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.

No abstract

Through the direct study of the analysis estimator we derive oracle inequalities with fast and slow rates by adapting the arguments involving projections by Dalalyan, Hebiri and Lederer (2017). We then extend the theory to the square root analysis estimator. Finally, we focus on (square root) total variation regularized estimators on graphs and obtain constant-friendly rates, which, up to log-terms, match previous results obtained by entropy calculations. We also obtain an oracle inequality for the (square root) total variation regularized estimator over the cycle graph.

We establish oracle inequalities for the least squares estimator f with penalty on the total variation of f or on its higher order differences. Our main tool is an interpolating vector that leads to upper bounds for the effective sparsity. This allows one to show that the penalty on the k th order differences leads to an estimator f that can adapt to the number of jumps in the (k − 1) th order differences. We present the details for k = 2, 3 and expose a framework for deriving the result for general k ∈ N.

Controlled hydrolysis via invertase action alters molecular shape and therefore lipid curvature, consequently triggering the release of encapsulated drug.