the numerical tuning parameter.