On the total variation regularized estimator over a class of tree graphs

Abstract

We generalize to tree graphs obtained by connecting path graphs an oracle result obtained for the Fused Lasso over the path graph. Moreover we show that it is possible to substitute in the oracle inequality the minimum of the distances between jumps by their harmonic mean. In doing so we prove a lower bound on the compatibility constant for the total variation penalty. Our analysis leverages insights obtained for the path graph with one branch to understand the case of more general tree graphs. As a side result, we get insights into the irrepresentable condition for such tree graphs.

arXiv comment42 pages

References in corpus (7)

• Pathwise coordinate optimization
• The solution path of the generalized lasso
• On the Prediction Performance of the Lasso
• Optimal rates for total variation denoising
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Cited by in corpus (3)

• New Risk Bounds for 2D Total Variation Denoising
• Adaptive Estimation of Multivariate Piecewise Polynomials and Bounded Variation Functions by Optimal Decision Trees
• Oracle inequalities for square root analysis estimators with application to total variation penalties