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 $k\in \mathbb{N}$.Comment: 28 page

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.Comment: 42 page

Hybrid plasmonic–photonic whispering gallery mode resonators for sensing: a critical review

In this review we present the state of the art and the most recent advances in the field of optical sensing with hybrid plasmonic–photonic whispering gallery mode (WGM) resonators

On the regularity of curvature fields in stress-driven nonlocal elastic beams

AbstractElastostatic problems of Bernoulli–Euler nanobeams, involving internal kinematic constraints and discontinuous and/or concentrated force systems, are investigated by the stress-driven nonlocal elasticity model. The field of elastic curvature is output by the convolution integral with a special averaging kernel and a piecewise smooth source field of elastic curvature, pointwise generated by the bending interaction. The total curvature is got by adding nonelastic curvatures due to thermal and/or electromagnetic effects and similar ones. It is shown that fields of elastic curvature, associated with piecewise smooth source fields and bi-exponential kernel, are continuously differentiable in the whole domain. The nonlocal elastic stress-driven integral law is then equivalent to a constitutive differential problem equipped with boundary and interface constitutive conditions expressing continuity of elastic curvature and its derivative. Effectiveness of the interface conditions is evidenced by the solution of an exemplar assemblage of beams subjected to discontinuous and concentrated loadings and to thermal curvatures, nonlocally associated with discontinuous thermal gradients. Analytical solutions of structural problems and their nonlocal-to-local limits are evaluated and commented upon

Parallel decomposition of persistence modules through interval bases

We introduce an algorithm to decompose any finite-type persistence module with coefficients in a field into what we call an {em interval basis}. This construction yields both the standard persistence pairs of Topological Data Analysis (TDA), as well as a special set of generators inducing the interval decomposition of the Structure theorem. The computation of this basis can be distributed over the steps in the persistence module. This construction works for general persistence modules on a field $mathbb{F}$, not necessarily deriving from persistent homology. We subsequently provide a parallel algorithm to build a persistent homology module over $mathbb{R}$ by leveraging the Hodge decomposition, thus providing new motivation to explore the interplay between TDA and the Hodge Laplacian

Parallel decomposition of persistence modules through interval bases

We introduce an algorithm to decompose any finite-type persistence module with coefficients in a field into what we call an {\em interval basis}. This construction yields both the standard persistence pairs of Topological Data Analysis (TDA), as well as a special set of generators inducing the interval decomposition of the Structure theorem. The computation of this basis can be distributed over the steps in the persistence module. This construction works for general persistence modules on a field $\mathbb{F}$, not necessarily deriving from persistent homology. We subsequently provide a parallel algorithm to build a persistent homology module over $\mathbb{R}$ by leveraging the Hodge decomposition, thus providing new motivation to explore the interplay between TDA and the Hodge Laplacian.Comment: 37 pages, 6 figure

Towards realistic laparoscopic image generation using image-domain translation

none5openMarzullo, Aldo; Moccia, Sara; Catellani, Michele; Calimeri, Francesco; Momi, Elena DeMarzullo, Aldo; Moccia, Sara; Catellani, Michele; Calimeri, Francesco; Momi, Elena D