Estimating Unknown Sparsity in Compressed Sensing
arXiv:1204.4227
Abstract
In the theory of compressed sensing (CS), the sparsity ||x||_0 of the unknown signal x\in\R^p is commonly assumed to be a known parameter. However, it is typically unknown in practice. Due to the fact that many aspects of CS depend on knowing ||x||_0, it is important to estimate this parameter in a data-driven way. A second practical concern is that ||x||_0 is a highly unstable function of x. In particular, for real signals with entries not exactly equal to 0, the value ||x||_0=p is not a useful description of the effective number of coordinates. In this paper, we propose to estimate a stable measure of sparsity s(x):=||x||_1^2/||x||_2^2, which is a sharp lower bound on ||x||_0. Our estimation procedure uses only a small number of linear measurements, does not rely on any sparsity assumptions, and requires very little computation. A confidence interval for s(x) is provided, and its width is shown to have no dependence on the signal dimension p. Moreover, this result extends naturally to the matrix recovery setting, where a soft version of matrix rank can be estimated with analogous guarantees. Finally, we show that the use of randomized measurements is essential to estimating s(x). This is accomplished by proving that the minimax risk for estimating s(x) with deterministic measurements is large when n<<p.
This is version 2. Many aspects of the paper have been revised. The restriction to non-negative signals has been removed. 22 pages
References in corpus (18)
- Discussion of "Least angle regression" by Efron et al
- Discussion of "Least angle regression" by Efron et al
- Rejoinder to "Least angle regression" by Efron et al
- Least Angle Regression
- Discussion of "Least angle regression" by Efron et al
- Discussion of "Least angle regression" by Efron et al
- Discussion of "Least angle regression" by Efron et al
- Discussion of "Least angle regression" by Efron et al
- Discussion of "Least angle regression" by Efron et al
- Discussion of "Least angle regression" by Efron et al
- Generalized SURE for Exponential Families: Applications to Regularization
- Compressed sensing performance bounds under Poisson noise
- A Bernstein-type inequality for stochastic processes of quadratic forms of Gaussian variables
- On Verifiable Sufficient Conditions for Sparse Signal Recovery via Minimization
- On Low Rank Matrix Approximations with Applications to Synthesis Problem in Compressed Sensing
- Performance Analysis of Sparse Recovery Based on Constrained Minimal Singular Values
- A More Powerful Two-Sample Test in High Dimensions using Random Projection
- On the Fundamental Limits of Adaptive Sensing