23 research outputs found
DC Proximal Newton for Non-Convex Optimization Problems
We introduce a novel algorithm for solving learning problems where both the
loss function and the regularizer are non-convex but belong to the class of
difference of convex (DC) functions. Our contribution is a new general purpose
proximal Newton algorithm that is able to deal with such a situation. The
algorithm consists in obtaining a descent direction from an approximation of
the loss function and then in performing a line search to ensure sufficient
descent. A theoretical analysis is provided showing that the iterates of the
proposed algorithm {admit} as limit points stationary points of the DC
objective function. Numerical experiments show that our approach is more
efficient than current state of the art for a problem with a convex loss
functions and non-convex regularizer. We have also illustrated the benefit of
our algorithm in high-dimensional transductive learning problem where both loss
function and regularizers are non-convex
Non-convex regularization in remote sensing
In this paper, we study the effect of different regularizers and their
implications in high dimensional image classification and sparse linear
unmixing. Although kernelization or sparse methods are globally accepted
solutions for processing data in high dimensions, we present here a study on
the impact of the form of regularization used and its parametrization. We
consider regularization via traditional squared (2) and sparsity-promoting (1)
norms, as well as more unconventional nonconvex regularizers (p and Log Sum
Penalty). We compare their properties and advantages on several classification
and linear unmixing tasks and provide advices on the choice of the best
regularizer for the problem at hand. Finally, we also provide a fully
functional toolbox for the community.Comment: 11 pages, 11 figure
\ell_p-\ell_q penalty for sparse linear and sparse multiple kernel multi-task learning,
13International audienceRecently, there has been a lot of interest around multi-task learning (MTL) problem with the constraints that tasks should share a common sparsity profile. Such a problem can be addressed through a regularization framework where the regularizer induces a joint-sparsity pattern between task decision functions. We follow this principled framework and focus on (with and ) mixed-norms as sparsity- inducing penalties. Our motivation for addressing such a larger class of penalty is to adapt the penalty to a problem at hand leading thus to better performances and better sparsity pattern. For solving the problem in the general multiple kernel case, we first derive a variational formulation of the penalty which helps up in proposing an alternate optimization algorithm. Although very simple, the latter algorithm provably converges to the global minimum of the penalized problem. For the linear case, we extend existing works considering accelerated proximal gradient to this penalty. Our contribution in this context is to provide an efficient scheme for computing the proximal operator. Then, for the more general case when , we solve the resulting non-convex problem through a majorization-minimization approach. The resulting algorithm is an iterative scheme which, at each iteration, solves a weighted sparse MTL problem. Empirical evidences from toy dataset and real-word datasets dealing with BCI single trial EEG classification and protein subcellular localization show the benefit of the proposed approaches and algorithms
Optimal Transport with Adaptive Regularisation
Regularising the primal formulation of optimal transport (OT) with a strictly convex term leads to enhanced numerical complexity and a denser transport plan. Many formulations impose a global constraint on the transport plan, for instance by relying on entropic regularisation. As it is more expensive to diffuse mass for outlier points compared to central ones, this typically results in a significant imbalance in the way mass is spread across the points. This can be detrimental for some applications where a minimum of smoothing is required per point. To remedy this, we introduce OT with Adaptive RegularIsation (OTARI), a new formulation of OT that imposes constraints on the mass going in or/and out of each point. We then showcase the benefits of this approach for domain adaptation
Optimal Transport with Adaptive Regularisation
Regularising the primal formulation of optimal transport (OT) with a strictly convex term leads to enhanced numerical complexity and a denser transport plan. Many formulations impose a global constraint on the transport plan, for instance by relying on entropic regularisation. As it is more expensive to diffuse mass for outlier points compared to central ones, this typically results in a significant imbalance in the way mass is spread across the points. This can be detrimental for some applications where a minimum of smoothing is required per point. To remedy this, we introduce OT with Adaptive RegularIsation (OTARI), a new formulation of OT that imposes constraints on the mass going in or/and out of each point. We then showcase the benefits of this approach for domain adaptation
Optimal Transport with Adaptive Regularisation
Regularising the primal formulation of optimal transport (OT) with a strictly convex term leads to enhanced numerical complexity and a denser transport plan. Many formulations impose a global constraint on the transport plan, for instance by relying on entropic regularisation. As it is more expensive to diffuse mass for outlier points compared to central ones, this typically results in a significant imbalance in the way mass is spread across the points. This can be detrimental for some applications where a minimum of smoothing is required per point. To remedy this, we introduce OT with Adaptive RegularIsation (OTARI), a new formulation of OT that imposes constraints on the mass going in or/and out of each point. We then showcase the benefits of this approach for domain adaptation
Optimal Transport with Adaptive Regularisation
International audienceRegularising the primal formulation of optimal transport (OT) with a strictly convex term leads to enhanced numerical complexity and a denser transport plan. Many formulations impose a global constraint on the transport plan, for instance by relying on entropic regularisation. As it is more expensive to diffuse mass for outlier points compared to central ones, this typically results in a significant imbalance in the way mass is spread across the points. This can be detrimental for some applications where a minimum of smoothing is required per point. To remedy this, we introduce OT with Adaptive RegularIsation (OTARI), a new formulation of OT that imposes constraints on the mass going in or/and out of each point. We then showcase the benefits of this approach for domain adaptation
Optimal Transport with Adaptive Regularisation
Regularising the primal formulation of optimal transport (OT) with a strictly convex term leads to enhanced numerical complexity and a denser transport plan. Many formulations impose a global constraint on the transport plan, for instance by relying on entropic regularisation. As it is more expensive to diffuse mass for outlier points compared to central ones, this typically results in a significant imbalance in the way mass is spread across the points. This can be detrimental for some applications where a minimum of smoothing is required per point. To remedy this, we introduce OT with Adaptive RegularIsation (OTARI), a new formulation of OT that imposes constraints on the mass going in or/and out of each point. We then showcase the benefits of this approach for domain adaptation
Unsupervised variable selection for kernel methods in systems biology
National audienceKernel methods have proven to be useful and successful to analyse large-scale multi-omics datasets [Schölkopf et al., 2004]. However, as stated in [Hofmann et al., 2015, Mariette et al., 2017], these methods usually suffer from a lack of interpretability as the information of thousands descriptors is summarized in a few similarity measures, that can be strongly in uenced by a large number of irrelevant descriptors. To address this issue, feature selection is a widely used strategy: it consist in selecting the most promising features during or prior the analysis. However, most existing methods are proposed in a supervised framework [Tibshirani, 1996, Robnik-Sikonja and Kononenko, 2003, Lin and Tang, 2006]. In the unsupervised framework, the number of proposals is much less important, because there is no objective criterion or value on which to tune the quality of a given feature. Proposals thus aim at preserving at best the similarities between individuals like the SPEC approach [Zhao and Liu, 2007] or at recovering a latent cluster structure, like MCFS [Cai et al., 2010], NDFS [Li et al., 2012] and UDFS [Yang et al., 2011]. In this communication, we will present a feature selection algorithm that explicitly takes advantage of the kernel structure in an unsupervised fashion