We consider a regularized least squares problem, with regularization by structured sparsity-inducing norms, which extend the usual ℓ 1 and the group lasso penalty, by allowing the subsets to overlap. Such regularizations lead to nonsmooth problems that are difficult to optimize, and we propose in this paper a suitable version of an accelerated proximal method to solve them. We prove convergence of a nested procedure, obtained composing an accelerated proximal method with an inner algorithm for computing the proximity operator. By exploiting the geometrical properties of the penalty, we devise a new active set strategy, thanks to which the inner iteration is relatively fast, thus guaranteeing good computational performances of the overall algorithm. Our approach allows to deal with high dimensional problems without pre-processing for dimensionality reduction, leading to better computational and prediction performances with respect to the state-of-the art methods, as shown empirically both on toy and real data.

Background/Aim: Salivary gland tumors are mostly benign tumors. Whether a more conservative surgical approach at greater risk of recurrence, or a more radical intervention with an increased risk of facial paralysis is warranted is still under discussion. Our study addresses the opportunity for improving surgical outcome by employing platelet-rich plasma (PRP) gel at the surgical site. Patients and Methods: Twenty consecutive patients undergoing superficial parotidectomy were randomized and assigned to two groups, one with and one without PRP gel. Many parameters were evaluated after surgery and during follow-up, such as the duration of hospitalization, facial nerve deficit, onset of Frey's syndrome, relapse, cosmetic results, presence of keloid or scar depressions, behavior of several facial muscles. Results: Our explorative analysis suggests a positive effect of PRP on surgical outcome in patients undergoing parotidectomy, whereas no negative effects were detected. Conclusion: This work suggests that administration of PRP in patients undergoing parotidectomy is beneficial.

Objective: To describe the dynamics of thickness and internal reflectivity after proton beam therapy (PBT) in uveal melanoma.Participants: One hundred and ninety-eight consecutive patients with choroidal or ciliary body melanoma treated by PBT were retrospectively considered.Methods: The post-PBT follow-up included ophthalmologic examination, retinography, and B and A modes of standardized echography every 6 months. A total of 1393 examinations were performed. We take into account 4 tumour categories according to the seventh TNM classification.Results: Before FBI, tumour thickness ranged from 1.5 to 12.5 mm with a mean of 5.9 mm. Its decrease after radiotherapy was best fitted by the sum of a first-order exponential decay and a constant with a decay half-life of 15 months. Based on the fit, tumour thickness stabilized on a constant value representing, on average, 47% of the initial value. Mean internal reflectivity before PBT was 68%. The dynamics of the reflectivity were best fitted by an exponential and a constant, with rise half-life of 11 months, and stability value of 87%.Conclusions: We found that ultrasonographic dynamics of uveal melanoma treated by PBT resembles a function composed of the sum of a constant and a first-order exponential, as previously noted in studies on brachytherapy. Interestingly, after FBI, because of its shorter half-life, internal reflectivity has a faster dynamic response than thickness in large tumours, suggesting that increase of internal reflectivity is a more sensitive indicator of early response to therapy in larger tumours.

Paediatric low-grade gliomas (LGGs) encompass a heterogeneous set of tumours of different histologies, site of lesion, age and gender distribution, growth potential, morphological features, tendency to progression and clinical course. Among LGGs, Pilocytic astrocytomas (PAs) are the most common central nervous system (CNS) tumours in children. They are typically well-circumscribed, classified as grade I by the World Health Organization (WHO), but recurrence or progressive disease occurs in about 10-20% of cases. Despite radiological and neuropathological features deemed as classic are acknowledged, PA may present a bewildering variety of microscopic features. Indeed, tumours containing both neoplastic ganglion and astrocytic cells occur at a lower frequency.

In this work we are interested in the problems of supervised learning and variable selection when the input-output dependence is described by a nonlinear function depending on a few variables. Our goal is to consider a sparse nonparametric model, hence avoiding linear or additive models. The key idea is to measure the importance of each variable in the model by making use of partial derivatives. Based on this intuition we propose a new notion of nonparametric sparsity and a corresponding least squares regularization scheme. Using concepts and results from the theory of reproducing kernel Hilbert spaces and proximal methods, we show that the proposed learning algorithm corresponds to a minimization problem which can be provably solved by an iterative procedure. The consistency properties of the obtained estimator are studied both in terms of prediction and selection performance. An extensive empirical analysis shows that the proposed method performs favorably with respect to the state-of-the-art methods.

