C. Herzet

C. Herzet, M. Diallo, P. Héas

We consider an enhanced version of the well-kwown "Petrov-Galerkin" projection in Hilbert spaces. The proposed procedure, dubbed "multi-slice" projector, exploits the fact that the sought solution belongs to the intersection of several high-dimensional slices. This setup is for example of interest in model-order reduction where this type of prior may be computed off-line. In this note, we provide a mathematical characterization of the performance achievable by the multi-slice projector and compare the latter with the results holding in the Petrov-Galerkin setup. In particular, we illustrate the superiority of the multi-slice approach in certain situations.

C. Herzet, A. Drémeau

This paper focusses on "safe" screening techniques for the LASSO problem. Motivated by the need for low-complexity algorithms, we propose a new approach, dubbed "joint" screening test, allowing to screen a set of atoms by carrying out one single test. The approach is particularized to two different sets of atoms, respectively expressed as sphere and dome regions. After presenting the mathematical derivations of the tests, we elaborate on their relative effectiveness and discuss the practical use of such procedures.

C. Herzet, P. Héas, A. Drémeau

This paper deals with model-order reduction of parametric partial differential equations (PPDE). More specifically, we consider the problem of finding a good approximation subspace of the solution manifold of the PPDE when only partial information on the latter is available. We assume that two sources of information are available: i) a "rough" prior knowledge, taking the form of a manifold containing the target solution manifold, ii) partial linear measurements of the solutions of the PPDE (the term partial refers to the fact that observation operator cannot be inverted). We provide and study several tools to derive good approximation subspaces from these two sources of information. We first identify the best worst-case performance achievable in this setup and propose simple procedures to approximate the corresponding optimal approximation subspace. We then provide, in a simplified setup, a theoretical analysis relating the achievable reduction performance to the choice of the observation operator and the prior knowledge available on the solution manifold.

L. Jacques, C. De Vleeschouwer, Y. Boursier et al.

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.