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.
Highlights
The aim of this paper is to refine and extend to the more general case of a class of tree graphs the approach used by Dalalyan, Hebiri and Lederer (2017) to prove an oracle inequality for the Fused Lasso estimator, known as total variation regularized estimator
The main refinements we prove are an oracle theory for the total variation regularized estimators over trees when the first coefficient is not penalized, a proof of an lower bound for the compatibility constant and, as a consequence of this bound, the substitution in the oracle bound of the minimum of the distances between jumps by their harmonic mean
While to our knowledge there is no attempt in the literature to analyze the specific properties of the total variation regularized least squares estimator over general branched tree graphs, there is a lot of work in the field of the so called Fused Lasso estimator
Summary
The aim of this paper is to refine and extend to the more general case of a class of tree graphs the approach used by Dalalyan, Hebiri and Lederer (2017) to prove an oracle inequality for the Fused Lasso estimator, known as total variation regularized estimator. The paper is organized as follows: in Section 1 we expose the framework together with a review of the literature on the topic; in Section 2 we refine the proof of Theorem 3 of Dalalyan, Hebiri and Lederer (2017) and adapt it to the case where one coefficient of the Lasso is left unpenalized: this proof will be a working tool for establishing oracle inequalities for total variation penalized estimators; in Section 3 we introduce the notation needed for the rest of the article; in Section 4 we expose how to compute objects related to projections which are needed for finding explicit bounds on weighted compatibility constants and when the irrepresentable condition is satisfied; in Section 5 we present a tight lower bound for the (weighted) compatibility constant for the Fused Lasso and use it with the approach exposed in Section 2 to prove an oracle inequality; in Section 6 we generalize Section 5 to the case of the branched path graph; Section 7 presents further extensions to more general tree graphs; Section 8 handles the asymptotic signal pattern recovery properties of the total variation regularized estimator on the (branched) path graph and exposes an extension to more general tree graphs; Section 9 concludes the paper
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