Francesco Ortelli

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Ortelli

First Name

Francesco

ORCID

0000-0002-3148-228X

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Publications 1 - 9 of 9

Oracle Results for Total Variation Regularization
Ortelli, Francesco (2021)
On the total variation regularized estimator over a class of tree graphs
Ortelli, Francesco; van de Geer, Sara (2018)
Electronic Journal of Statistics
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.
Prediction bounds for higher order total variation regularized least squares
Ortelli, Francesco; van de Geer, Sara (2020)
arXiv
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 $k\in \mathbb{N}$.
Adaptive Rates for Total Variation Image Denoising
Ortelli, Francesco; van de Geer, Sara (2019)
arXiv
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 which 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.
Dynamic formation of nanostructured particles from vesicles via invertase hydrolysis for on-demand delivery
Fong, Wye-Khay; Sánchez-Ferrer, Antoni; Ortelli, Francesco; et al. (2017)
RSC Advances
The unique multicompartmental nanostructure of lipid-based mesophases can be triggered, on-demand, in order to control the release of encapsulated drugs. In this study, these nanostructured matrices have been designed to respond to a specific enzyme, invertase, an enzyme which catalyses the hydrolysis of sucrose. The effect of two sugar esters upon the phase behaviour of two different lipids which form cubic phases, phytantriol and monolinolein, was investigated. Factors affecting the hydrolysis of the sucrose headgroup are discussed in terms of the molecular structure of the sugar surfactant and also its ability to incorporate into the lipid bilayer. By hydrolysing the incorporated sugar esters, a dynamic change in mesophase nanostructure from vesicles to a cubic phase was observed. This phase change resulted in the triggered release of an encapsulated model drug, fluorescein. This investigation demonstrates, for the first time, that changes on a molecular level by subtly controlling the hydrophilic and hydrophobic features of an amphiphilic additive at the interface by enzymatic hydrolysis can result in a global change in the system and so paves the way towards the design and development of lipid-based matrices which are responsive to specific enzymes for the controlled delivery of pharmaceutically active molecules or functional foods.
Oracle inequalities for square root analysis estimators with application to total variation penalties
Ortelli, Francesco; van de Geer, Sara (2021)
Information and Inference: A Journal of the IMA
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 et al. (2017, Bernoulli, 23, 552–581). 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.
Tensor denoising with trend filtering
Ortelli, Francesco; van de Geer, Sara (2022)
Mathematical Statistics and Learning
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 ℓ0-rates and not-so-slow ℓ1-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 ℓ0-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 ℓ1-rate of estimation n−H(d)+2k−12H(d)+2k−1, up to logarithmic terms, where H(d) is the dth 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 ℓ0-rates, mesh grids for ℓ1-rates and, in the background, the projection arguments by Dalalyan, Hebiri, and Lederer (2017).
Synthesis and analysis in total variation regularization
Ortelli, Francesco; van de Geer, Sara (2019)
arXiv
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.
Adaptive rates for total variation image denoising
Ortelli, Francesco; van de Geer, Sara (2020)
Journal of Machine Learning Research
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. (© 2020 Microtome Publishing)

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Francesco Ortelli

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