In this work we are interested in the problems of supervised learning and variable selection when the input-output dependence is described by a nonlinear function depending on a few variables. Our goal is to devise a sparse nonparametric model, avoiding linear or additive models. The key intuition is to measure the importance of each variable in the model by making use of partial derivatives. Based on this idea we propose and study a new regularizer and a corresponding least squares regularization scheme. Using concepts and results from the theory of reproducing kernel Hilbert spaces and proximal methods, we show that the proposed learning algorithm induces a minimization problem which can be provably solved by an iterative procedure. The consistency properties of the obtained estimator are studied both in terms of prediction and selection performance.

Purpose: To evaluate survival and clinical outcome for patients with a large uveal melanoma treated by either enucleation or proton beam radiotherapy (PBRT). Procedures: This retrospective non-randomized study evaluated 132 consecutive patients with T3 and T4 choroidal melanoma classified according to TNM stage grouping. Results: Cumulative all-cause mortality, melanoma-related mortality and metastasis-free survival were not statistically different between the two groups (log-rank test, p = 0.56, p = 0.99 and p = 0.25, respectively). Eye retention of the tumours treated with PBRT at 5 years was 74% (SD 6.2%). In these patients at diagnosis, 73% of eyes had a best-corrected visual acuity (BCVA) of 0.1 or better. After 12 and 60 months, BCVA of 0.1 or better was observed in 47.5 and 32%, respectively. Conclusion and Message: Although enucleation is the most common primary treatment for large uveal melanomas, PBRT is an eye-preserving option that may be considered for some patients.

Proximal methods have recently been shown to provide effective optimization procedures to solve the variational problems defining the ℓ1 regularization algorithms. The goal of the paper is twofold. First we discuss how proximal methods can be applied to solve a large class of machine learning algorithms which can be seen as extensions of ℓ1 regularization, namely structured sparsity regularization. For all these algorithms, it is possible to derive an optimization procedure which corresponds to an iterative projection algorithm. Second, we discuss the effect of a preconditioning of the optimization procedure achieved by adding a strictly convex functional to the objective function. Structured sparsity algorithms are usually based on minimizing a convex (not strictly convex) objective function and this might lead to undesired unstable behavior. We show that by perturbing the objective function by a small strictly convex term we often reduce substantially the number of required computations without affecting the prediction performance of the obtained solution.

In this paper we consider a regularization approach to variable selection when the regression function depends nonlinearly on a few input variables. The proposed method is based on a regularized least square estimator penalizing large values of the partial derivatives. An efficient iterative procedure is proposed to solve the underlying variational problem, and its convergence is proved. The empirical properties of the obtained estimator are tested both for prediction and variable selection. The algorithm compares favorably to more standard ridge regression and `1 regularization schemes.

Hypoxia is a condition of low oxygen tension occurring in the tumor and negatively correlated with the progression of the disease. We studied the gene expression profiles of nine neuroblastoma cell lines grown under hypoxic conditions to define gene signatures that characterize hypoxic neuroblastoma. The l(1)-l(2) regularization applied to the entire transcriptome identified a single signature of 11 probesets discriminating the hypoxic state. We demonstrate that new hypoxia signatures, with similar discriminatory power, can be generated by a prior knowledge-based filtering in which a much smaller number of probesets, characterizing hypoxia-related biochemical pathways, are analyzed. l(1)-l(2) regularization identified novel and robust hypoxia signatures within apoptosis, glycolysis, and oxidative phosphorylation Gene Ontology classes. We conclude that the filtering approach overcomes the noisy nature of the microarray data and allows generating robust signatures suitable for biomarker discovery and patients risk assessment in a fraction of computer time.

