arXiv subjects

Francesco Ortelli

Publications and source records attributed to Francesco Ortelli.

6 recordsLinked to original sources

arXiv2021

Adaptive Rates for Total Variation Image Denoising

We study the theoretical properties of image denoising via total variation penalized least-squares. We define the total vatiation in terms of the two-dimensional total discrete derivative of the image and show that it gives rise to denoised images that are piecewise constant on rectangular sets. We prove that, if the true image is piecewise constant on just a few rectangular sets, the denoised image converges to the true image at a parametric rate, up to a log factor. More generally, we show that the denoised image enjoys oracle properties, that is, it is almost as good as if some aspects of the true image were known. In other words, image denoising with total variation regularization leads to an adaptive reconstruction of the true image.

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arXiv2021

Tensor denoising with trend filtering

We extend the notion of trend filtering to tensors by considering the $k^{\rm th}$-order Vitali variation, a discretized version of the integral of the absolute value of the $k^{\rm th}$-order total derivative. We prove adaptive $\ell^0$-rates and not-so-slow $\ell^1$-rates for tensor denoising with trend filtering. For $k=\{1,2,3,4\}$ we prove that the $d$-dimensional margin of a $d$-dimensional tensor can be estimated at the $\ell^0$-rate $n^{-1}$, up to logarithmic terms, if the underlying tensor is a product of $(k-1)^{\rm th}$-order polynomials on a constant number of hyperrectangles. For general $k$ we prove the $\ell^1$-rate of estimation $n^{- \frac{H(d)+2k-1}{2H(d)+2k-1}}$, up to logarithmic terms, where $H(d)$ is the $d^{\rm th}$ 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 $\ell^0$-rates, mesh grids for $\ell^1$-rates and, in the background, the projection arguments by Dalalyan et al.

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arXiv2020

Prediction bounds for higher order total variation regularized least squares

We establish adaptive results for trend filtering: least squares estimation with a penalty on the total variation of $(k-1)^{\rm th}$ order differences. Our approach is based on combining a general oracle inequality for the $\ell_1$-penalized least squares estimator with "interpolating vectors" to upper-bound the "effective sparsity". This allows one to show that the $\ell_1$-penalty on the $k^{\text{th}}$ order differences leads to an estimator that can adapt to the number of jumps in the $(k-1)^{\text{th}}$ order differences of the underlying signal or an approximation thereof. We show the result for $k \in \{1,2,3,4\}$ and indicate how it could be derived for general $k\in \mathbb{N}$.

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arXiv2019

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

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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arXiv2019

Synthesis and analysis in total variation regularization

We generalize the bridge between analysis and synthesis estimators by Elad, Milanfar and Rubinstein (2007) to rank deficient cases. This is a starting point for the study of the connection between analysis and synthesis for total variation regularized estimators. In particular, the case of first order total variation regularized estimators over general graphs and their synthesis form are studied. We give a definition of the discrete graph derivative operator based on the notion of line graph and provide examples of the synthesis form of $k^{\text{th}}$ order total variation regularized estimators over a range of graphs.

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arXiv2018

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

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.

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