New Risk Bounds for 2D Total Variation Denoising

Chatterjee, Sabyasachi; Goswami, Subhajit

doi:10.1109/tit.2021.3059657

Cited by 12 publications

(3 citation statements)

References 37 publications

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“…@x i / k are studied by Hütter and Rigollet [11], and Sadhanala et al [22] for general d . For d D 2, Chatterjee and Goswami [3] show the fast rate n 3=4 for estimating axis-aligned rectangles. Sadhanala et al [22], and Sadhanala et al [20] call the estimator for general k Kronecker trend filtering.…”

Section: Literature Review: Adaptive Results For Tv Regularizationmentioning

confidence: 99%

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%

Tensor denoising with trend filtering

Ortelli

Geer

2022

Math. Stat. Learn.

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“…It would also be interesting to examine other versions of the Fused Lasso estimator such as 2D total variation denoising (Hütter and Rigollet, 2016;Chatterjee and Goswami, 2021), and the graph fused lasso (Hallac et al, 2015;Tansey and Scott, 2015;Barbero and Sra, 2014). For these extensions, all existing analyses bound a global loss.…”

Section: Discussionmentioning

confidence: 99%

Element-wise Estimation Error of Generalized Fused Lasso

Zhang¹,

Chatterjee²

2022

Preprint

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The main result of this article is that we obtain an elementwise error bound for the Fused Lasso estimator for any general convex loss function ρ. We then focus on the special cases when either ρ is the square loss function (for mean regression) or is the quantile loss function (for quantile regression) for which we derive new pointwise error bounds. Even though error bounds for the usual Fused Lasso estimator and its quantile version have been studied before; our bound appears to be new. This is because all previous works bound a global loss function like the sum of squared error, or a sum of huber losses in the case of quantile regression in Padilla and Chatterjee (2021). Clearly, element wise bounds are stronger than global loss error bounds as it reveals how the loss behaves locally at each point. Our element wise error bound also has a clean and explicit dependence on the tuning parameter λ which informs the user of a good choice of λ. In addition, our bound is nonasymptotic with explicit constants and is able to recover almost all the known results for Fused Lasso (both mean and quantile regression) with additional improvements in some cases.

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“…Hütter and Rigollet (2016), Sadhanala et al (2016) derive error bounds for total variation denoising (trend filtering with k = 0) on lattice graphs. Chatterjee and Goswami (2021), Ortelli and van de Geer (2020) show stronger error bounds when the signal has axis-parallel patches. Sadhanala et al (2017Sadhanala et al ( , 2021, extend the analysis to higher-order trend filtering on lattice graphs of arbitrary dimension.…”

Section: Related Workmentioning

confidence: 98%

Exponential Family Trend Filtering on Lattices

Sadhanala¹,

Bassett²,

Sharpnack³

et al. 2022

Preprint

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Trend filtering is a modern approach to nonparametric regression that is more adaptive to local smoothness than splines or similar basis procedures. Existing analyses of trend filtering focus on estimating a function corrupted by homoskedastic Gaussian noise, but our work extends this technique to general exponential family distributions. This extension is motivated by the need to study massive, gridded climate data derived from polar-orbiting satellites. We present algorithms tailored to large problems, theoretical results for general exponential family likelihoods, and principled methods for tuning parameter selection without excess computation.

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New Risk Bounds for 2D Total Variation Denoising

Cited by 12 publications

References 37 publications

Tensor denoising with trend filtering

Tensor denoising with trend filtering

Element-wise Estimation Error of Generalized Fused Lasso

Exponential Family Trend Filtering on Lattices

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