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

Ortelli, Francesco; Geer, Sara van de

doi:10.1214/18-ejs1519

Cited by 19 publications

(36 citation statements)

References 15 publications

(22 reference statements)

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“…Corollary 4.1 below, is a result already found inOrtelli and van de Geer (2018). It is reported here for comparison with the analogous result obtained for the square root analysis estimator on the path graph (s. Corollary 4.3).…”

supporting

confidence: 80%

“…Ortelli and van de Geer (2019a)) to prove oracle inequalities. However, these studies were confined to restrictive graph structures: the path in Dalalyan, Hebiri and Lederer (2017) and a class of tree graphs in Ortelli and van de Geer (2018). Other studies focusing on the fused lasso and not directly involving its synthesis form also implicitly relied on some kind of dictionary to handle the error term by projections onto some columns of this dictionary, see for instance the lower interpolant by Lin et al (2017).…”

Section: Total Variation Regularized Estimatorsmentioning

confidence: 99%

“…In Ortelli and van de Geer (2018) it is explained that to bound the weak weighted compatibility constant for the path graph one needs to cut it into s smaller modules. These modules lie around an edge in S and consist of at least one additional edge on each side of the edge in S, see Figure 1.…”

Section: Path Graphmentioning

confidence: 99%

“…we need |S| ≤ n/4 to hope to be able to upper bound κ 2 (S, W ) by using the method proposed by van de Geer (2018). Moreover, a vertex not involved in an edge in S can only be involved in one module to obtain the bounds exposed in Ortelli and van de Geer (2018).…”

Section: Path Graphmentioning

confidence: 99%

“…. As pointed out in Ortelli and van de Geer (2018), X has the meaning of the rooted path matrix of the tree graph considered. Thus, the columns of X −1 contain a minimum of 1 and a maximum of (n − 1) entries having value 1, while the remaining entries are zeroes.…”

Section: C4 Proof Of the Oracle Inequalitiesmentioning

confidence: 99%

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Oracle inequalities for square root analysis estimators with application to total variation penalties

Ortelli

Geer

2020

Information and Inference: A Journal of the IMA

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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.

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supporting

confidence: 80%

Section: Total Variation Regularized Estimatorsmentioning

confidence: 99%

Section: Path Graphmentioning

confidence: 99%

Section: Path Graphmentioning

confidence: 99%

Section: C4 Proof Of the Oracle Inequalitiesmentioning

confidence: 99%

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Oracle inequalities for square root analysis estimators with application to total variation penalties

Ortelli

Geer

2020

Information and Inference: A Journal of the IMA

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Adaptive nonparametric regression with the K-nearest neighbour fused lasso

et al. 2020

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Summary The fused lasso, also known as total-variation denoising, is a locally adaptive function estimator over a regular grid of design points. In this article, we extend the fused lasso to settings in which the points do not occur on a regular grid, leading to a method for nonparametric regression. This approach, which we call the $K$-nearest-neighbours fused lasso, involves computing the $K$-nearest-neighbours graph of the design points and then performing the fused lasso over this graph. We show that this procedure has a number of theoretical advantages over competing methods: specifically, it inherits local adaptivity from its connection to the fused lasso, and it inherits manifold adaptivity from its connection to the $K$-nearest-neighbours approach. In a simulation study and an application to flu data, we show that excellent results are obtained. For completeness, we also study an estimator that makes use of an $\epsilon$-graph rather than a $K$-nearest-neighbours graph and contrast it with the $K$-nearest-neighbours fused lasso.

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New Risk Bounds for 2D Total Variation Denoising

Chatterjee

Goswami

2021

IEEE Trans. Inform. Theory

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2D Total Variation Denoising (TVD) is a widely used technique for image denoising. It is also an important nonparametric regression method for estimating functions with heterogenous smoothness. Recent results have shown the TVD estimator to be nearly minimax rate optimal for the class of functions with bounded variation. In this paper, we complement these worst case guarantees by investigating the adaptivity of the TVD estimator to functions which are piecewise constant on axis aligned rectangles. We rigorously show that, when the truth is piecewise constant with few pieces, the ideally tuned TVD estimator performs better than in the worst case. We also study the issue of choosing the tuning parameter.In particular, we propose a fully data driven version of the TVD estimator which enjoys similar worst case risk guarantees as the ideally tuned TVD estimator.

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

Cited by 19 publications

References 15 publications

Oracle inequalities for square root analysis estimators with application to total variation penalties

Oracle inequalities for square root analysis estimators with application to total variation penalties

Adaptive nonparametric regression with the K-nearest neighbour fused lasso

New Risk Bounds for 2D Total Variation Denoising

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