Signal Recovery in Unions of Subspaces with Applications to Compressive Imaging
arXiv:1209.3079
Abstract
In applications ranging from communications to genetics, signals can be modeled as lying in a union of subspaces. Under this model, signal coefficients that lie in certain subspaces are active or inactive together. The potential subspaces are known in advance, but the particular set of subspaces that are active (i.e., in the signal support) must be learned from measurements. We show that exploiting knowledge of subspaces can further reduce the number of measurements required for exact signal recovery, and derive universal bounds for the number of measurements needed. The bound is universal in the sense that it only depends on the number of subspaces under consideration, and their orientation relative to each other. The particulars of the subspaces (e.g., compositions, dimensions, extents, overlaps, etc.) does not affect the results we obtain. In the process, we derive sample complexity bounds for the special case of the group lasso with overlapping groups (the latent group lasso), which is used in a variety of applications. Finally, we also show that wavelet transform coefficients of images can be modeled as lying in groups, and hence can be efficiently recovered using group lasso methods.
arXiv commentarXiv admin note: substantial text overlap with arXiv:1106.4355
References in corpus (9)
- Simultaneous analysis of Lasso and Dantzig selector
- Message Passing Algorithms for Compressed Sensing
- Consistency of the group Lasso and multiple kernel learning
- Compressive Imaging using Approximate Message Passing and a Markov-Tree Prior
- Compressed Sensing of Analog Signals in Shift-Invariant Spaces
- Group Lasso with Overlaps: the Latent Group Lasso approach
- Performance Analysis of Sparse Recovery Based on Constrained Minimal Singular Values
- Uncertainty Relations for Shift-Invariant Analog Signals
- Holographic Dual of Linear Dilaton Black Hole in Einstein-Maxwell-Dilaton-Axion Gravity