Compressed Sensing and Best Approximation from Unions of Subspaces: Beyond Dictionaries

Tomer Peleg (1) , Rémi Gribonval (2) , Mike E. Davies (3)

1 EE-Technion - Department of Electrical Engineering - Technion [Haïfa]
2 PANAMA - Parcimonie et Nouveaux Algorithmes pour le Signal et la Modélisation Audio
3 Institute for Digital Communication Joint Research Institute for Signal & Image Processing School of Engineering and Electronics - University of Edinburgh

We propose a theoretical study of the conditions guaranteeing that a decoder will obtain an optimal signal recovery from an underdetermined set of linear measurements. This special type of performance guarantee is termed instance optimality and is typically related with certain properties of the dimensionality-reducing matrix M. Our work extends traditional results in sparse recovery, where instance optimality is expressed with respect to the set of sparse vectors, by replac- ing this set with an arbitrary finite union of subspaces. We show that the suggested instance optimality is equivalent to a generalized null space property of M and discuss possible relations with generalized restricted isometry properties.

Mots clés: Instance optimality; null space property; restricted isometry property; union-of-subspaces

Domaines: Traitement du signal et de l'image [eess.SP]; Analyse numérique [math.NA]; Traitement du signal et de l'image [eess.SP]

Tomer Peleg, Rémi Gribonval, Mike E. Davies. Compressed Sensing and Best Approximation from Unions of Subspaces: Beyond Dictionaries. 21st European Signal Processing Conference (EUSIPCO 2013), Sep 2013, Marrakech, Morocco. ⟨hal-00812858v2⟩