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Graph total variation denoising is a penalized least-squares estimator for noisy vertex signals on a graph. The squared loss is strictly convex, so the fitted signal is unique. Its basic inequality for a penalized least-squares estimator separates a noise inner product from the change in penalty. The incidence pseudoinverse decomposition controls the nonconstant noise through the geometry of the graph.

Decompose $h={\Pi }_{1}h+D^{†}Dh$ using the incidence pseudoinverse decomposition. The Cauchy-Schwarz inequality controls the constant component, and the duality of the ${\ell }_{\infty }$ and ${\ell }_1$ norms controls the remaining component:On the stipulated event, the second term contributes at most $\lambda \Vert Dh{\Vert }_1$ after multiplication by $2/n$. The triangle inequality gives $\Vert Dh{\Vert }_1\le \Vert D\hat{\theta }{\Vert }_1+\Vert D{\theta }^{\ast }{\Vert }_1$, so the negative penalty in the basic inequality for a penalized least-squares estimator cancels. We obtainon an event of probability at least $1-\delta$.