We deal with the problem of variable selection when variables must be selected group-wise, with possibly overlapping groups defined a priori. In particular we propose a new optimization procedure for solving the regularized algorithm presented in [12], where the group lasso penalty is generalized to overlapping groups of variables. While in [12] the proposed implementation requires explicit replication of the variables belonging to more than one group, our iterative procedure is based on a combination of proximal methods in the primal space and projected Newton method in a reduced dual space, corresponding to the active groups. This procedure provides a scalable alternative with no need for data duplication, and allows to deal with high dimensional problems without pre-processing for dimensionality reduction. The computational advantages of our scheme with respect to state-of-the-art algorithms using data duplication are shown empirically with numerical simulations.

Background: Hypoxia is a condition of low oxygen tension occurring in the tumor microenvironment and it is related to poor prognosis in human cancer. To examine the relationship between hypoxia and neuroblastoma, we generated and tested an in vitro derived hypoxia gene signature for its ability to predict patients' outcome. Results: We obtained the gene expression profile of 11 hypoxic neuroblastoma cell lines and we derived a robust 62 probesets signature (NB-hypo) taking advantage of the strong discriminating power of the l1-l2 feature selection technique combined with the analysis of differential gene expression. We profiled gene expression of the tumors of 88 neuroblastoma patients and divided them according to the NB-hypo expression values by K-means clustering. The NBhypo successfully stratifies the neuroblastoma patients into good and poor prognosis groups. Multivariate Cox analysis revealed that the NB-hypo is a significant independent predictor after controlling for commonly used risk factors including the amplification of MYCN oncogene. NB-hypo increases the resolution of the MYCN stratification by dividing patients with MYCN not amplified tumors in good and poor outcome suggesting that hypoxia is associated with the aggressiveness of neuroblastoma tumor independently from MYCN amplification. Conclusions: Our results demonstrate that the NB-hypo is a novel and independent prognostic factor for neuroblastoma and support the view that hypoxia is negatively correlated with tumors' outcome. We show the power of the biology-driven approach in defining hypoxia as a critical molecular program in neuroblastoma and the potential for improvement in the current criteria for risk stratification.

PURPOSE:To evaluate the results of 15 years of experience with proton beam radiotherapy in the treatment of intraocular melanoma, and to determine univariate and multivariate risk factors for local failure, eye retention, and survival.METHODS:A total of 368 cases of intraocular melanoma were treated with proton beam radiotherapy at Centre Lacassagne Cyclotron Biomedical of Nice, France, between 1991 and 2006. Actuarial methods were used to evaluate rate of local tumor control, eye retention, and survival after proton beam radiotherapy. Cox regression models were extracted to evaluate univariate risk factors, while regularized least squares algorithm was used to have a multivariate classification model to better discriminate risk factors.RESULTS:Tumor relapse occurred in 8.4% of the eyes, with a median recurrence time of 46 months. Enucleation was performed on 11.7% of the eyes after a median time of 49 months following proton beam; out of these, 29 eyes were enucleated due to relapse and 16 due to other causes. The univariate regression analysis identified tumor height and diameter as primary risk factors for enucleation. Regularized least squares analysis demonstrated the higher effectiveness of a multivariate model of five risk factors (macula distance, optic disc distance, tumor height, maximum diameter, and age) in discriminating relapsed vs nonrelapsed patients.CONCLUSIONS:This data set, which is the largest in Italy with relatively long-term follow-up, demonstrates that a high rate of tumor control, survival, and eye retention were achieved after proton beam irradiation, as in other series.

In this paper we propose a general framework to characterize and solve the optimization problems underlying a large class of sparsity based regularization algorithms. More precisely, we study the minimization of learning functionals that are sums of a differentiable data term and a convex non differentiable penalty. These latter penalties have recently become popular in machine learning since they allow to enforce various kinds of sparsity properties in the solution. Leveraging on the theory of Fenchel duality and subdifferential calculus, we derive explicit optimality conditions for the regularized solution and propose a general iterative projection algorithm whose convergence to the optimal solution can be proved. The generality of the framework is illustrated, considering several examples of regularization schemes, including `1 regularization (and several variants), multiple kernel learning and multi-task learning. Finally, some features of the proposed framework are empirically studied.