Linear embeddings of low-dimensional subsets of a Hilbert space to R^m
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Linear embeddings of low-dimensional subsets of a Hilbert space to R^m
Thu, 20/08/2015 - 16:00 to 17:00
Location:
Speaker:
Gilles Puy
Affiliation:
INRIA Rennes, France
Synopsis:
The restricted isometry property (RIP) is at the core of many of the theoretical developments in compressed sensing (CS). For example, if a matrix A satisfies the RIP on the set of 2s-sparse signals then one can show that any s-sparse vector x can be accurately and stably recovered from its noisy compressed measurements z = Ax + n by solving the Basis Pursuit problem.
In this talk, we go beyond sparsity and consider generic low-dimensional signal models in a Hilbert space. We consider the problem of embedding a low-dimensional set, M, from an infinite-dimensional Hilbert space, H, to a finite-dimensional space. Defining appropriate random linear projections, we propose two constructions of linear maps that have the RIP on the secant set of M with high probability, i.e., both linear maps stably embed the set M with high probability. The first linear map is optimal in the sense that it only needs a number of projections essentially proportional to the intrinsic dimension of M to satisfy the RIP. The second one, which is based on a variable density sampling technique, is computationally more efficient, while potentially requiring more measurements.
An aperitif will follow at 17.00.
Biography:
Gilles Puy is a postdoctoral researcher at INRIA, France, where he is a member of the PANAMA Research team. He received the Ph.D. degree in Electrical Engineering from EPFL in January 2014. He also received the M.Sc. degree in Electrical and Electronics Engineering from Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland, and the Engineering Diploma from Supélec, France, in 2009. In 2013, he was awarded the EPFL Chorafas Foundation award, whose purpose is to distinguish innovative and high level research in the fields of advanced data processing technology, life sciences and/or sustainability. His research focuses on compressive sampling and, more generally, on the development of efficient acquisition and processing methods for signals in high dimension with applications in, e.g., magnetic resonance imaging, computer vision or radio-interferometry.