Prediction bounds for higher order total variation regularized least squares

Ortelli, Francesco; Geer, Sara van de

doi:10.1214/21-aos2054

Cited by 11 publications

(29 citation statements)

References 21 publications

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“…This extension is called "trend filtering" [12,25,28]. Adaptivity of trend filtering is established in [10,19]. Wang et al [32] consider trend filtering on graphs, Sadhanala et al [20,22] in higher-dimensional situations and [21] for additive models.…”

Section: Literature Review: Adaptive Results For Tv Regularizationmentioning

confidence: 99%

“…In this paper we will prove adaptivity of tensor denoising with kth-order Vitali TV regularization for k 2 ¹1; 2; 3; 4º and general dimension d 1. The results obtained for k 2 ¹1; 2; 3; 4º and d D 1 in [19] and for k D 1 and d D 2 in [17] will then be retrieved as special cases.…”

Section: Literature Review: Adaptive Results For Tv Regularizationmentioning

confidence: 99%

“…Typically rem.D; f 0 ; S 0 / D O 2 2 D .S 0 / ; where 2 D .S 0 / is called "effective sparsity" and depends both on D and S 0 . Proving adaptivity therefore translates into proving a bound for the effective sparsity: a task which depends on the structure imposed by D. To bound the effective sparsity for tensor denoising with trend filtering we use an interpolating tensor, in analogy to the interpolating vector and the interpolating matrix by Ortelli and van de Geer [17,19].…”

Section: Adaptation For Structured Problemsmentioning

confidence: 99%

“…Previously, we have derived tools like interpolating vectors and matching derivatives to prove adaptivity for trend filtering (d D 1 and k 2 ¹1; 2; 3; 4º, see [19]). In [17] we have come up with tools to extend our results for adaptation of the fused Lasso (d D 1 and k D 1) to the two-dimensional case of image denoising (d D 2 and k D 1).…”

Section: Contributionsmentioning

confidence: 99%

“…The integration of the arguments by Ortelli and van de Geer [19] with the ones by Ortelli and van de Geer [17], the general bounds on the antiprojections and the ANOVA decomposition allow us to present general risk bounds for tensor denoising with trend filtering.…”

Section: Contributionsmentioning

confidence: 99%

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Tensor denoising with trend filtering

Ortelli

Geer

2022

Math. Stat. Learn.

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We extend the notion of trend filtering to tensors by considering the kth-order Vitali variation -a discretized version of the integral of the absolute value of the kth-order total derivative. We prove adaptive `0-rates and not-so-slow `1-rates for tensor denoising with trend filtering.For k D ¹1; 2; 3; 4º we prove that the d -dimensional margin of a d -dimensional tensor can be estimated at the `0-rate n 1 , up to logarithmic terms, if the underlying tensor is a product of .k 1/th-order polynomials on a constant number of hyperrectangles. For general k we prove the `1-rate of estimation n H.d /C2k 1 2H.d /C2k 1 , up to logarithmic terms, where H.d / is the d th harmonic number.Thanks to an ANOVA-type of decomposition we can apply these results to the lower dimensional margins of the tensor to prove bounds for denoising the whole tensor. Our tools are interpolating tensors to bound the effective sparsity for `0-rates, mesh grids for `1-rates and, in the background, the projection arguments by Dalalyan, Hebiri, and Lederer (2017).

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Section: Literature Review: Adaptive Results For Tv Regularizationmentioning

confidence: 99%

Section: Literature Review: Adaptive Results For Tv Regularizationmentioning

confidence: 99%

Section: Adaptation For Structured Problemsmentioning

confidence: 99%

Section: Contributionsmentioning

confidence: 99%

Section: Contributionsmentioning

confidence: 99%

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Tensor denoising with trend filtering

Ortelli

Geer

2022

Math. Stat. Learn.

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The Lasso with Structured Design and Entropy of (Absolute) Convex Hulls

Geer

Hinz

2023

Springer Proceedings in Mathematics &Amp; Statistics

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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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Prediction bounds for higher order total variation regularized least squares

Cited by 11 publications

References 21 publications

Tensor denoising with trend filtering

Tensor denoising with trend filtering

The Lasso with Structured Design and Entropy of (Absolute) Convex Hulls

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

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